text
"page_content='POSE: E FFICIENT CONTEXT WINDOW EXTENSION OF\nLLM S VIA POSITIONAL SKIP-WISE TRAINING\nDawei Zhu∗†‡Nan Yang‡Liang Wang‡Yifan Song†Wenhao Wu†\nFuru Wei‡Sujian Li†B\n†School of Computer Science, Peking University\n‡Microsoft Corporation\nhttps://github.com/dwzhu-pku/PoSE\nABSTRACT\nLarge Language Models (LLMs) are trained with a pre-defined context length,\nrestricting their use in scenarios requiring long inputs. Previous efforts for adapting' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='LLMs to a longer length usually requires fine-tuning with this target length ( Full-\nlength fine-tuning), suffering intensive training cost. To decouple train length\nfrom target length for efficient context window extension, we propose Positional\nSkip-wis E(PoSE) training that smartly simulates long inputs using a fixed context\nwindow. This is achieved by first dividing the original context window into several\nchunks, then designing distinct skipping bias terms to manipulate the position' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='indices of each chunk. These bias terms and the lengths of each chunk are altered\nfor every training example, allowing the model to adapt to all positions within\ntarget length. Experimental results show that PoSE greatly reduces memory and\ntime overhead compared with Full-length fine-tuning, with minimal impact on per-\nformance. Leveraging this advantage, we have successfully extended the LLaMA\nmodel to 128k tokens using a 2k training context window. Furthermore, we empir-' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='ically confirm that PoSE is compatible with all RoPE-based LLMs and position\ninterpolation strategies. Notably, our method can potentially support infinite length,\nlimited only by memory usage in inference. With ongoing progress for efficient\ninference, we believe PoSE can further scale the context window beyond 128k.\n1 I NTRODUCTION\nLarge Language Models (LLMs) have revolutionized language modeling and demonstrated impres-' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='sive abilities to perform various tasks (Brown et al., 2020). However, even with their remarkable\ncapacity, these LLMs remain restricted by pre-defined context window sizes, suffering from notable\nperformance decline when input tokens exceeds these limits. Nevertheless, numerous application\nscenarios demand extremely long input sequences, including long document summarization (Huang\net al., 2021), in-context learning with numerous examples (Li et al., 2023), and long document' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='retrieval (Zhou et al., 2022), etc. This naturally poses a significant challenge of context window\nextension : Extending the context window of a pre-trained LLM to accommodate longer sequences.\nNaively fine-tuning LLMs on inputs of target length for window extension has received limited success\ndue to the large disruption introduced by new position indices (Chen et al., 2023a). Addressing\nthis, Position Interpolation (Chen et al., 2023a; kaiokendev, 2023; Peng et al., 2023) propose to' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='down-scale the position indices to match the original window size, yielding improved results for\ncontext extension. However, these methods still rely on Full-length fine-tuning, i.e., fine-tuning with\ncontext of target length, which is memory and time-intensive due to the computational complexity\nthat increases quadratically with input length. For example, Chen et al. (2023a) use 32 A100 GPUs\nto extend LLaMA models from 2k to 8k context, and 128 A100 GPUs for even larger context. These' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='computational cost has made it impossible to extend context window to extreme lengths.\nIn this paper, we introduce Positional Skip-wis E(PoSE) fine-tuning to decouple the fine-tuning\nlength from the target context window length, unleashing the possibility of efficiently extending\n∗Work done during internship at MSRA.\n1arXiv:2309.10400v2  [cs.CL]  10 Oct 2023' metadata={'source': 'pdfs/paper_2.pdf', 'page': 0}"
"page_content='0 511 ... 6560 8095 ...\n0 1023 ... 3552 4575 ...... 0 8191 1Full-length: 8192 tok ens\nPoSE: 2048 tok ens\nskip\nskip target  / original context siz e = 8192 / 2048\n1\n2AssetsTrain Example Relative Positions\n# 1 [1,1535] ∪ [6049,8095]\n# 2 [1,1024] ∪ [2529,4575]\n... ...Figure 1: Position indices of Full-length fine-tuning v.s. PoSE fine-tuning for extending the context\nwindow size from 2,048 to 8,192. At each iteration, the former directly takes 8,192 tokens for' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='fine-tuning, while PoSE manipulates the position indices of 2,048 tokens to simulate longer inputs.\nFor example, we partition the original context window of 2,048 tokens into two chunks, and adjust\nthe position indices of the second chunk by adding a distinct skipping bias term. These bias terms, as\nwell as the length of each chunk, are altered for each training example, so that the model can adapt to\nall relative positions of the target context window through fine-tuning.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='context window to an extreme size. The key idea of PoSE is to simulate long inputs by manipulating\nposition indices within a fixed context window. As depicted in Figure 1, we partition the original\ncontext window into several chunks, and adjust the position indices of each chunk by adding a distinct\nskipping bias term. These bias terms, as well as the length of each chunk, are altered for each training' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='example, so that the model can adapt to all positions (including both absolute and relative) within\nthe target context window through fine-tuning. Meanwhile, by maintaining continuous position\nindices within each chunk, PoSE bears a close resemblance to pre-training. As a result, the model’s\npre-trained capacity for language modeling and comprehension is retained to the greatest degree.\nThe advantages of our PoSE are threefold: 1) Memory and Time Efficiency: By only requiring' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='the original context size for fine-tuning, PoSE circumvents the quadratic increase in computational\ncomplexity with respect to target length during the fine-tuning stage, thereby significantly reducing\nmemory and time overhead. 2) Potential for Extremely-Long Context: We manage to extend the\ncontext window of LLaMA (Touvron et al., 2023a) by up to 64 times (2k − →128k, k=1,024) while pre-\nserving decent ability of language modeling and understanding. 3) Compatible with all RoPE-based' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='LLMs and PI strategies: The effectiveness of PoSE has been empirically validated across several\nrepresentative RoPE-based LLMs, including LLaMA, LLaMA2 (Touvron et al., 2023b), GPT-J (Wang\n& Komatsuzaki, 2021), and Baichuan (Baichuan, 2023). Additionally, PoSE has been demonstrated\nto be compatible with a variety of position interpolation methods, including Linear (Chen et al.,\n2023a), NTK (Peng & Quesnelle, 2023), and YaRN (Peng et al., 2023) interpolation.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='Notably, by decoupling the fine-tuning and target length, PoSE can theoretically extend context\nwindow to an infinite length. The only constraint is the memory usage during the inference phase.\nHopefully, with the continuous advancements in efficient inference techniques, including Flash\nAttention (Dao et al., 2022; Dao, 2023), xFormers (Lefaudeux et al., 2022), vLLM (Kwon et al.,\n2023), etc, we believe PoSE can promisingly push the context window size to a even larger scale.\n2 R ELATED WORK' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='Training Length-Extrapolatable Models. Length extrapolation aims to ensure that the model\ncontinues to perform well, even when the number of input tokens during inference exceeds the size\nof the context window on which the model is trained (Press et al., 2021). To this end, a series of\npositional embedding schemes have been proposed, including ALibi (Press et al., 2021), xPos (Sun\net al., 2023), NoPos (Haviv et al., 2022), etc.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='Similar to our work, Ruoss et al. (2023) also attempted to simulate longer sequences during\ntraining time to mitigate out-of-distribution lengths. They proposed randomized positional encoding\n(RandPos), which randomly selected an ordered subset of position indices from longer sequences.\n2' metadata={'source': 'pdfs/paper_2.pdf', 'page': 1}"
"page_content='Our proposed method, PoSE, diverges from their approach in several key aspects: First, RandPos\nis a positional embedding scheme designed for pre-training encoder-only models from scratch to\nenhance length generalization ability. In contrast, PoSE is a fine-tuning method that aims to efficiently\nextend the context window of pre-trained LLMs, the majority of which follow a decoder-only\narchitecture. Second, in RandPos, the position indices between adjacent tokens are not continuous.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='However, in PoSE, the position indices within each chunk are intentionally made continuous to closely\nresemble the pre-training phase, therefore reducing the risk of disrupting the language modeling and\nunderstanding abilities learned during the pre-training stage.\nFine-tuning LLMs for Longer Context. Differing from length extrapolation, which primarily\ninvolves training a model from scratch to support lengths exceeding those it was initially trained for,' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='context window extension focuses on extending the context window of a pre-trained LLM. Directly\nfine-tuning an existing LLM with a longer context window has been shown to progress slowly (Chen\net al., 2023a). To expedite and stabilize training, Chen et al. (2023a) first down-scaled position\nindices to match original context size through Linear Position Interpolation. Subsequently, a range\nof Positional Interpolation (PI) strategies have been introduced, including NTK (Peng & Quesnelle,' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='2023) and YaRN (Peng et al., 2023). More recently, LongLora (Chen et al., 2023b) propose shift short\nattention to approximate full attention. However, all these methods require Full-length fine-tuning,\nsuffering computational cost that grows with target context size. By contrast, our method managed to\ndecouple train / target length, requiring only the original context size for fine-tuning.\nMemory Transformers. An alternative strategy for managing extremely long input sequences' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='involves the adoption of memory mechanisms. Typically, there are two lines of research for utilizing\nmemory: the recurrence-based approach (Dai et al., 2019; Bulatov et al., 2022) and the retrieval-based\napproach (Wu et al., 2022; Wang et al., 2023; Tworkowski et al., 2023). The recurrence-based\napproach involves segmenting long inputs and reusing the hidden states obtained from preceding\nsegments to serve as memory for the current segment. Nonetheless, this architecture is hindered' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='by information loss and limited capacity for random access. On the other hand, the retrieval-based\nparadigm entails encoding prior sequences as (key, value) pairs and utilizing a memory retriever\nand reader to extract previously encoded information. The primary limitation of this approach is\nthe absence of interaction between discrete memory segments. More recently, Mohtashami & Jaggi\n(2023) introduced landmark attention, which facilitates random access to any chunk of the input by' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='introducing landmark tokens. In contrast, our method achieves full access to the entire input without\nany modifications to the attention mechanism.\n3 M ETHODOLOGY\n3.1 P RELIMINARIES\nRotary Position Embedding (RoPE). The use of RoPE (Su et al., 2021) has become pervasive\nin contemporary LLMs, including LLaMA (Touvron et al., 2023a), GPT-J (Wang & Komatsuzaki,\n2021), etc. It encodes position information of tokens with a rotation matrix that naturally incorporates' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='explicit relative position dependency. To elucidate, given a hidden vector h= [h0, h1, ..., h d−1],\nwhere dis the hidden dimension, and a position index m, RoPE operates as follows:\nf(h, m) =\uf8eb\n\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edh0\nh1\nh2\nh3\n...\nhd−2\nhd−1\uf8f6\n\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8⊗\uf8eb\n\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edcosmθ0\ncosmθ0\ncosmθ1\ncosmθ1\n...\ncosmθd/2−1\ncosmθd/2−1\uf8f6\n\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8+\uf8eb\n\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−h1\nh0\n−h3\nh2\n...\n−hd−1\nhd−2\uf8f6\n\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8⊗\uf8eb\n\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsinmθ0\nsinmθ0\nsinmθ1\nsinmθ1\n...\nsinmθd/2−1\nsinmθd/2−1\uf8f6\n\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8(1)' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='where θj= 10000−2j/d, j∈ {0,1, ..., d/ 2−1}. Unlike previous absolute position encodings that\nare directly applied to the input vector x, RoPE is employed on the query and key vectors at each\nlayer. Considering a query vector qat position mand a key vector kat position n, the attention score\n3' metadata={'source': 'pdfs/paper_2.pdf', 'page': 2}"
"page_content='a(q,k)is defined as follows:\na(q,k) =< f(q, m), f(k, n)>\n=d/2−1X\nj=0[(q2jk2j+q2j+1k2j+1) cos ( m−n)θj+ (q2jk2j+1−q2j+1k2j) sin ( m−n)θj]\n:=g(q,k,θ, m−n) (2)\nHence, RoPE encodes position information in a relative manner, as the attention score depends on the\nrelative distances between positions rather than their absolute position values.\nProblem Formulation. Given a Large Language Model pre-trained with a context window size of' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='Lc, our objective is to extend this context size to a target length Lt, so that the model maintains good\nperformance when processing input sequences containing a maximum of Lttokens.\nPosition Interpolation (PI). In contrast to directly extending the position indices to Lt−1when\ndealing with an input text x={x0, x1, ..., x Lt}, position interpolation down-scales the position\nindices to align with the original context window size Lc. This approach effectively mitigates the' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='risk of encountering extreme values and has been empirically demonstrated to enhance stability\nduring fine-tuning. Various interpolation strategies have been proposed, with α=Lt/Lcdenoting\nthe scaling factor:\n•Linear Interpolation. As described by Chen et al. (2023a) and kaiokendev (2023), linear interpola-\ntion involves a proportional down-scaling of the position index mtom/α . Consequently, the atten-\ntion score between a query qat position mand a key kat position nbecomes g(q,k,θ,(m−n)/α),' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='as defined in Equation 2. Theoretical analysis has substantiated that the interpolated attention score\nexhibits significantly greater stability compared to the extrapolated counterpart.\n•Neural Tangent Kernel (NTK) Interpolation. In contrast to linear interpolation, NTK Interpola-\ntion alters the base of RoPE, effectively modifying the rotational ""speed"" of each dimension of\nRoPE (Peng & Quesnelle, 2023). Specifically, the original θj= 10000−2j/d, j∈ {0,1, ..., d/ 2−1}' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='in RoPE is transformed into θ′\nj= (10000 λ)−2j/d, where λ=αd/d−2. It is noteworthy that the\nvalue of λis chosen to ensure that mθ′\nd/2−1= (m/α)θd/2−1.\n•YaRN Interpolation. Different from Linear and NTK interpolation that treat each dimension of\nRoPE equally, YaRN (Peng et al., 2023) employs a ramp function to combine Linear and NTK\ninterpolation at varying proportions across different dimensions. Simultaneously, it introduces a' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='temperature factor to mitigate distribution shift of attention matrix caused by long inputs.\n3.2 P ROPOSED APPROACH : POSITIONAL SKIP-WISE TRAINING (POSE)\nAlthough position interpolation effectively addresses out-of-distribution position indices, extending\nto an extreme length by fine-tuning on context window of this size remains impractical, owing to the\nquadratic growth in computational complexity of attention as sequence length increases. Instead, we' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='explore to train within the original context window Lcand achieve context window extension via\nmanipulating position indices to simulate longer inputs.\nThere are two designing desiderata for this endeavor: First, to avoid out-of-distribution positions\nduring inference, the relative distance of manipulated position indices should comprehensively cover\nthe range of {1, . . . , L t−1}. Second, fine-tuning with the manipulated position indices should not' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='harm the original abilities of LLMs, so the structure of manipulated position indices should closely\nadhere to the original structure to the greatest extent possible.\nInitially, we randomly divide the original context window LcintoNchunks c0, c1, . . . , c N−1, each\nwith lengths l0, l1, . . . , l N−1, wherePN−1\ni=0li=Lc. We introduce the starting index stifor each\nchunk ci, which facilitates the formulation of its position indices as follows:\nPos(ci) ={sti, sti+ 1, . . . , st i+li−1}, st i=i−1X' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='j=0lj (3)\nSubsequently, we employ the discrete uniform distribution U(S)to sample a skipping bias term\nui∼ U({ui−1, . . . , L t−Lc})for each chunk ci. This bias term is applied to the corresponding\n4' metadata={'source': 'pdfs/paper_2.pdf', 'page': 3}"
"page_content='chunk to transform the original position indices into:\nPoSE (ci) ={ui+sti, ui+sti+ 1, . . . , u i+sti+li−1} (4)\nNote that the constraint of ui≥ui−1is applied to prevent position index overlaps between chunks.\nIntuitively, the introduction of skipping bias terms exposes model to a more diverse range of relative\npositions. To achieve comprehensive coverage of the target context window, we re-sample both the' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='length and skipping bias term of every chunk for each training example. Moreover, the continuity\nof position indices within each chunk closely resembles the structure employed during pre-training.\nConsequently, fine-tuning the model on these new position indices for language modeling does not\ncompromise its original capabilities.\nConcerning the text contained within each chunk, a similar procedure is followed to select continuous' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='spans of tokens from the input text x={x0, x1, ..., x Lx}. To elaborate, we begin by sampling a bias\ntermvi∼ U({vi−1, . . . , L x−Lc)followed by assigning the content of chunk cias below:\nci=x[vi+sti:vi+sti+li] (5)\nNotably, we have also explored other assigning strategy of vi, including scenarios where vi= 0,\nwhich results in genuinely continuous content for the chunks, or vi=ui, aligning the manipulated' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='position indices with actual positions in the original text. However, we observe that these variations\nhave relatively little impact on the outcomes of fine-tuning.\nAfter position indices and content for each chunk are settled, we perform position interpolation for\nstabilized fine-tuning. For simplicity, We set the initial bias terms u0andv0to 0. In terms of chunk\nnumber N, we view it as an trade-off between efficiency and effectiveness. Because an increase in the' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='number of chunks will further deviates from the position structure of pre-training, which may harm\nthe ability acquired during pre-training. Hence, in this paper we set Nto 2, exposing the models to a\nwider range of relative positions, while adhering as close to the original position structure as possible.\n(See Appendxi A and B for further discussion of viandN.)\n4 E XPERIMENTS\nIn this section, we conduct experiments to verify the effectiveness of PoSE for context window' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='extension. Our method demonstrates impressive results on context lengths of both 16k and 32k for\nlanguage modeling as well as passkey retrieval. Other advantages of PoSE are discussed in Section 5.\n4.1 S ETUPS\nTraining Procedure. For each setting in the main experiments, we train LLaMA-7B with the next\ntoken prediction objective. This training process comprises 1,000 steps, employing a global batch size' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='of 64 on 8 V100 GPUs using Deepspeed ZeRO stage 3 (Rajbhandari et al., 2020). The fine-tuning\ndataset is sourced from The Pile (Gao et al., 2020), with a minimum length requirement of 2,048\ntokens. Our default choice for interpolation strategies is linear interpolation. For evaluation, we use a\nsingle A100 GPU. Flash Attention V2 (Dao, 2023) is applied, making it possible to evaluate long\ndocuments of up to 128k tokens (k=1,024)' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='Evaluation Tasks and Datasets. We examine the ability of long text modeling on two tasks:\nlanguage modeling and passkey retrieval. The language modeling task is a fundamental task that\nreflects the overall capability of a model in handling long text. Passkey retrieval, on the other hand,\ncan effectively measure the maximum distance that a token can attend to during the inference stage.\nWe evaluate language modeling on GovReport (Huang et al., 2021) and Proof-pile (Zhangir et al.,' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='2022) datasets. For passkey retrieval, we follow Mohtashami & Jaggi (2023) to construct synthetic\nprompts for evaluation.\nBaseline Methods. We compare our PoSE training method against following baselines:\n•Full-length fine-tuning takes input tokens of target length for fine-tuning. For this method, compu-\ntation complexity scales quadratically with target context window size.\n5' metadata={'source': 'pdfs/paper_2.pdf', 'page': 4}"
"page_content='Table 1: Perplexity of models trained with different methods. We conduct evaluation on the GovReport\nand Proof-pile datasets, varying evaluation context window size from 2k to 32k. Our PoSE, with\na fixed training window size of 2k, effectively extended to a target context size of 16k / 32k for\ninference while receiving only minimal performance degradation compared to Full-length.\nMethodContext size GovReport Proof-pile\nTrain / Target 2k 4k 8k 16k 32k 2k 4k 8k 16k 32k' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='None - / - 4.74 >103>103>103>1032.83 >103>103>103>103\nFull-length 16k / 16k 4.87 4.70 4.61 4.59 - 2.93 2.71 2.58 2.53 -\nRandPos2k / 16k 11.63 11.17 11.54 15.16 - 7.26 6.83 6.76 7.73 -\n2k / 32k 93.43 95.85 91.79 93.22 97.57 60.74 63.54 60.56 63.15 66.47\nPoSE (Ours)2k / 16k 4.84 4.68 4.60 4.60 - 2.95 2.74 2.61 2.60 -\n2k / 32k 4.91 4.76 4.68 4.64 4.66 3.01 2.78 2.66 2.60 2.59\n•RandPos (Ruoss et al., 2023) is initially designed to train an encoder-only model from scratch' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='for length extrapolation. However, since it shares similar idea of simulating longer sequences via\nchanging position indices, we include it for a comprehensive comparison. Given the original /\ntarget context window length Lc/Lt, it uniquely samples Lcpositions from the set {0, ..., L t−1},\narranges them in ascending order, and employs them as new position indices for training.\n4.2 L ANGUAGE MODELING' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='First, we investigate the impacts of different fine-tuning methods on long sequence language modeling\nusing the GovReport and Proof-pile datasets. GovReport is a summarization dataset comprising\n19,402 reports published by the Congress and the U.S. Government, with an average document length\nof 7,866 tokens. We randomly select 50 reports containing more than 32,768 tokens for evaluation.\nSimilarly, Proof-pile is a 13GB mathematical dataset of long mathematical documents. In line with' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='the approach taken for GovReport, we choose 50 samples from Proof-pile that contain more than\n32,768 tokens for evaluation.\nTable 1 presents the results of scaling to 16k and 32k using Full-length training, RandPos, and PoSE.\nFor each scaled model, as well as the non-fine-tuned LLaMA model (None), we report perplexity\nscores at various evaluation context window sizes, ranging from 2k to 32k, employing the sliding' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='window approach proposed by Press et al. (2021). For evaluation efficiency, we set the stride of the\nsliding window to 1,024.\nFirst, we observe an overall decreasing trend of perplexity for both models scaled to 16k and 32k via\nPoSE as evaluation context window size increases, proving their abilities to leverage longer context.\nSecond, with significantly shorter context length during fine-tuning, our PoSE achieves comparable' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='results with Full-length, consolidating its effectiveness. Third, our method achieves much stronger\nresults than RandPos. We suppose it is because our manipulated position indices closely resembles\nthat of pre-training, hereby preserving the pre-trained language modeling ability to the greatest extent.\nWe also notice that all the scaling methods suffers certain performance degradation as the supported' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='context length increases. We perceive this as a trade-off between the quantity of tokens the model can\nprocess and the level of granularity in the attention the model can pay to each individual token.\n4.3 P ASSKEY RETRIEVAL FOR EFFECTIVE CONTEXT WINDOW\nTo effectively measure the maximum distance that a token can attend to during the inference stage,\nwe adopt the passkey retrieval test proposed by Mohtashami & Jaggi (2023). In this test, models are' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='tasked with recovering a random passkey hidden within a lengthy document. Prompt template used\nfor this task is presented in Figure 2a.\nSpecifically, we compare the non-fine-tuned LLaMA model (denoted as None ) with the PoSE-\nextended version for 16k and 32k context window size. For each model, we vary the prompt length\nfrom 2k to 32k. In each case, we conduct the passkey retrieval test for 50 times, with a random passkey\n6' metadata={'source': 'pdfs/paper_2.pdf', 'page': 5}"
"page_content='There is an important info hidden inside a lot of irrelevant \ntext. Find it and memorize them. I will quiz you about the \nimportant information there.\nThe grass is green. The sky is blue. The sun is yellow. Here \nwe go. There and back again. (repeat x times)\nThe pass key is 81501 . Remember it. 81501  is the pass key.\nThe grass is green. The sky is blue. The sun is yellow. Here \nwe go. There and back again. (repeat y times)\nWhat is the pass key? The pass key is(a)' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='2k 8k 16k 24k 32k / Tokens020406080100Accuracy (%)None\nPoSE-16k\nPoSE-32k (b)\nFigure 2: (a) Prompt template used for passkey retrieval; (b) retrieval accuracy for the non-fine-tuned\nLLaMA model ( None ), and the PoSE-extended counterparts for 16k / 32k window size. Both PoSE-\nextended models maintain a high retrieval accuracy ( ≥90%) within their respective context window.\nof 5 digits generated and placed at a random position inside the prompt for each trial. Figure 2b' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='illustrates the results. For the none-fine-tuned LLaMA model ( None ), the retrieval accuracy rapidly\ndrops to 0 when the prompt length exceeds 2k. In contrast, both PoSE-extended models managed\nto maintain a high retrieval accuracy ( ≥90%) within their respective target context window. This\nindicates that models trained via PoSE genuinely possess the capability to attend to all tokens within\nthe extended context windows.\n5 A NALYSIS' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='In this section, we analyze the advantages of PoSE, including 1) memory and time efficiency;\n2) compatibility with all RoPE-based LLMs and diverse interpolation strategies; 3) potential for\nextremely-long context. In Section 5.4, We also verify that model performance within the original\ncontext window only receives minimal degradation.\n5.1 M EMORY AND TIMEEFFICIENCY\nWe study the memory and time efficiency of PoSE compared with Full-length fine-tuning. For each' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='method, we scale LLaMA-7B to 4k / 8k / 16k through 1,000 training steps with a global batch size of\n16 on 8 V100 GPUs. Experiment results are demonstrated in Figure 3. Figure 3(a) and (b) respectively\nillustrates memory and time consumption for 1,000 steps of Full-length versus PoSE. While the\ntraining cost of Full-length increases rapidly with target window length, PoSE only requires a fixed\nquota of memory and time for context extension, which is significantly lower. Figure 3(c) further' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='compares model perplexity of the two training methods at different steps on GovReport. Notably,\nboth models achieve relatively low perplexity levels within the initial 100 training steps. Moreover,\nat each step, our proposed PoSE, while requiring only a training context size of 2k tokens, exhibits\nvery close language modeling ability to Full-length fine-tuning, which requires an extended training\ncontext of 16k. We did not experiment with context window of 32k or above, because V100 machines' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='cannot afford full fine-tuning of these lengths. But it can be expected that the overhead ration between\nFull-leng and PoSE will become more exaggerated as target length increases. Consequently, we can\nconfidently assert that our proposed approach is both memory and time-efficient.\n5.2 COMPATIBILITY WITH ROPE-B ASED LLM S AND DIVERSE INTERPOLATION STRATEGIES\nWe also delve into the effectiveness of PoSE when applied to different RoPE-based LLMs, as' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='well as various interpolation strategies. Specifically, we employ PoSE on four distinct models:\nLLaMA-7B, LLaMA2-7B, GPT-J-6B, and Baichuan2-7B, all of which encompasses RoPE in their\narchitectures. The original context size of LLaMA-7B and GPT-J-6B is 2k, while that of LLaMA2-7B\nand Baichuan2-7B is 4k. For each model, we examine the integration with Linear, NTK, and YaRN\ninterpolation, as well as the non-fine-tuned original version for comparative purposes. The same' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='GovReport dataset as described in Section 4.2 is utilized. The test set is truncated to the first 1k to\n7' metadata={'source': 'pdfs/paper_2.pdf', 'page': 6}"
"page_content='4k 8k 16k\n(a)101520253035\nOOMMemory (GB)\nFull-length\nPoSE\n4k 8k 16k\n(b)051015Time (h)\nFull-length\nPoSE\n10 30 100 300 1000 / Steps\n(c)4.75.15.45.7\nPerplexity\nFull-length\nPoSEFigure 3: Full-length fine-tuning v.s. PoSE in terms of (a) Memory and (b) Time consumption\nfor extending LLaMA-7B from 2k to 4k / 8k / 16k context, each finishing 1000 training steps.\n(c) Perplexity of both 16k-context models at every training steps. We show that PoSE takes a' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='constantly reduced time and memory for context extension, while attaining a comparable level of\nPPL performance with Full-length fine-tuning at each step.\n1k 2k 4k 8k 16k4.85.05.25.4Perplexity\nLLaMA-7B\n1k 2k 4k 8k 16k4.64.85.05.2\nLLaMA2-7B\n1k 2k 4k 8k 16k7.58.08.59.09.5\nGPT-J-6B\n1k 2k 4k 8k 16k5.86.36.87.3\nBaichuan2-7B\nOriginal PoSE-Linear PoSE-NTK PoSE-YaRN\nFigure 4: Perplexity of LLaMA-7B, LLaMA2-7B, GPT-J-6B, Baichuan2-7B extended to 16k via' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='PoSE with Linear / NTK / YaRN interpolation, along with the non-fine-tuned Original model. The\nconsistently low perplexity observed across all nine combinations serves as an indication of the\neffectiveness of our method across RoPE-based LLMs and diverse interpolation strategies.\n16k tokens for plotting the perplexity curve, as depicted in Figure 4. First, it is evident that PoSE is\neffective across all four models and three interpolation strategies, as evidenced by the low perplexities' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='achieved by all 12 combinations in comparison to the non-fine-tuned original model. Second, we\nobserve that NTK and YaRN interpolation generally yields superior results compared to Linear\ninterpolation. However, it is noteworthy that NTK exhibits a significant increase in perplexity after\na certain turning point, which occurs prior to reaching the target context length. This behavior is\nconsistent with previous findings, indicating that for a given scaling factor α, NTK cannot genuinely' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='expand the context window by αtimes (Peng & Quesnelle, 2023; Quesnelle, 2023; Peng et al., 2023).\n5.3 P OTENTIAL FOR EXTREMELY -LONG CONTEXT\nBecause PoSE only takes a fixed context window at training stage to extend to target context window\nsize, we can promisingly extend LLMs to support infinite input lengths using this method. In this\nsection, we extend context window size to 96k and 128k to explore PoSE’s potential for extreme' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='context window extension. Given the need to evaluate on extremely long documents, we have opted to\nemploy two book datasets, namely Books3 (Presser, 2020) and Gutenberg (PG-19) (Rae et al., 2019).\nBoth of these datasets consist of extensive collections of literary works, rendering them well-suited\nsubjects for the assessment of long-range modeling. For our evaluation, we randomly selected 20\nbooks from each dataset, each containing more than 128k tokens.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='Fine-tuning LLaMA models using PoSE, we experimented with Linear / NTK / YaRN interpolation\nfor both the 96k and 128k models. To calculate perplexity, we adhere to the sliding window strategy\nadopted in Section 4.2, with an increased sliding window step of 16k to enhance evaluation efficiency.\n8' metadata={'source': 'pdfs/paper_2.pdf', 'page': 7}"
"page_content='Table 2: Perplexity of models extended to extreme context size via PoSE on PG-19 and Books3. We\nshow that our training method can effectively extend context window size to 128k when combined\nwith YaRN interpolation.\nModelGutenberg (PG-19) Books3\n32k 64k 96k 128k 32k 64k 96k 128k\nPoSE-Linear-96k 10.18 11.11 13.57 - 9.98 10.90 13.42 -\nPoSE-NTK-96k 7.98 20.39 38.73 - 8.29 20.82 40.39 -\nPoSE-YaRN-96k 8.31 8.65 9.36 - 8.90 9.40 10.38 -\nPoSE-Linear-128k 16.90 22.47 26.77 31.18 26.20 43.62 57.08 70.87' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='PoSE-NTK-128k 8.04 14.84 29.48 34.80 8.34 16.04 31.42 37.00\nPoSE-YaRN-128k 9.32 10.36 10.77 11.33 10.56 12.30 13.07 13.81\nTable 3: Performance of PoSE-extended LLaMA model on standard benchmarks in comparison with\nFull-length fine-tuning and the original LLaMA. We show that PoSE-extended models exhibit only\nmarginal performance degradation compared with Full-length fine-tuning and the original version.\nModelZero-Shot Few-Shot\nBoolQ PIQA WinoGrande TruthfulQA ARC-C HellaSwag' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='LLaMA 75.11 78.67 69.85 34.08 51.19 77.75\nFull-Linear-16k 70.95 77.64 69.06 31.89 48.55 74.19\nFull-NTK-16k 75.80 78.08 68.98 33.83 48.81 76.57\nFull-YaRN-16k 73.88 77.64 68.15 34.12 50.60 77.18\nPoSE-Linear-16k 74.50 78.13 68.59 32.05 48.29 75.56\nPoSE-NTK-16k 74.28 78.24 68.90 33.89 49.83 76.82\nPoSE-YaRN-16k 74.28 78.02 69.06 34.00 49.23 77.04\nPoSE-Linear-128k 67.71 76.22 67.56 36.16 39.93 66.04\nPoSE-NTK-128k 75.35 78.18 68.98 32.71 49.66 76.19\nPoSE-YaRN-128k 73.61 77.80 70.01 34.47 48.46 75.54' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='The outcomes of these experiments are detailed in Table 2. It is observe that, PoSE successfully\nextends the model’s context window to 96k when coupled with Linear interpolation, and further\nextends the context window to 128k when paired with YaRN. These promising results consolidates\nthe effectiveness of PoSE for extreme context window extension.\n5.4 E VALUATION OF CAPABILITY ON ORIGINAL CONTEXT WINDOW' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='In this section, we examine the capabilities of the PoSE-extended models on the original context win-\ndow using standard benchmarks. We combine the Hugging Face Open LLM Leaderboard (Face, 2023)\nwith a subset of LLaMA benchmarks to assess zero-shot and few-shot performance. For zero-shot\nevaluation, we employ BoolQ (Clark et al., 2019), PIQA (Bisk et al., 2020), WinoGrande (Keisuke\net al., 2019), and TruthfulQA (Lin et al., 2022). For few-shot evaluation, we utilize 25-shot ARC-' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='Challenge (Clark et al., 2018) and 10-shot HellaSwag (Zellers et al., 2019). Our evaluation metrics\nare benchmark-specific: for BoolQ, PIQA, and WinoGrande, we report accuracy; for TruthfulQA, we\nreport mc2; and for ARC-C and HellaSwag, we report normalized accuracy.\nTable 3 summarizes the results. It is observed that, PoSE-extended models exhibit only marginal\nperformance degradation compared with Full-length fine-tuning and the original LLaMA, with the' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='only exception of the 128k model employing linear interpolation. This indicates that while extending\ncontext window size, PoSE effectively preserves original language comprehension ability.\n9' metadata={'source': 'pdfs/paper_2.pdf', 'page': 8}"
"page_content='6 C ONCLUSION\nIn this paper, we introduce Positional Skip-wis E(PoSE) training to efficiently extend the context\nwindow of Large Language Models. PoSE simulates long inputs by manipulating position indices,\nthereby requiring only the original context window for fine-tuning, successfully decoupling train\nlength and target length. Experiments have shown that, compared with fine-tuning on the full length,\nPoSE greatly reduces memory and time overhead. Taking advantage of this, we have managed to' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='extend LLaMA model to 128k on 8 V100 GPUs, observing only minimal performance degradation on\nstandard benchmarks. We have also empirically verified that PoSE is compatible with all RoPE-based\nLLMs and position interpolation strategies.\nREFERENCES\nBaichuan. Baichuan 2: Open large-scale language models. arXiv preprint arXiv:2309.10305 , 2023.\nURLhttps://arxiv.org/abs/2309.10305 .\nYonatan Bisk, Rowan Zellers, Ronan Le Bras, Jianfeng Gao, and Yejin Choi. Piqa: Reasoning' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='about physical commonsense in natural language. In Thirty-Fourth AAAI Conference on Artificial\nIntelligence , 2020.\nTom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal,\nArvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are\nfew-shot learners. Advances in neural information processing systems , 33:1877–1901, 2020.\nAydar Bulatov, Yury Kuratov, and Mikhail Burtsev. Recurrent memory transformer. Advances in' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='Neural Information Processing Systems , 35:11079–11091, 2022.\nShouyuan Chen, Sherman Wong, Liangjian Chen, and Yuandong Tian. Extending context window of\nlarge language models via positional interpolation. ArXiv , abs/2306.15595, 2023a.\nYukang Chen, Shengju Qian, Haotian Tang, Xin Lai, Zhijian Liu, Song Han, and Jiaya Jia. Longlora:\nEfficient fine-tuning of long-context large language models. arXiv preprint arXiv:2309.12307 ,\n2023b.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='Christopher Clark, Kenton Lee, Ming-Wei Chang, Tom Kwiatkowski, Michael Collins, and Kristina\nToutanova. Boolq: Exploring the surprising difficulty of natural yes/no questions. In NAACL ,\n2019.\nPeter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and\nOyvind Tafjord. Think you have solved question answering? try arc, the ai2 reasoning challenge.\narXiv preprint arXiv:1803.05457 , 2018.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='Zihang Dai, Zhilin Yang, Yiming Yang, Jaime G Carbonell, Quoc Le, and Ruslan Salakhutdinov.\nTransformer-xl: Attentive language models beyond a fixed-length context. In Proceedings of the\n57th Annual Meeting of the Association for Computational Linguistics , pp. 2978–2988, 2019.\nTri Dao. FlashAttention-2: Faster attention with better parallelism and work partitioning. 2023.\nTri Dao, Daniel Y . Fu, Stefano Ermon, Atri Rudra, and Christopher Ré. FlashAttention: Fast and' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='memory-efficient exact attention with IO-awareness. In Advances in Neural Information Processing\nSystems , 2022.\nHugging Face. Open llm leaderboard. https://huggingface.co/spaces/\nHuggingFaceH4/open_llm_leaderboard , 2023.\nLeo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang,\nHorace He, Anish Thite, Noa Nabeshima, Shawn Presser, and Connor Leahy. The Pile: An 800gb\ndataset of diverse text for language modeling. arXiv preprint arXiv:2101.00027 , 2020.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='Adi Haviv, Ori Ram, Ofir Press, Peter Izsak, and Omer Levy. Transformer language models without\npositional encodings still learn positional information. In Findings of the Association for Computa-\ntional Linguistics: EMNLP 2022 , pp. 1382–1390, Abu Dhabi, United Arab Emirates, December\n2022. Association for Computational Linguistics. doi: 10.18653/v1/2022.findings-emnlp.99.\n10' metadata={'source': 'pdfs/paper_2.pdf', 'page': 9}"
"page_content='Luyang Huang, Shuyang Cao, Nikolaus Parulian, Heng Ji, and Lu Wang. Efficient attentions for long\ndocument summarization. In Proceedings of the 2021 Conference of the North American Chapter\nof the Association for Computational Linguistics: Human Language Technologies , pp. 1419–1436,\nOnline, June 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.naacl-main.\n112.\nkaiokendev. Things i’m learning while training superhot. https://kaiokendev.github.io/' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='til#extending-context-to-8k , 2023.\nSakaguchi Keisuke, Le Bras Ronan, Bhagavatula Chandra, and Choi Yejin. Winogrande: An\nadversarial winograd schema challenge at scale. 2019.\nWoosuk Kwon, Zhuohan Li, Siyuan Zhuang, Ying Sheng, Lianmin Zheng, Cody Hao Yu, Joseph E.\nGonzalez, Hao Zhang, and Ion Stoica. Efficient memory management for large language model\nserving with pagedattention, 2023.\nBenjamin Lefaudeux, Francisco Massa, Diana Liskovich, Wenhan Xiong, Vittorio Caggiano, Sean' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='Naren, Min Xu, Jieru Hu, Marta Tintore, Susan Zhang, Patrick Labatut, and Daniel Haziza.\nxformers: A modular and hackable transformer modelling library. https://github.com/\nfacebookresearch/xformers , 2022.\nMukai Li, Shansan Gong, Jiangtao Feng, Yiheng Xu, Jun Zhang, Zhiyong Wu, and Lingpeng Kong.\nIn-context learning with many demonstration examples. arXiv preprint arXiv:2302.04931 , 2023.\nStephanie Lin, Jacob Hilton, and Owain Evans. Truthfulqa: Measuring how models mimic human' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='falsehoods. In Proceedings of the 60th Annual Meeting of the Association for Computational\nLinguistics (Volume 1: Long Papers) , pp. 3214–3252, 2022.\nAmirkeivan Mohtashami and Martin Jaggi. Landmark attention: Random-access infinite context\nlength for transformers, 2023.\nBowen Peng and Jeffrey Quesnelle. Ntk-aware scaled rope allows llama models\nto have extended (8k+) context size without any fine-tuning and minimal perplex-\nity degradation. https://www.reddit.com/r/LocalLLaMA/comments/14lz7j5/' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='ntkaware_scaled_rope_allows_llama_models_to_have , 2023.\nBowen Peng, Jeffrey Quesnelle, Honglu Fan, and Enrico Shippole. Yarn: Efficient context window\nextension of large language models, 2023.\nOfir Press, Noah A Smith, and Mike Lewis. Train short, test long: Attention with linear biases\nenables input length extrapolation. arXiv preprint arXiv:2108.12409 , 2021.\nShawn Presser. https://twitter.com/theshawwn/status/\n1320282149329784833 , 2020.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='Jeffrey Quesnelle. Dynamically scaled rope further increases performance of long context llama with\nzero fine-tuning. https://www.reddit.com/r/LocalLLaMA/comments/14mrgpr/\ndynamically_scaled_rope_further_increases/ , 2023.\nJack W Rae, Anna Potapenko, Siddhant M Jayakumar, Chloe Hillier, and Timothy P Lillicrap.\nCompressive transformers for long-range sequence modelling. arXiv preprint , 2019.\nSamyam Rajbhandari, Jeff Rasley, Olatunji Ruwase, and Yuxiong He. Zero: Memory optimizations' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='toward training trillion parameter models. In SC20: International Conference for High Performance\nComputing, Networking, Storage and Analysis , pp. 1–16. IEEE, 2020.\nAnian Ruoss, Grégoire Delétang, Tim Genewein, Jordi Grau-Moya, Róbert Csordás, Mehdi Bennani,\nShane Legg, and Joel Veness. Randomized positional encodings boost length generalization of\ntransformers. In Proceedings of the 61st Annual Meeting of the Association for Computational' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='Linguistics (Volume 2: Short Papers) , pp. 1889–1903, Toronto, Canada, July 2023. Association for\nComputational Linguistics. doi: 10.18653/v1/2023.acl-short.161.\nJianlin Su, Yu Lu, Shengfeng Pan, Ahmed Murtadha, Bo Wen, and Yunfeng Liu. Roformer: Enhanced\ntransformer with rotary position embedding. arXiv preprint arXiv:2104.09864 , 2021.\n11' metadata={'source': 'pdfs/paper_2.pdf', 'page': 10}"
"page_content='Yutao Sun, Li Dong, Barun Patra, Shuming Ma, Shaohan Huang, Alon Benhaim, Vishrav Chaudhary,\nXia Song, and Furu Wei. A length-extrapolatable transformer. In Proceedings of the 61st\nAnnual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) , pp.\n14590–14604, Toronto, Canada, July 2023. Association for Computational Linguistics. doi:\n10.18653/v1/2023.acl-long.816.\nHugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée' metadata={'source': 'pdfs/paper_2.pdf', 'page': 11}"
"page_content='Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and\nefficient foundation language models. arXiv preprint arXiv:2302.13971 , 2023a.\nHugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay\nBashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. Llama 2: Open foundation\nand fine-tuned chat models. arXiv preprint arXiv:2307.09288 , 2023b.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 11}"
"page_content='Szymon Tworkowski, Konrad Staniszewski, Mikołaj Pacek, Yuhuai Wu, Henryk Michalewski,\nand Piotr Miło ´s. Focused transformer: Contrastive training for context scaling. arXiv preprint\narXiv:2307.03170 , 2023.\nBen Wang and Aran Komatsuzaki. GPT-J-6B: A 6 Billion Parameter Autoregressive Language Model.\nhttps://github.com/kingoflolz/mesh-transformer-jax , May 2021.\nWeizhi Wang, Li Dong, Hao Cheng, Xiaodong Liu, Xifeng Yan, Jianfeng Gao, and Furu Wei.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 11}"
"page_content='Augmenting language models with long-term memory. arXiv preprint arXiv:2306.07174 , 2023.\nYuhuai Wu, Markus N Rabe, DeLesley Hutchins, and Christian Szegedy. Memorizing transformers.\narXiv preprint arXiv:2203.08913 , 2022.\nRowan Zellers, Ari Holtzman, Yonatan Bisk, Ali Farhadi, and Yejin Choi. Hellaswag: Can a machine\nreally finish your sentence? In Proceedings of the 57th Annual Meeting of the Association for\nComputational Linguistics , pp. 4791–4800, 2019.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 11}"
"page_content='Azerbayev Zhangir, Ayers Edward, and Bartosz Piotrowski. Proof-pile. https://github.com/\nzhangir-azerbayev/proof-pile , 2022.\nYucheng Zhou, Tao Shen, Xiubo Geng, Chongyang Tao, Guodong Long, Can Xu, and Daxin Jiang.\nFine-grained distillation for long document retrieval. arXiv preprint arXiv:2212.10423 , 2022.\n12' metadata={'source': 'pdfs/paper_2.pdf', 'page': 11}"
"page_content='Table 4: Comparison of different methods for choosing vi. We report perplexity with evaluation\ncontext window ranging from 2k to 16k. We show that these variations have relatively little impact\non the outcomes of fine-tuning.\nMethodGovReport Proof-pile\n2k 4k 8k 16k 2k 4k 8k 16k\nvi∼ U (. . .) 4.84 4.68 4.60 4.60 2.95 2.74 2.61 2.60\nvi= 0 4.85 4.72 4.64 4.68 2.96 2.75 2.63 2.61\nvi=ui 4.84 4.68 4.60 4.60 2.95 2.73 2.60 2.56\nA A BLATION OF TEXT CONTAINED WITHIN EACH CHUNK' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='PoSE divide the original context window into several chunks, and modify the position indices of each\nchunk to cover a wider range of relative positions in a fixed window. However, it does not impose a\nparticular constraint on the text contained within each chunk. Recall that in Equation 5, we assign the\ncontent of chunk cias below:\nci=x[vi+sti:vi+sti+li]\nIn this section, we explore several strategies for determining vi: 1) sampling from uniform distribution,' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='vi∼ U({vi−1, . . . , L x−Lc), which is the one used in PoSE; 2) vi= 0, which results in genuinely\ncontinuous content for the chunks; 3) vi=ui, aligning the manipulated position indices with actual\npositions in the original text. We use the same test setting as Section 4.2, extending LLaMA-7B from\n2k to 16k context. As can be seen in Table 4, we show that these variations have relatively little\nimpact on the outcomes of fine-tuning.\nB A NALYSIS OF CHUNK NUMBER N' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='0 2500 5000 7500 10000 12500 15000\nRelative Position0.00.20.40.60.81.0ProbabilityOriginal\n2 chunks\n3 chunks\nRandPos\nFigure 5: Coverage probability for each relative position in a single training example (2k -> 16k).\nUtilizing multiple chunks reduces coverage probability within the original [0,2,048] context window,\nwhile enhancing the coverage likelihood of relative positions in the range of [2,048,16,383]. Proba-' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='bility of coverage increases with the number of chunks. Pushing the chunk number to the limit is\nRandPos, utilizing 2048 chunks, capable of covering every relative position in each training example\nby expectation.\nPoSE achieves coverage of all positions within the target context window by randomly sampling\nthe chunk sizes and skipping bias terms for each training example. In this section, we explore the' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='probability of each relative position being covered by a training example, using a context extension\nof 2,048 to 16,384 as an example. For the unextended original version, the probability of a relative\nposition within 2048 being covered is 1, and the probability of a relative position above 2,048 being\ncovered is 0. For the cases where the number of chunks is 2, 3, or 2,048 (i.e., RandPos), we use the\n13' metadata={'source': 'pdfs/paper_2.pdf', 'page': 12}"
"page_content='visit_prob_list = np.array ([0] * Lt)\niter_times = 10000\nfor_ inrange(iter_times ):\nl0 = random.randint (1, Lc-1)\nu1 = random.randint (0, Lt-Lc)\nl1 = Lc -l0\nrng1 = set(range( 1, max(l0,l1))\nrng2 = set(range(u1+ 1, u1+))\nrng= rng1 | rng2\nforx inrng:\nvisit_prob_list [x] += 1\nvisit_prob_list /= iter_timesvisit_prob_list = np.array ([0] * Lt)\niter_times = 10000\nfor_ inrange(iter_times ):\nl0 = random.randint (1, Lc-2)\nl1 = random.randint (1, Lc-l0-1)\nl2 = Lc -l0 -l1\nu1 = random.randint (0, Lt-Lc)' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='u2 = random.randint (u1, Lt-Lc)\nrng1 = set(range( 1, max(l0,l1,l2)))\nrng2 = set(range(u1+ 1, u1+l0+l1))\nrng3 = set(range(u2 -u1+1, u2-\nu1+l1+l2))\nrng4 = set(range(u2+l1+ 1, u2+Lc))\nrng= rng1 | rng2 | rng3 | rng4\nforx inrng:\nvisit_prob_list [x] += 1\nvisit_prob_list / = iter_timesvisit_prob_list = np.array ([0] * Lt)\niter_times = 100\nfor_ inrange(iter_times ):\ntot_pos_list = list(range(Lt))\nnew_pos_list = random.sample (tot_pos_list , Lc)\nnew_pos_list.sort ()\ndistance_rng = set()' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='foriinrange(0, len(new_pos_list )-1):\nforj inrange(i+ 1, len(new_pos_list )):\ndistance_rng.add (new_pos_list [j] -\nnew_pos_list [i])\nforx indistance_rng :\nvisit_prob_list [x] += 1\nvisit_prob_list /= iter_timesFigure 6: Python Code used for calculating coverage probability of each relative position in Figure 5.\nTable 5: Comparison of different chunk numbers. We report perplexity with evaluation context' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='window ranging from 2k to 16k. By increasing chunk number, relative positions in [2,048,16,383]\nreceive an increased chance of being trained, rendering better results for context extension. However,\nextremely large chunk number also damages model performance.\nChunk numberProof-pile\n2k 4k 8k 16k\n1 2.83 >103>103>103\n2 2.95 2.74 2.61 2.60\n3 2.93 2.72 2.60 2.59\n2048 7.26 6.83 6.76 7.73\nMonte Carlo method to estimate this coverage probability. The code used is demonstrated in Figure 6.' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='The estimated results are shown in Figure 5. It can be seen that PoSE reduces the coverage probability\nof positions within the original context window, while all relative positions in [2,048,16,383]\nreceives a certain increase in chance of being covered, and the probability of coverage increases as\nthe number of chunks increases. For the case where the number of chunks is equal to 2,048, the\nprobability of each relative position being covered is close to 1. With this observation, we further' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='compare the impact of chunk number on language modeling capability, as presented in Table 5.\nIncreasing chunk number efficiently renders better results for context extension. However, extremely\nlarge chunk number also damages model performance, due to the severe deviation from the position\nencoding structure used in pre-training phase. We believe that the choice of the number of chunks is\na trade-off between training efficiency and performance.\n14' metadata={'source': 'pdfs/paper_2.pdf', 'page': 13}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nZichang Liu1Jue Wang2Tri Dao3Tianyi Zhou4Binhang Yuan5Zhao Song6Anshumali Shrivastava1\nCe Zhang5Yuandong Tian7Christopher Ré3Beidi Chen8 7\nAbstract\nLarge language models (LLMs) with hundreds of\nbillions of parameters have sparked a new wave\nof exciting AI applications. However, they are\ncomputationally expensive at inference time. Spar-\nsity is a natural approach to reduce this cost, but' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='existing methods either require costly retraining,\nhave to forgo LLM’s in-context learning ability, or\ndo not yield wall-clock time speedup on modern\nhardware. We hypothesize that contextual sparsity ,\nwhich are small, input-dependent sets of attention\nheads and MLP parameters that yield approxi-\nmately the same output as the dense model for a\ngiven input, can address these issues. We show that\ncontextual sparsity exists, that it can be accurately' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='predicted, and that we can exploit it to speed up\nLLM inference in wall-clock time without compro-\nmising LLM’s quality or in-context learning ability.\nBased on these insights, we propose DEJAVU , a\nsystem that uses a low-cost algorithm to predict\ncontextual sparsity on the fly given inputs to each\nlayer, along with an asynchronous and hardware-\naware implementation that speeds up LLM\ninference. We validate that DEJAVU can reduce the\ninference latency of OPT-175B by over 2 ×com-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='pared to the state-of-the-art FasterTransformer,\nand over 6 ×compared to the widely used Hugging\nFace implementation, without compromising\nmodel quality. The code is available at https:\n//github.com/FMInference/DejaVu .\n1 Introduction\nLarge language models (LLMs), such as GPT-3, PaLM,\nand OPT have demonstrated that an immense number of\n1Rice University2Zhe Jiang University3Stanford Uni-\nversity4University of California, San Diego5ETH Zurich\n6Adobe Research7Meta AI (FAIR)8Carnegie Mellon Univer-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='sity. Correspondence to: Zichang Liu <zl71@rice.edu>, Tri Dao\n<trid@stanford.edu>, Tianyi Zhou <t8zhou@ucsd.edu>, Zhao Song\n<zsong@adobe.com>, Beidi Chen <beidic@andrew.cmu.edu>.\nProceedings of the 40thInternational Conference on Machine\nLearning , Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright\n2023 by the author(s).parameters unleashes impressive performance and emergent\nin-context-learning abilities—they can perform a task by\nconditioning on input-output examples, without updating' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='their parameters (Bommasani et al., 2021; Liang et al.,\n2022; Brown et al., 2020; Min et al., 2022; Chan et al.,\n2022). However, they are very expensive at inference time,\nespecially for latency-sensitive applications (Pope et al.,\n2022). An ideal inference-time model should use less com-\nputation and memory while maintaining the performance\nand special abilities of pre-trained LLMs. The simplest and\nmost natural approach is sparsification or pruning, which' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='has a long history before the LLM era (LeCun et al., 1989).\nUnfortunately, speeding up inference-time sparse LLMs in\nwall-clock time while maintaining quality and in-context\nlearning abilities remains a challenging problem.\nWhile sparsity and pruning have been well-studied, they\nhave not seen wide adoption on LLMs due to the poor\nquality and efficiency trade-offs on modern hardware such\nas GPUs. First, it is infeasible to retrain or iteratively prune' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='models at the scale of hundreds of billions of parameters.\nThus, methods in iterative pruning and lottery ticket\nhypothesis (Lee et al., 2018; Frankle & Carbin, 2018) can\nonly be applied to smaller-scale models. Second, it is\nchallenging to find sparsity that preserves the in-context\nlearning ability of LLMs. Many works have shown the\neffectiveness of task-dependent pruning (Michel et al., 2019;\nBansal et al., 2022), but maintaining different models for' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='each task conflicts with the task independence goal of LLMs.\nLastly, it is hard to achieve wall-clock time speed-up with\nunstructured sparsity due to its well-known difficulty with\nmodern hardware (Hooker, 2021). For example, recent\ndevelopment in zero-shot pruning like SparseGPT (Frantar\n& Alistarh, 2023) finds 60% unstructured sparsity but does\nnot yet lead to any wall-clock time speedup.\nAn ideal sparsity for LLMs should (i) not require model' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='retraining, (ii) preserve quality and in-context learning\nability, and (iii) lead to speed-up in wall-clock time on\nmodern hardware. To achieve such demanding requirements,\nwe go beyond static sparsity in previous works (e.g., struc-\ntured/unstructured weight pruning). We instead envision\ncontextual sparsity , which are small, input-dependent\nsets of attention heads and MLP parameters that lead to\n(approximately) the same output as the full model for an' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='input. Inspired by the connections between LLMs, Hidden\n1arXiv:2310.17157v1  [cs.LG]  26 Oct 2023' metadata={'source': 'pdfs/paper_3.pdf', 'page': 0}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0 20 40 60 80\nTransformer Layer 0.00 0.20 0.40 0.60 0.80 1.00Contextual SparsityOPT-175B\n(a) Contextual Sparsity\n1 2 3 4 5 6 7 8\nTheoretical Reduction0.7940.7960.7980.8000.8020.8040.8060.8080.810Accuracy\nStatic Sparsity\nNon-Contextual Sparsity\nContextual Sparsity\n(b) Accuracy-Efficiency Trade-offs\nFigure 1. (1) LLMs have up to 85% contextual sparsity for a given\ninput. (2) Contextual sparsity has much better efficiency-accuracy' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='trade-offs (up to 7 ×) than non-contextual sparsity or static sparsity.\nMarkov Models (Xie et al., 2022; Baum & Petrie, 1966), and\nthe classic Viterbi algorithm (Viterbi, 1967), we hypothesize\nthat for pre-trained LLMs,\ncontextual sparsity exists given any input.\nThe hypothesis, if true, would enable us to cut off specific\nattention heads and MLP parameters (structured sparsity)\non the fly for inference-time, without modifying pre-trained\nmodels. However, there are three challenges.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='Existence : It is nontrivial to verify if such contextual sparsity\nexists, and naive verification can be prohibitively expensive.\nPrediction : Even if contextual sparsity exists, it is challeng-\ning to predict the sparsity for a given input in advance.\nEfficiency : Even if the sparsity can be predicted, it might\nbe difficult to achieve end-to-end wall-clock time speedup.\nTaking OPT-175B as an example, the latency of one MLP\nblock is only 0.2 ms on an 8 ×A100 80GB machine. Without' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='a fast prediction and optimized implementation, the overhead\ncan easily increase the LLM latency rather than reduce it.\nIn this work, we address these challenges as follows:\nExistence : Fortunately, we verify the existence of contextual\nsparsity with a surprisingly simple approach. To achieve\nessentially the same output, contextual sparsity is on average\n85% structured sparse and thereby potentially leads to a 7×\nparameter reduction for each specific input while maintain-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='ing accuracy (Figure 1(a)). During explorations of contextual\nsparsity, we make important empirical observations and build\na theoretical understanding of major components in LLMs\nthat help address the prediction and efficiency challenge.\nDeja Vu\nAttentionkMLPkPredictorPredictorPredictorAttentionk+1……Figure 2. DEJAVU uses lookahead predictors to side-step prediction\ncosts: given the input to the attention layer at block k, they (asyn-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='chronously) predict the contextual sparsity for the MLP at block k,\nand given the input to the MLP at block k, they predict the sparsity\nfor the attention head at the next layer.\nPrediction : We discover that contextual sparsity depends\nnot only on individual input tokens (i.e., non-contextual\ndynamic sparsity) but also on their interactions ( contextual\ndynamic sparsity). Figure 1(b) shows that with pure dynamic\ninformation, sparsity prediction is inaccurate. Only with' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='token embeddings with sufficient contextual information\ncan we predict sparsity accurately. Another finding is that\ncontextual dynamic sparsity for every layer can be predicted\nbased on the “similarity” between layer parameters (head-\ns/MLP) and the output from the previous layer, which carries\nthe immediate contextual mixture of token embeddings.\nEfficiency : Because at inference time, model parameters are\nstatic, inspired by the classical nearest neighbor search (NNS)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='literature and its applications in efficient deep learning, it is\npossible to formulate the above similarity-based prediction\nas an NNS problem (Indyk & Motwani, 1998b; Zhang et al.,\n2018; Chen et al., 2020a). However, as mentioned, the over-\nhead might be difficult to overcome as we would need to\nperform on-the-fly predictions before every layer. Luckily,\nwe exploit a phenomenon of LLM where token embeddings\nchange slowly across layers due to residual connections (well-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='known in computer vision (He et al., 2016)). Since the inputs\nto a few consecutive layers are very similar, we can design\nan asynchronous lookahead predictor (Figure 2).\nBased on our findings, we present a system, DEJAVU , that\nexploits contextual sparsity and realizes efficient LLMs for\nlatency-sensitive applications.\n•In Section 4.1 and Section 4.2, we present a low-cost\nlearning-based algorithm to predict sparsity on the fly.\nGiven the input to a specific layer, it predicts a relevant' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='subset of attention (heads) or MLP parameters in the next\nlayer and only loads them for the computation.\n•In Section 4.3, we propose an asynchronous predictor (simi-\nlar to classic branch predictor (Smith, 1998)) to avoid the se-\nquential overhead. A theoretical guarantee justifies that the\n2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 1}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\ncross-layer design suffices for accurate sparsity prediction.\nAfter integrating hardware-aware implementation of sparse\nmatrix multiply (Section 4.4), DEJAVU (written mostly in\nPython) can reduce latency of open-source LLMs such\nas OPT-175B by over 2 ×end-to-end without quality\ndegradation compared to the state-of-the-art library Faster-\nTransformer from Nvidia (written entirely in C++/CUDA),' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='and over 2 ×compared to the widely used Hugging Face\nimplementation at small batch sizes. Furthermore, we show\nseveral ablations on different components of DEJAVU and\nits compatibility with quantization techniques.\n2 Related Work and Problem Formulation\nWe first briefly discuss the rich literature on efficient\ninference. Then, we introduce the latency breakdown in our\nsetting. Last, we provide a formal problem formulation.\n2.1 Quantization, Pruning, Distillation for Inference' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='Various relaxations have been studied for decades for\nmodel inference in machine learning. There are three main\ntechniques: quantization (Han et al., 2015; Jacob et al.,\n2018; Nagel et al., 2019; Zhao et al., 2019), pruning or\nsparsity (Molchanov et al., 2016; Liu et al., 2018; Hoefler\net al., 2021), and distillation (Hinton et al., 2015; Tang et al.,\n2019; Touvron et al., 2021). They are orthogonal areas and\nusually excel in different settings. Recently, there is active' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='research attempting to apply one or a combination of such\ntechniques in LLM inference (Yao et al., 2022; Park et al.,\n2022; Dettmers et al., 2022; Frantar et al., 2022; Frantar &\nAlistarh, 2023; Bansal et al., 2022; Xiao et al., 2022). More\ndiscussion is presented in Appendix A.\n2.2 LLM Inference Latency Breakdown\nThe generative procedure of LLMs consists of two phases: (i)\ntheprompt phase takes an input sequence to generate the keys\nand values (KV cache) for each transformer block of LLMs,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='which is similar to the forwarding pass of LLMs training;\nand (ii) the token generation phase utilizes and updates the\nKV cache to generate tokens step by step, where the current\ntoken generation depends on previously generated tokens.\nThis paper studies the setting where the token generation\nphase easily dominates the end-to-end inference time. As\nshown in Table 1, generating a sequence of length 128 takes\nmuch longer time than processing a sequence of length 128' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='as prompt due to I/O latency of loading model parameters.\nIn addition, Table 2 shows that attention and MLP are both\nbottlenecks in LLMs, e.g., in 175B models, loading MLP\nparameters takes around2\n3of the total I/O and attention\nheads take the other1\n3. Further, in the tensor-parallel regime,\nthere are two communications between GPUs, one after\nthe attention block, and the other one after the MLP block.\nAs shown in Table 3, communication between GPUs takes' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='around 15 % token generation latency. This paper focuses on\nmaking attention and MLP more efficient. Communicationcost implies that the upper bound of such speed-up is around\n6×when skipping all transformer blocks.\nTable 1. Theoretical breakdown for prompting versus token genera-\ntion (tensor model parallelism on 8 A100-80G GPUs).\nTFLOPs I/O Compute Latency (ms) I/O Latency (ms)\nPrompting 128 44.6 330 GB 17.87 20.6\nToken Generation 128 44.6 41 TB 17.87 2600' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='Table 2. Theoretical breakdown for Attention block versus MLP\nblock in one transformer layer when generating one token (tensor\nmodel parallelism on 8 A100-80G GPUs).\nGFLOPs I/O (GB) Compute Latency (ms) I/O Latency (ms)\nAttention Block 1.21 1.12 0.00048 0.07\nMLP Block 2.41 2.25 0.00096 0.14\nTable 3. Latency breakdown of generating 1 token under the setting\nof batch size 1 and prompt length 128 on 8 A100-80GB.\nAll Reduce MLP Block Attention Block (ms) Others\n6 ms 19ms 13ms 2ms' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='2.3 Problem Formulation\nThe goal is to reduce the generation latency of LLMs by\nexploiting contextual sparsity. In the following, we formally\ndefine the sparsified attention and MLP blocks.\nSparsified MLP: There are two linear layers in one MLP\nblock, W1,W2∈Rd×4d. Denote y∈R1×das the input to the\nMLP block in the current generation step. Let each column\n(the weight of i-th neuron) of linear layers be W1\ni,W2\ni∈\nRd×1. With contextual sparsity, only a small set of them are' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='required for computation. Let SM⊆[4d]denote such set of\nneurons for input y. The sparsified MLP computation is\nMLP SM(y)=σ(yW1\nSM)(W2\nSM)⊤, (1)\nwhere σis the activation function, e.g., ReLU, GeLU. Note\nthat since the computation in the first linear results in sparse\nactivations, the second linear layer is also sparsified.\nSparsified Attention: LetX∈Rn×ddenote the embeddings\nof all tokens (e.g., prompts and previously generated tokens).\nLety∈R1×dbe the input to the Multi-Head-Attention' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='(MHA) in the current generation step. Suppose there are h\nheads. For each i∈[h], we use WK\ni,WQ\ni,WV\ni∈Rd×dhto\ndenote key, query, value projections for the i-th head, and\nWO\ni∈Rdh×dfor output projections. With contextual spar-\nsity, we denote SAas a small set of attention heads leading to\napproximately the same output as the full attention for input\ny. Following the notation system in (Alman & Song, 2023),\nsparsified MHA computation can be formally written as\nMHA SA(y)=X\ni∈SAHi(y)|{z}\n1×dhWO' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='i|{z}\ndh×d,\nwhere Hi(y):Rd→RdhandDi(y)∈Rcan be written as\nHi(y):=Di(y)−1exp(yWQ\ni(WK\ni)⊤X⊤)XWV\ni,(2)\nDi(y):=exp( yWQ\ni(WK\ni)⊤X⊤)1n.\nFor both MLP and Attention, given a compute budget, the\ngoal is to find SMandSAthat minimize the error between\nthe sparse approximation and full computation.\n3' metadata={'source': 'pdfs/paper_3.pdf', 'page': 2}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0 20 40 60 80\nTransformer Layer20%40%60%80%100%% Not Activated HeadOPT-30B\nOPT-66B\nOPT-175B\n(a) Contextual sparsity in Attention Head\n0 20 40 60 80\nTransformer Layer90%92%94%96%98%100%% Not Activated NeuronsOPT-30B\nOPT-66B\nOPT-175B\n(b) Contextual sparsity in MLP Block\nFigure 3. In Figure (a), we plot the percentage of not-activated\nattention heads. By only keeping heads that yield large output' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='norms, we can silence over 80% attention heads for a given token.\nIn Figure (b), we plot the average sparsity we impose on MLP layers.\nWe can zero out over 95% of MLP parameters for a given token.\n3Pre-trained LLMs are Contextually Sparse\nIn this section, we present several key observations and the-\noretical understandings of sparsity in LLMs, upon which the\nDEJAVU design is based. We first test the contextual sparsity\nhypothesis and verify that contextual sparsity exists in pre-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='trained LLMs in Section 3.1. Then, we build an understand-\ning of why contextual sparsity happens naturally even when\nLLMs are densely trained in Section 3.2. Finally, we present\nan observation on residual connections and explain their\nrelationship to contextual sparsity analytically in Section 3.3.\n3.1 Contextual Sparsity Hypothesis\nInspired by prior pruning literature (Molchanov et al., 2016),\nwe find a surprisingly simple method is sufficient to study and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='verify our hypothesis. In this section, we describe the testing\nprocedure, observation details, and insights of this study.\nVerification: Our test is performed on OPT-175B, 66B, and\n30B models and various downstream datasets such as Open-\nBookQA (Mihaylov et al., 2018) and Wiki-Text (Merity et al.,\n2016). We find the contextual sparsity for every input exam-\nple with two forward passes of the model. In the first pass, we\nrecord a subset of parameters, specifically which attention' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='heads and MLP neurons yield large output norms for the input.\nIn the second pass, each input example only uses the recorded\nsubset of parameters for the computation. Surprisingly, these\ntwo forward passes lead to similar prediction or performance\non all in-context learning and language modeling tasks.Observation: Figure 3 shows that on average, we can impose\nup to 80% sparsity on attention heads and 95% sparsity on\nMLP neurons. As mentioned in Section 2, OPT-175B model' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='has2×MLP parameters than those of attention blocks.\nTherefore total sparsity here is around 85%. Since these are\nall structured sparsity (heads and neurons), predicting them\naccurately could potentially lead to 7×speedup.\nInsight: It is intuitive that we can find contextual sparsity in\nMLP blocks at inference time because of their activation func-\ntions, e.g., ReLU or GeLU (Kurtz et al., 2020). Similar obser-\nvations were made by (Li et al., 2022). However, it is surpris-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='ing that we can find contextual sparsity in attention layers.\nNote that, finding contextual sparsity in attention is not the\nsame as head pruning. We cross-check that different exam-\nples have different contextual sparsity. Although 80% of the\nparameters are not included in the paths for a given example,\nthey might be used by other examples. Next, we will try to\nunderstand why contextual sparsity exists in attention blocks.\n3.2 Token Clustering in Attention Layers' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='In the previous section, we have verified that there exists\ncontextual sparsity for a given input in LLMs. In this\nsection, we try to understand the reason for such phenomena,\nespecially in attention layers. We first show an in-depth\nobservation of attention. Then we present a hypothesis that\nself-attentions are conceptually clustering algorithms. Last\nwe show analytical evidence to support this hypothesis.\nObservation: Figure 4 shows the attention map of three' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='different heads from the same layer for an example input.\nThe next token it should predict is “Truck”. Darker color\nrepresents higher attention scores. We observe that the\nmiddle head is a relatively uniform token-mixing head\nwhile the top and bottom ones are “heavy hitter” attention\nheads (with high attention to “like” and “shipping”).\nUnsurprisingly, only selecting heavy hitter heads but not\nuniform heads does not affect the prediction, since uniform' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='heads do not model or encode important token interactions.\nIn the next section, we will also explain in detail how the\ncriteria for selecting uniform attention heads and heads with\nsmall output norms are highly correlated.\nHypothesis: We hypothesize that the attention head is\nperforming mean-shift clustering (Derpanis, 2005).\nRecall the notation defined in Section 2.3. For i-th head\nat current layer, X= [x1,...,x n]⊤∈Rn×dare the token\nembeddings in the previous time steps. XWK\niandXWV\ni' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='are the projection of embedding. For an input embedding\ny, the output ˜yi=Hi(y), where Hi(y)is defined in Eq. 2.\nFor each i∈[h], if we let Ki(xj,y):=exp( yWQ\ni(WK\ni)⊤xj)\nmeasure the similarity between xjandy, and define\nmi(y):=P\njKi(xj,y)xjP\njKi(xj,y), then we have ˜yi=mi(y)WV\ni. Fur-\nther, if we set WV\ni=Iand consider the residue connection\nfollowed by layer norm, then in the next layer, the embedding\n4' metadata={'source': 'pdfs/paper_3.pdf', 'page': 3}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nLayer LThis fruit shipping company provide different vehicle options like car and [MASK]Truck\nFigure 4. We visualize the attention scores of three different heads\nfor an exemplary sentence. Head 42 and Head 44 give heavy atten-\ntion scores on particular tokens while Head 43 is more uniform.\nˆyiof the current token becomes ˆyi= Normalize( y+ ˜yi) =\nNormalize( y+mi(y)), which has a fixed point y=γmi(y)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='for any scalar γ. This iteration bears a resemblance to mean-\nshift clustering, which simply performs iteration y←mi(y)\nuntil convergence. This has an obvious fixed point y=mi(y).\nTherefore, the self-attention head can be regarded as one\nmean-shift step to push input embeddings of different tokens\ntogether, if they are already neighbors in a projection space\nspecified by WQ\ni(WK\ni)⊤. Different heads learn different\nprojection spaces to perform clustering. These dynamics' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='explain the precise reason why token embeddings tend to\ncluster after going through more layers, resulting in high\nattention scores among cluster members, and low scores for\nnon-members. Furthermore, the cluster patterns are different\nat different heads (More details in Appendix K).\nThe above analysis not only provides an understanding of\nwhy contextual sparsity exists naturally in pre-trained LLMs,\nbut also inspires our design of “similarity”-based sparsity\nprediction for DEJAVU in Section 4.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='3.3 Slowly Changing Embeddings across Layers\nWe first present our observation that embeddings change\nslowly across consecutive layers. Then we provide a detailed\nanalysis on the phenomenon. Finally, we show its close\nconnection with contextual sparsity. Details are in Section B.\nHigh similar embeddings in consecutive layers: In\nFigure 5(a), we show that for the same given input, the cosine\nsimilarity between embeddings or activations in two consec-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='utive layers is exceptionally high on 7 different sizes of OPT\nmodels. Specifically, we collect activations from each layer\nwhile performing OPT model inference on C4 validation\nset (Raffel et al., 2019). Taking OPT-175B as an example,\nstarting from the second layer, the similarity between any\ntwo consecutive layers is around 0.99, which indicates that\nwhen an input is passed through the model, the direction of\nits embedding changes slowly. Interestingly, the most drastic' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='change happens in the first layer. Furthermore, we increase\nthe gap and investigate the similarity between the embedding\nat layer land at layer l+nshown in Figure 5(b). As we\nincrease the gap, the similarity decreases as expected while\nthe differences in cosine similarity between various choices\n125m1.3b 6.7b 13b 30b 66b175b0.950.960.970.980.991.00Cosine Similarity\n(a) Model Comparison\n0 20 40 60 80\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8 (b) Across Layer' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='0 20 40 60 80\nTransformer Layer0500100015002000Norm||X||\n||F(X)||\n(c) Residual Around Attention\n0 20 40 60 80\nTransformer Layer05001000150020002500Norm||X||\n||F(X)|| (d) Residual Around MLP\nFigure 5. Slowly Changing Embedding. Figure (a) shows the\nmedian cosine similarity between representations at two consecutive\nlayers across all layers for different OPT models. All models show\na similarity greater than 95%. Figure (b) shows cosine similarity' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='stays high even a few layers apart. For the residual connection\nX′=X+F(X)inside each block, we plot the ℓ2norm of Xand\nF(X)in Figure (c) and Figure (d). ∥X∥is significantly higher than\n∥F(X)∥, which explains the slowly changing embedding.\nofnare smaller at the shallower layer. We plot the mean sim-\nilarity, and the standard deviation is indicated by the shading.\nSimilar plots on more models are presented in Appendix B.\nConnection to residuals: We verify that the high similarity' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='in embeddings in LLM inference is due to the residual\nconnection. We first dissect the computation graph inside\neach transformer layer to understand the cause behind\nthis phenomenon. There are two residual connections\ninside a transformer layer, one around the attention block,\nand the other one around the MLP block. The residual\nconnection can be written as X+F(X), where Fis either\nthe Multi-Head Attention or two MLP Layers. In Figure 5(c)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='and Figure 5(d), indeed we can see that ∥X∥is significantly\ngreater than ∥F(X)∥, confirming that embeddings are\nchanging slowly because the residual norm is large.\nConnection to Contextual Sparsity: We take a step deeper\ntrying to understand the reason behind the large residual\nnorm with mathematical modeling. We discover that one pos-\nsible reason for small ∥F(X)∥is due to high sparsity. For the\nMLP Block, high sparsity may contribute to the small norm' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='ofF(X)because a large portion of outputs have small norms.\nSimilar reasoning applies to the Attention Block, and thus\na large number of attention heads yield small norm outputs.\nResidual Two Sides Bound: Besides empirical reasoning,\n5' metadata={'source': 'pdfs/paper_3.pdf', 'page': 4}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nwe formally define the computation of LLMs mathematically.\nUnder our computation model, we can show that a shrinking\nproperty which is observed by our practical experiments.\nProofs are in Appendix G, H, I.\nLemma 3.1 (Informal) .Let0<ϵ1<ϵ2<1be the lower and\nupper bound of the shrinking factor. Let xbe the ybe the\noutput. We have the residual connection y=x+F(x). For\nthe MLP block F(x), we have ϵ1≤ ∥y−x∥2≤ϵ2. For the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='attention block F(x), we have ϵ1≤∥y−x∥2≤ϵ2.\n4 DEJAVU\nIn this section, we present our framework for inference-time\ncontextual sparsity search for LLMs. We introduce the\nsparsity predictor for MLPs in Section 4.1 and for attention\nheads in Section 4.2. DEJAVU ’s workflow is shown in\nFigure 2. Section 4.3 discusses exploiting our observation\non LLMs to avoid the sparse prediction overhead with\ntheoretical guarantees. In Section 4.4, we present our' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='optimized implementation that enables end-to-end latency\nreduction. More details are presented in Section D.\n4.1 Contextual Sparsity Prediction in MLP Blocks\nAs explained in Section 2, MLP blocks are one of the major\nbottlenecks for the LLM generation (2\n3of the FLOPs and\nIOs). In this section, we discuss how we achieve wall-clock\ntime speed-up with contextual sparsity in the MLP blocks.\nChallenge Figure 3(b) shows that for a given token, the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='contextual sparsity of 95% is possible. The contextual\nsparsity in the MLP block can be identified after computing\nthe activation. However, this only demonstrates the existence\nof contextual sparsity but brings no benefits in terms of\nefficiency. A fast and precise prediction is needed to exploit\ncontextual sparsity for end-to-end efficiency. The naive way\nis to select a subset of neurons randomly. Unsurprisingly,\nrandom selection fails to identify the accurate contextual' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='sparsity, resulting in drastic model degradation.\nA Near-Neighbor Search Problem: Recall that we verify\nthe existence of contextual sparsity by recording which\nneurons yield significant norms. Essentially, given the input,\nthe goal is to search for the neurons that have high inner prod-\nucts with the input, because the activation function “filters""\nlow activation. Thus, we formulate the contextual sparsity\nprediction of an MLP layer as the classical near-neighbor' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='search problem under the inner product metric.\nDefinition 4.1 (Approximate MaxIP in MLP) .Letc∈(0,1)\nandτ∈(0,1)denote two parameters. Given an n-vector\ndataset W1⊂Sd−1on a unit sphere, the objective of the\n(c,τ)-MaxIP is to construct a data structure that, given a\nquery y∈Sd−1such that max w∈W1⟨y,w⟩≥τ, it retrieves a\nvector zfromW1that satisfies ⟨y,z⟩≥c·max w∈W1⟨y,w⟩.\nRemark 4.2.OurW1(first linear layer) and y(input\nembedding) in MLP blocks can be viewed as the dataset and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='query in Definition 4.1 respectively.Design The standard state-of-the-art near-neighbor search\nmethods and implementations slow down the computa-\ntion. Take OPT-175B where dis 12288 as an example.\nHNSW (Malkov & Yashunin, 2018) requires more than 10ms,\nand FAISS (Johnson et al., 2019) requires more than 4ms,\nwhile the MLP computation is only 0.2ms. The high dimen-\nsionality and complications of data structure implementation\non GPU make the search time longer than the MLP computa-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='tion. Therefore, we choose a neural network classifier as our\nnear-neighbor search method to exploit the fast matrix multi-\nplication on GPU. For each MLP block, we train a small two-\nlayer fully connected network to predict contextual sparsity.\nCollecting training data is straightforward because we know\nthe contextual sparsity using dense computation. The training\nalgorithm is summarized in Algorithm 1. The sparsified com-\nputation in W1has two steps: (1) Given y, the sparsity predic-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='torSPMpredicts a set SMof important neurons in weights\nW1. (2) Compute the sparsified MLP defined in Eq. equa-\ntion 1. Note here the sparsity in MLP is highly structured.\nAlgorithm 1 Sparse Predictor Training\nInput : A pre-trained LLM block with parameter set M,\ntoken embedding set at block M={xi}i∈[N], threshold t\nSparse Predictor SP\nP+←∅,P−←∅\nfori=1→Ndo\nP+←P +∪{(xi,mr)|mr∈M,m r(xi)≥t}\nP−←P−∪{(xi,mr)|mr∈M,m r(xi)<t}\nend for\nSP ← TRAIN(P+,P−,L) ▷Lis a loss function' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='4.2 Contextual Sparsity Prediction in Attention Blocks\nAttention blocks take around 30% I/Os in the generation. In\nthis section, we describe how DEJAVU exploits contextual\nsparsity to speed up the Attention blocks.\nChallenge: As discussed in Section 3.1, only a few heads per-\nform important computations for a given input token. Similar\nto the MLP blocks, a fast selection of attention heads without\nfull computation is required to reduce end-to-end latency.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='Furthermore, one particular challenge of sparse prediction in\nattention blocks is attention’s dependence on previous tokens.\nOn the one hand, it is unclear whether the past token’s key and\nvalue caches are needed for sparse prediction. On the other\nhand, it is unclear how to handle the missing KV cache of past\ntokens for the current token computation at the selected head.\nA Near-Neighbor Search Problem: Head prediction\ncan also be formulated as a near-neighbor search problem' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='based on our understanding in Section 3.2. Since each\nhead is performing mean-shift clustering, after the first\nfew layers, the current token embedding alone is sufficient\nfor the prediction thanks to the token-mixing nature of the\ntransformer. Therefore, the prediction can be based on the\nsimilarity between yand head parameters.\n6' metadata={'source': 'pdfs/paper_3.pdf', 'page': 5}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nApproach: We design our attention sparse predictor to be\nthe same architecture as the MLP sparse predictor. Each head\nis regarded as one class and a similar training process is used\n(Algorithm 1). Then, similar to how MLP prediction is per-\nformed, the attention sparsity predictor SPAselects a set SA\nof heads Hi(see Eq. equation 2). To address the problem of\nmissing KV cache for a past token, we exploit the fact that the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='generation latency is I/O bounded while computation is essen-\ntially “free"". Specifically, for the predicted attention head of\ninput y, we compute the corresponding keys, and values and\nstore them in the KV cache. But we also save a copy of yfor\nall the other non-selected heads. Then during the future token\ngeneration, if there is missing KV cache in the selected heads,\nwe could load stored token embeddings and compute the\nkeys and values together. This requires almost minimal extra' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='memory access (the main cost is loading the weight matrices).\n4.3 Reducing Overhead with Asynchronous Execution\nSparse prediction overhead may easily increase the end-to-\nend latency rather than reduce it despite the reduction in\nFLOPs. Therefore, we introduce a look-ahead sparse pre-\ndiction method, inspired by our observations in Section 3.3.\nChallenge: Denote yl∈Rdas the input to transformer\nlayer l. We can write the computation at layer laseyl←\nMHAl(yl),byl←MLPl(eyl). With predictors SPl' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='AandSPl\nM,\nthe computation at the transformer layer lcan be re-written as\nSl\nA←SPl\nA(yl),eyl←MHAl\nSl\nA(yl),\nSl\nM←SPl\nM(eyl),byl←MLPl\nSl\nM(eyl)\nwhere set Sl\nAis the contextual sparsity for the Attention\nblock, and set Sl\nMis the contextual sparsity for the MLP\nblock at l-th layer. Note that the computation at Attention\nand MLP blocks have to wait for the sparse predictor\ndecision. This overhead potentially outweighs the saving\nfrom Attention and MLP blocks in terms of latency.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='Approach: In Section 3.3, we present the slowly evolving\nembedding phenomenon, which provides opportunities to\nrelax the sequential computation to parallel computation.\nAlong with the observation of low computation intensity\nduring generation, we parallel the sparse prediction with the\ncomputation of each block ( See Figure 2). The computation\ncan be written as follows:\neyl←MHAl\nSl\nA(yl),byl←MLPl\nSl\nM(eyl),\nSl+1\nA←SPl\nA(yl), Sl+1\nM←SPl\nM(yl),\nWe remark Sl+1\nAandSl+1' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='Mcan be computed in parallel with\neylorbyl, while the previous 4 steps are sequential.\nTheoretical guarantee: The sparse predictor can make fur-\nther cross-layer decisions because of the residual connection.\nWe present an informal lemma statement regarding cross-\nlayer prediction. It is well-known that MaxIP is equivalent to\nℓ2nearest neighbor search. For convenience, we use MaxIP\nhere. We include more discussions and proofs in Section J.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='Lemma 4.3 (Informal) .Letϵ∈(0,1). Let ylbe input atl-th layer. Let yl−1be the input at (l−1)-th layer. Suppose\nthat∥yl−yl−1∥2≤ϵ. For any parameters c,τsuch that\nϵ < O (cτ). Then we can show that, solving MaxIP (c,τ)is\nsufficient to solve MaxIP (0.99c,τ).\n4.4 Hardware-efficient Implementation\nWe describe how DEJAVU is implemented in a hardware-\nefficient manner to realize the theoretical speedup of contex-\ntual sparsity. Taking into account hardware characteristics' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='leads to over 2 ×speedup compared to an optimized dense\nmodel, and 4 ×faster than a standard sparse implementation.\nWe highlight some hardware characteristics of GPUs:\n•Small-batch generation is bottlenecked by GPU memory\nI/Os (NVIDIA, 2022; Ivanov et al., 2021; Dao et al., 2022).\nThis is because of low arithmetic intensity. For each\nelement loaded from GPU memory, only a small number\nof floating point operations are performed.\n•GPUs are block-oriented devices: loading a single byte of' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='memory takes the same time as loading a block of memory\naround that same address (Harris, 2013). The block size\nis usually 128 bytes for NVIDIA GPUs (Cook, 2012).\nThese characteristics present some challenges in implement-\ning contextual sparsity. However, they can be addressed with\nclassical techniques in GPU programming.\nKernel fusion: A standard implementation of sparse\nmatrix-vector multiply (e.g., in PyTorch) that separately\nindexes a subset of the matrix W1\nSMbefore multiplying' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='with input ywould incur 3 ×the amount of memory I/Os.\nTherefore, to avoid such overhead, we fuse the indexing\nand the multiplication step. Specifically, we load a subset\nofW1\nSMto memory, along with y, perform the multiply,\nthen write down the result. This fused implementation (in\nTriton (Tillet et al., 2019)) yields up to 4 ×speedup compared\nto a standard PyTorch implementation (Appendix E).\nMemory coalescing: In the dense implementation, the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='weight matrices of two linear layers in MLP are stored\nas(W1)⊤andW2so that no extra transpose operation is\nneeded. They are conventionally stored in row-major format.\nIn the sparse implementation, it allows us to load (W1\nSM)⊤\noptimally (the second dimension is contiguous in memory).\nHowever, for cases where we need to load (W2\nSM), this\nformat significantly slows down memory loading, as indices\ninSMpoint to non-contiguous memory. We simply store' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='these matrices in column-major format (i.e., store (W2)⊤\nin row-major format), then use the same fused kernel above.\nSimilarly, in attention blocks, we store attention output\nprojection WOcolumn-major format.\nThese two techniques (kernel fusion and memory-\ncoalescing) make DEJAVU hardware-efficient, yielding up\nto 2×speedup end-to-end compared to the state-of-the-art\nFasterTransformer (Section 5.1).\n7' metadata={'source': 'pdfs/paper_3.pdf', 'page': 6}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0 20 40 60 80 100\nSparsity percentage (%)891011 PerplexityWikiText\nc4\n(a) Language Modeling\n020 40 60 80100\nSparsity percentage (%)0.40.50.60.70.80.9Accuracy\n020 40 60 80100\nSparsity percentage (%)0.40.50.60.70.80.9\nCB\nCOPA\nLambada\nOpenbookQA\nPIQA\nRTE\nWinogrande (b) Zero-Shot(Left). Five-Shot(Right)\nFigure 6. Accuracy Trend for DEJAVU -OPT-175B . This figure shows the accuracy of DEJAVU -OPT-175B on language modeling datasets' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='and downstream tasks when we set different sparsity at test time. In general, DEJAVU -OPT-175B incurs no accuracy drop until 75% sparsity.\n128 256 512 1024\nSequence Length020406080100Latency(ms)HuggingFace\nFasterTransformer\nDejaVu\nFigure 7. Average per-token latency (ms) with batch size 1 on 8\nA100-80GB with NVLink when generating sequences with prompt\nlengths 128, 256, 512, and 1024, using FP16. DEJAVU speeds up\ngeneration by 1.8-2 ×compared to the state-of-the-art FT and by' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='4.8-6×compared to the widely used HF implementation.\n5 Empirical Evaluation\nIn Section 5.1, we present the end-to-end results that show\nDEJAVU achieves over 2 ×reduction in token generation\nlatency compared to the state-of-the-art FasterTransformer\nand over 6 ×compared to Hugging Face with no accuracy\nloss. In Section 5.2, we perform a list of ablation studies such\nas independent evaluation on the inference-time contextual\nsparsity of the MLP block and the Attention block (Details' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='are presented in Section C). At last, we present the additional\nresults to demonstrate the future possibility of sparsifying\nthe entire LLMs via layer skipping in Section C.3.\n5.1 End-to-End Result\nExperiment Setting: We compare the accuracy of DE-\nJAVU -OPT against the original OPT model on two lan-\nguage modeling datasets Wiki-Text (Merity et al., 2016)\nand C4 (Raffel et al., 2019) and seven few-shot downstream\ntasks: CB (de Marneffe et al., 2019), COPA (Gordon et al.,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='2012), Lambada (Radford et al., 2019), OpenBookQA (Mi-\nhaylov et al., 2018), PIQA (Bisk et al., 2020), RTE (Giampic-\ncolo et al., 2007), Winogrande (ai2, 2019). We use lm-eval-\nharness (Gao et al., 2021) for zero-shot and five-shot tasks.\nWe collect training data for the sparsity predictor using 500\nrandom data points from the C4 training dataset. Our exper-\niments are conducted on NVIDIA A100 80GB GPU servers.\nNo accuracy drop until 75% sparsity: In Figure 6, we' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='present DEJAVU -OPT-175B’s accuracy trend. In a zero-shotsetting, the average accuracy across tasks does not drop\nuntil 75% sparsity. A similar trend can be observed for\nthe five-shot setting, which verifies the model’s ability for\nin-context learning. This result is exceptionally encouraging\ngiven our observation in Figure 1(a), where we could impose\n85% sparsity when allowed full computation.\nOver 2 ×latency reduction: Figure 7 presents the latency' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='speed-up for the token generation with OPT-175B at batch\nsize 1, where DEJAVU achieves the best performance. At\naround 75% sparsity, DEJAVU speeds up generation by\n1.8-2×compared to the state-of-the-art FasterTransformers\n(FT)1and by 4.8-6 ×to Hugging Face (HF) implementation2.\n5.2 Ablation Results\nContextual Sparsity for Larger Batches: Although this\npaper focuses on latency-sensitive settings, we demonstrate\nthat DEJAVU generalizes to larger batches. we present the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='Union contextual sparsity (fraction of neurons/heads that are\nnot used by any of the inputs in the batch) of different batches\nsizes for MLP and Attention blocks, respectively, in Fig-\nure 8 and 11. The union operation is essential to realize a fast\nsparse GEMM. Surprisingly the number of MLP neurons and\nAttention heads that DEJAVU activated does not grow linearly\nwith the batch size. This suggests a power law distribution\nrather than a uniform distribution of parameter access from' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='all input examples. This provides an opportunity for poten-\ntially extending Dejavu to the high-throughout setting. For\nexample, we can first pre-process the inputs and batch similar\ninputs to enjoy a higher level of union contextual sparsity.\nContextual sparsity on MLP blocks: We study the contex-\ntual sparsification of the MLP block in OPT-175B. We leave\nthe Attention block as dense computation. Table 4 shows\nthe model performance at 85% sparsity. The MLP sparse' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='predictor introduces no accuracy loss on both zero-shot tasks\nand language modeling. In the training of the MLP sparse\npredictor, we observe that the sparse predictor achieves high\nvalidation accuracy. The shallow layer seems easier to model\n1http://github.com/NVIDIA/FasterTransformer\n2http://github.com/huggingface/transformers\n8' metadata={'source': 'pdfs/paper_3.pdf', 'page': 7}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nTable 4. Accuracy of zero-shot tasks and language modeling when sparsifying the MLP block and the Attention block separately. The\nsparsity is set at 85% for MLP-block and 50% for Attention-block. DEJAVU incurs no accuracy drop across the boards.\nModel CB COPA Lambada OpenBookQA PIQA RTE Winogrande Wikitext C4\nOPT-175B 0.3523 0.86 0.7584 0.446 0.8096 0.6029 0.7261 10.8221 7.7224' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='DEJAVU -MLP-OPT-175B 0.3544 0.85 0.7619 0.446 0.8096 0.6065 0.7206 10.7988 7.7393\nDEJAVU -Attention-OPT-175B 0.3544 0.86 0.7586 0.4460 0.8063 0.5921 0.7245 10.8696 7.7393\nTable 5. DEJAVU -OPT66B on zero-shot downstream task.\nModel CB COPA Lambada OpenBookQA PIQA RTE Winogrande\nOPT-66B 0.3928 0.87 0.7508 0.426 0.7921 0.6028 0.6890\nDEJAVU -OPT-66B 0.4285 0.87 0.7458 0.434 0.7933 0.5884 0.6898\nTable 6. DEJAVU -BLOOM on zero-shot downstream task.\nCB COPA OpenBookQA PIQA RTE Winogrande Lambada' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='BLOOM 0.455 0.8 0448 0.79 0.617 0.704 0.677\nDejavu-BLOOM 0.448 0.8 0.44 0.787 0.606 0.710 0.675\n0 20 40 60 80 96\nTransformer Layer0.50.60.70.80.91.0Union Contextual Sparsity\nBatch size\n2\n4\n8\n16\n32\nFigure 8. Union contextual sparsity with larger batch size.\nbecause the predictor has validation accuracy over 99% in the\nshallow layers and drops to around 93% in the ending layers.\nContextual sparsity on attention blocks: In this section,\nwe study the sparse predictor for the Attention block on OPT-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='175B and leave the MLP block as dense computation. Table 4\ndisplays the test accuracy on zero-shot tasks and perplexity on\nthe language modeling datasets. In summary, the Attention\nsparse predictor introduces no accuracy loss at around 50%\nsparsity. During the training of the Attention sparse predictor,\nwe observe different trends compared to the MLP sparse\npredictor. The validation accuracy is around 93% in the\nmiddle layers and near 99% in the shallow and deep layers.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='Contextual Sparsity on Smaller Models: Our main exper-\niments focus on OPT-175B. Here, we verify DEJAVU ’s effec-\ntiveness on a smaller model, specifically OPT-66B. In Table 5,\nwe summarize the accuracy on zero-shot task at 50% sparsity.\nSimilar to DEJAVU -OPT-175B, we notice no accuracy loss.\nContextual Sparsity on Other Models: We expand\nthe evaluation to another model family. In Table 6, we\nsummarize the accuracy at attention sparsity 50% and MLP' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='sparsity 30%. Similar to OPT family, we notice no accuracy\nloss. The lower sparsity level in MLP is due to the difference\nin activation function.\nNon-Contextual Sparsity: As we mentioned in Section 1,\none could predict sparsity without contextual information.\nFor non-contextual sparsity, we rely on the originalTable 7. DEJAVU -OPT-175B with 4-bit quantization.\nCB COPA OpenBookQA PIQA RTE Winogrande Lambada\nOPT-175B 0.352 0.86 0.446 0.809 0.602 0.726 0.758' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='Dejavu-OPT-175B 0.402 0.85 0.450 0.802 0.592 0.726 0.753\nOPT-175B + W4A16 0.356 0.85 0.44 0.806 0.574 0.714 0.757\nDejavu-OPT-175B + W4A16 0.365 0.86 0.452 0.805 0.592 0.726 0.754\nembedding at the input layer. At every block, we first pass\nthe original embedding to record a subset of parameters\nyielding a large norm. In the second pass, the embedding\nat every layer only uses the recorded subset. As shown in\nFigure 1, non-contextual prediction is not sufficient and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='leads to accuracy losses even at 50% sparsity. This result\nverifies our design choices of relying on the activation at\nevery layer as input to make contextual sparsity predictions.\nCompatibility with Quantization: Quantization is another\npromising direction for efficient language models. We inves-\ntigate the possibility of combining contextual sparsity with\nquantization techniques. For DEJAVU -OPT-175B, we set\nthe entire model sparsity at 75%. For quantization, we apply' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='4-bit quantization on model weights (W4A16). As shown\nin Table 7, the combination of quantization and DEJAVU\nalmost always achieves better accuracy than DEJAVU or\nquantization alone. This suggests that the approximation\nerrors from these two directions do not get compounded.\n6 Conclusion\nOur main goal is to make LLM inference efficient so that\ntheir powerful in-context learning abilities can be used\nin more application domains. We observe that contextual' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='sparsity can be accurately predicted with lightweight\nlearning-based algorithms. This motivated us to design\nDEJAVU that uses asynchronous lookahead predictors and\nhardware-efficient sparsity to speed up LLM inference in\nwall-clock time. Our encouraging empirical results validate\nthat contextual sparsity can reduce inference latency by\nover 2×compared to the state-of-the-art FasterTransformer\nwithout model quality drops. Our method is a step towards' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='making LLMs more accessible to the general community,\nwhich could unlock exciting new AI applications.\nAcknowledgements\nWe would like to thank Ryan Spring, Laurel Orr, Guangxuan\nXiao, Eric Han, Xun Huang, Daniel Y . Fu, Benjamin Spector,\nRuan Silva, Diana Liskovich, and the anonymous reviewers\nfor helpful discussions and feedback. We acknowledge the\ngenerous support by Together Computer, which enabled the\nnecessary partial computations in this work.\n9' metadata={'source': 'pdfs/paper_3.pdf', 'page': 8}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nReferences\nWinogrande: An adversarial winograd schema challenge\nat scale. 2019.\nAllen-Zhu, Z. and Li, Y . What can resnet learn efficiently,\ngoing beyond kernels? Advances in Neural Information\nProcessing Systems , 32, 2019.\nAlman, J. and Song, Z. Fast attention requires bounded\nentries. arXiv preprint arXiv:2302.13214 , 2023.\nAlman, J., Liang, J., Song, Z., Zhang, R., and Zhuo, D. By-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='pass exponential time preprocessing: Fast neural network\ntraining via weight-data correlation preprocessing. arXiv\npreprint arXiv:2211.14227 , 2022.\nAlon, N., Matias, Y ., and Szegedy, M. The space complexity\nof approximating the frequency moments. In Proceedings\nof the twenty-eighth annual ACM symposium on Theory\nof computing , pp. 20–29, 1996.\nAminabadi, R. Y ., Rajbhandari, S., Awan, A. A., Li, C.,\nLi, D., Zheng, E., Ruwase, O., Smith, S., Zhang, M.,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='Rasley, J., et al. Deepspeed-inference: Enabling efficient\ninference of transformer models at unprecedented scale.\nIn2022 SC22: International Conference for High Per-\nformance Computing, Networking, Storage and Analysis\n(SC), pp. 646–660. IEEE Computer Society, 2022.\nAndoni, A. and Razenshteyn, I. Optimal data-dependent\nhashing for approximate near neighbors. In Proceedings\nof the forty-seventh annual ACM symposium on Theory\nof computing (STOC) , pp. 793–801, 2015.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='Andoni, A., Indyk, P., Nguyen, H. L., and Razenshteyn, I.\nBeyond locality-sensitive hashing. In Proceedings of the\ntwenty-fifth annual ACM-SIAM symposium on Discrete\nalgorithms , pp. 1018–1028. SIAM, 2014.\nAndoni, A., Indyk, P., Laarhoven, T., Razenshteyn, I., and\nSchmidt, L. Practical and optimal lsh for angular distance.\nInAdvances in Neural Information Processing Systems\n(NIPS) , pp. 1225–1233. Curran Associates, 2015.\nAndoni, A., Laarhoven, T., Razenshteyn, I., and Waingarten,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='E. Optimal hashing-based time-space trade-offs for\napproximate near neighbors. In Proceedings of the\nTwenty-Eighth Annual ACM-SIAM Symposium on\nDiscrete Algorithms (SODA) , pp. 47–66. SIAM, 2017.\nAndoni, A., Indyk, P., and Razenshteyn, I. Approximate\nnearest neighbor search in high dimensions. arXiv\npreprint arXiv:1806.09823 , 7, 2018.\nArya, S. and Mount, D. M. Approximate nearest neighbor\nqueries in fixed dimensions. In SODA , volume 93, pp.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='271–280. Citeseer, 1993.Balduzzi, D., Frean, M., Leary, L., Lewis, J., Ma, K. W.-D.,\nand McWilliams, B. The shattered gradients problem:\nIf resnets are the answer, then what is the question? In\nInternational Conference on Machine Learning , pp.\n342–350. PMLR, 2017.\nBansal, H., Gopalakrishnan, K., Dingliwal, S., Bodapati, S.,\nKirchhoff, K., and Roth, D. Rethinking the role of scale for\nin-context learning: An interpretability-based case study' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='at 66 billion scale. arXiv preprint arXiv:2212.09095 , 2022.\nBaum, L. E. and Petrie, T. Statistical inference for proba-\nbilistic functions of finite state markov chains. The annals\nof mathematical statistics , 37(6):1554–1563, 1966.\nBello, I., Fedus, W., Du, X., Cubuk, E. D., Srinivas, A., Lin,\nT.-Y ., Shlens, J., and Zoph, B. Revisiting resnets: Im-\nproved training and scaling strategies. Advances in Neural\nInformation Processing Systems , 34:22614–22627, 2021.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='Bengio, Y ., Ducharme, R., Vincent, P., and Jauvin, C. A\nneural probabilistic language model. Journal of machine\nlearning research (JMLR) , 3(Feb):1137–1155, 2003.\nBisk, Y ., Zellers, R., Bras, R. L., Gao, J., and Choi, Y .\nPiqa: Reasoning about physical commonsense in natural\nlanguage. In Thirty-Fourth AAAI Conference on Artificial\nIntelligence , 2020.\nBlack, S., Biderman, S., Hallahan, E., Anthony, Q., Gao,\nL., Golding, L., He, H., Leahy, C., McDonell, K., Phang,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='J., Pieler, M., Prashanth, U. S., Purohit, S., Reynolds, L.,\nTow, J., Wang, B., and Weinbach, S. GPT-NeoX-20B:\nAn open-source autoregressive language model. In\nProceedings of the ACL Workshop on Challenges &\nPerspectives in Creating Large Language Models , 2022.\nURLhttps://arxiv.org/abs/2204.06745 .\nBommasani, R., Hudson, D. A., Adeli, E., Altman, R., Arora,\nS., von Arx, S., Bernstein, M. S., Bohg, J., Bosselut, A.,\nBrunskill, E., et al. On the opportunities and risks of foun-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='dation models. arXiv preprint arXiv:2108.07258 , 2021.\nBoutsidis, C., Woodruff, D. P., and Zhong, P. Optimal\nprincipal component analysis in distributed and streaming\nmodels. In STOC’16—Proceedings of the 48th Annual\nACM SIGACT Symposium on Theory of Computing , 2016.\nBoytsov, L., Novak, D., Malkov, Y ., and Nyberg, E. Off the\nbeaten path: Let’s replace term-based retrieval with k-nn\nsearch. In Proceedings of the 25th ACM international on\nconference on information and knowledge management' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='(CIKM) , pp. 1099–1108, 2016.\nBrand, J. v. d., Peng, B., Song, Z., and Weinstein, O. Training\n(overparametrized) neural networks in near-linear time.\nInITCS , 2021.\n10' metadata={'source': 'pdfs/paper_3.pdf', 'page': 9}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nBrand, J. v. d., Song, Z., and Zhou, T. Algorithm and\nhardness for dynamic attention maintenance in large\nlanguage models. arXiv preprint arXiv:2304.02207 , 2023.\nBrown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D.,\nDhariwal, P., Neelakantan, A., Shyam, P., Sastry, G.,\nAskell, A., et al. Language models are few-shot learners.\nAdvances in neural information processing systems , 33:\n1877–1901, 2020.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Chan, S. C., Santoro, A., Lampinen, A. K., Wang, J. X.,\nSingh, A. K., Richemond, P. H., McClelland, J., and\nHill, F. Data distributional properties drive emergent\nin-context learning in transformers. In Advances in Neural\nInformation Processing Systems , 2022.\nChang, W.-C., Yu, F. X., Chang, Y .-W., Yang, Y ., and Kumar,\nS. Pre-training tasks for embedding-based large-scale\nretrieval. arXiv preprint arXiv:2002.03932 , 2020.\nCharikar, M., Chen, K., and Farach-Colton, M. Finding' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='frequent items in data streams. In International Collo-\nquium on Automata, Languages, and Programming , pp.\n693–703. Springer, 2002.\nChen, B., Xu, Y ., and Shrivastava, A. Fast and accurate\nstochastic gradient estimation. Advances in Neural\nInformation Processing Systems , 32, 2019.\nChen, B., Medini, T., Farwell, J., Tai, C., Shrivastava, A., et al.\nSlide: In defense of smart algorithms over hardware accel-\neration for large-scale deep learning systems. Proceedings' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='of Machine Learning and Systems , 2:291–306, 2020a.\nChen, B., Dao, T., Winsor, E., Song, Z., Rudra, A., and Ré,\nC. Scatterbrain: Unifying sparse and low-rank attention.\nAdvances in Neural Information Processing Systems , 34:\n17413–17426, 2021a.\nChen, B., Liu, Z., Peng, B., Xu, Z., Li, J. L., Dao, T., Song,\nZ., Shrivastava, A., and Re, C. Mongoose: A learnable lsh\nframework for efficient neural network training. In Inter-\nnational Conference on Learning Representations , 2021b.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Chen, H., Chillotti, I., Dong, Y ., Poburinnaya, O., Razen-\nshteyn, I., and Riazi, M. S. {SANNS }: Scaling up\nsecure approximate k-nearest neighbors search. In 29th\n{USENIX }Security Symposium ( {USENIX }Security 20) ,\npp. 2111–2128, 2020b.\nChen, L. On the hardness of approximate and exact (bichro-\nmatic) maximum inner product. In 33rd Computational\nComplexity Conference (CCC) , 2018.\nCho, J. H. and Hariharan, B. On the efficacy of knowledge' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='distillation. In Proceedings of the IEEE/CVF international\nconference on computer vision , pp. 4794–4802, 2019.Chowdhery, A., Narang, S., Devlin, J., Bosma, M., Mishra,\nG., Roberts, A., Barham, P., Chung, H. W., Sutton, C.,\nGehrmann, S., et al. PaLM: Scaling language modeling\nwith pathways. arXiv preprint arXiv:2204.02311 , 2022.\nClarkson, K. L. and Woodruff, D. P. Low-rank approximation\nand regression in input sparsity time. In STOC , 2013.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Cohen, M. B. Nearly tight oblivious subspace embeddings\nby trace inequalities. In Proceedings of the twenty-seventh\nannual ACM-SIAM symposium on Discrete algorithms ,\npp. 278–287. SIAM, 2016.\nCook, S. CUDA Programming: A Developer’s Guide to\nParallel Computing with GPUs . Morgan Kaufmann\nPublishers Inc., San Francisco, CA, USA, 1st edition,\n2012. ISBN 9780124159334.\nCox, M. and Cox, T. Multidimensional scaling, 315–347.\nHandbook of data visualization. Springer, Berlin,\nGermany , 2008.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Dao, T., Fu, D. Y ., Ermon, S., Rudra, A., and Ré, C.\nFlashattention: Fast and memory-efficient exact attention\nwith io-awareness. In Advances in Neural Information\nProcessing Systems , 2022.\nDatar, M., Immorlica, N., Indyk, P., and Mirrokni, V . S.\nLocality-sensitive hashing scheme based on p-stable distri-\nbutions. In Proceedings of the twentieth annual symposium\non Computational geometry (SoCG) , pp. 253–262, 2004.\nde Marneffe, M.-C., Simons, M., and Tonhauser, J. The' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='commitmentbank: Investigating projection in naturally\noccurring discourse. 2019.\nDeng, Y ., Li, Z., and Song, Z. Attention scheme inspired soft-\nmax regression. arXiv preprint arXiv:2304.10411 , 2023a.\nDeng, Y ., Mahadevan, S., and Song, Z. Randomized and\ndeterministic attention sparsification algorithms for\nover-parameterized feature dimension. arxiv preprint:\narxiv 2304.03426 , 2023b.\nDerpanis, K. G. Mean shift clustering. Lecture Notes , 32:\n1–4, 2005.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Dettmers, T., Lewis, M., Belkada, Y ., and Zettlemoyer, L.\nLlm. int8 (): 8-bit matrix multiplication for transformers\nat scale. arXiv preprint arXiv:2208.07339 , 2022.\nDong, S., Lee, Y . T., and Ye, G. A nearly-linear time\nalgorithm for linear programs with small treewidth: A\nmultiscale representation of robust central path. In\nProceedings of the 53rd Annual ACM SIGACT Symposium\non Theory of Computing , pp. 1784–1797, 2021.\nDong, Y ., Indyk, P., Razenshteyn, I., and Wagner, T. Learning' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='space partitions for nearest neighbor search. In Interna-\ntional Conference on Learning Representations , 2019.\n11' metadata={'source': 'pdfs/paper_3.pdf', 'page': 10}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nFang, J., Yu, Y ., Zhao, C., and Zhou, J. Turbotransformers:\nan efficient gpu serving system for transformer models.\nInProceedings of the 26th ACM SIGPLAN Symposium\non Principles and Practice of Parallel Programming , pp.\n389–402, 2021.\nFrankle, J. and Carbin, M. The lottery ticket hypothesis:\nFinding sparse, trainable neural networks. arXiv preprint\narXiv:1803.03635 , 2018.\nFrantar, E. and Alistarh, D. Massive language models' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='can be accurately pruned in one-shot. arXiv preprint\narXiv:2301.00774 , 2023.\nFrantar, E., Ashkboos, S., Hoefler, T., and Alistarh, D. Gptq:\nAccurate post-training quantization for generative pre-\ntrained transformers. arXiv preprint arXiv:2210.17323 ,\n2022.\nFrei, S., Cao, Y ., and Gu, Q. Algorithm-dependent gen-\neralization bounds for overparameterized deep residual\nnetworks. Advances in neural information processing\nsystems , 32, 2019.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='Gao, L., Tow, J., Biderman, S., Black, S., DiPofi, A., Foster,\nC., Golding, L., Hsu, J., McDonell, K., Muennighoff, N.,\nPhang, J., Reynolds, L., Tang, E., Thite, A., Wang, B.,\nWang, K., and Zou, A. A framework for few-shot lan-\nguage model evaluation, September 2021. URL https:\n//doi.org/10.5281/zenodo.5371628 .\nGao, Y ., Mahadevan, S., and Song, Z. An over-parameterized\nexponential regression. arXiv preprint arXiv:2303.16504 ,\n2023a.\nGao, Y ., Song, Z., and Yang, X. Differentially private' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='attention computation. arXiv preprint arXiv:2305.04701 ,\n2023b.\nGiampiccolo, D., Magnini, B., Dagan, I., and Dolan, B. The\nthird PASCAL recognizing textual entailment challenge.\nInProceedings of the ACL-PASCAL Workshop on Textual\nEntailment and Paraphrasing , pp. 1–9, Prague, June\n2007. Association for Computational Linguistics. URL\nhttps://aclanthology.org/W07-1401 .\nGionis, A., Indyk, P., Motwani, R., et al. Similarity search\nin high dimensions via hashing. In Vldb , volume 99, pp.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='518–529, 1999.\nGordon, A., Kozareva, Z., and Roemmele, M. SemEval-2012\ntask 7: Choice of plausible alternatives: An evaluation\nof commonsense causal reasoning. In *SEM 2012: The\nFirst Joint Conference on Lexical and Computational Se-\nmantics – Volume 1: Proceedings of the main conference\nand the shared task, and Volume 2: Proceedings of the\nSixth International Workshop on Semantic Evaluation\n(SemEval 2012) , pp. 394–398, Montréal, Canada, 7-8June 2012. Association for Computational Linguistics.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='URLhttps://aclanthology.org/S12-1052 .\nGu, Y . and Song, Z. A faster small treewidth sdp solver.\narXiv preprint arXiv:2211.06033 , 2022.\nGu, Y ., Song, Z., Yin, J., and Zhang, L. Low rank matrix\ncompletion via robust alternating minimization in nearly\nlinear time. arXiv preprint arXiv:2302.11068 , 2023.\nHall, R. and Attenberg, J. Fast and accurate maximum inner\nproduct recommendations on map-reduce. In Proceedings\nof the 24th International Conference on World Wide Web' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='(WWW) , pp. 1263–1268, 2015.\nHan, S., Mao, H., and Dally, W. J. Deep compression:\nCompressing deep neural networks with pruning, trained\nquantization and huffman coding. arXiv preprint\narXiv:1510.00149 , 2015.\nHarris, M. How to access global memory efficiently in\nCUDA C/C++ kernels. NVIDIA, Jan , 2013.\nHe, K., Zhang, X., Ren, S., and Sun, J. Deep residual\nlearning for image recognition. In Proceedings of\nthe IEEE conference on computer vision and pattern\nrecognition , pp. 770–778, 2016.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='He, Y ., Liu, P., Wang, Z., Hu, Z., and Yang, Y . Filter pruning\nvia geometric median for deep convolutional neural\nnetworks acceleration. In Proceedings of the IEEE/CVF\nconference on computer vision and pattern recognition ,\npp. 4340–4349, 2019.\nHinton, G., Vinyals, O., Dean, J., et al. Distilling\nthe knowledge in a neural network. arXiv preprint\narXiv:1503.02531 , 2(7), 2015.\nHoefler, T., Alistarh, D., Ben-Nun, T., Dryden, N., and Peste,\nA. Sparsity in deep learning: Pruning and growth for' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='efficient inference and training in neural networks. J.\nMach. Learn. Res. , 22(241):1–124, 2021.\nHooker, S. The hardware lottery. Communications of the\nACM , 64(12):58–65, 2021.\nHu, H., Song, Z., Weinstein, O., and Zhuo, D. Training\noverparametrized neural networks in sublinear time.\narXiv preprint arXiv:2208.04508 , 2022.\nIndyk, P. and Motwani, R. Approximate nearest neighbors:\ntowards removing the curse of dimensionality. In\nProceedings of the thirtieth annual ACM symposium on' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='Theory of computing (STOC) , pp. 604–613, 1998a.\nIndyk, P. and Motwani, R. Approximate nearest neighbors:\ntowards removing the curse of dimensionality. In\nProceedings of the thirtieth annual ACM symposium on\nTheory of computing , pp. 604–613, 1998b.\n12' metadata={'source': 'pdfs/paper_3.pdf', 'page': 11}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nIndyk, P. and Wagner, T. Approximate nearest neighbors\nin limited space. In Conference On Learning Theory , pp.\n2012–2036. PMLR, 2018.\nIvanov, A., Dryden, N., Ben-Nun, T., Li, S., and Hoefler,\nT. Data movement is all you need: A case study on\noptimizing transformers. Proceedings of Machine\nLearning and Systems , 3:711–732, 2021.\nJacob, B., Kligys, S., Chen, B., Zhu, M., Tang, M., Howard,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='A., Adam, H., and Kalenichenko, D. Quantization and\ntraining of neural networks for efficient integer-arithmetic-\nonly inference. In Proceedings of the IEEE conference on\ncomputer vision and pattern recognition , pp. 2704–2713,\n2018.\nJiang, S., Song, Z., Weinstein, O., and Zhang, H. A faster\nalgorithm for solving general lps. In Proceedings of the\n53rd Annual ACM SIGACT Symposium on Theory of\nComputing , pp. 823–832, 2021.\nJohnson, J., Douze, M., and Jégou, H. Billion-scale' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='similarity search with GPUs. IEEE Transactions on Big\nData , 7(3):535–547, 2019.\nJohnson, W. B. and Lindenstrauss, J. Extensions of\nlipschitz mappings into a hilbert space. Contemporary\nmathematics , 26(189-206):1, 1984.\nKitaev, N., Kaiser, Ł., and Levskaya, A. Reformer: The\nefficient transformer. In ICLR , 2020.\nKurtz, M., Kopinsky, J., Gelashvili, R., Matveev, A., Carr, J.,\nGoin, M., Leiserson, W., Moore, S., Shavit, N., and Alis-\ntarh, D. Inducing and exploiting activation sparsity for fast' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='inference on deep neural networks. In III, H. D. and Singh,\nA. (eds.), Proceedings of the 37th International Confer-\nence on Machine Learning , volume 119 of Proceedings\nof Machine Learning Research , pp. 5533–5543. PMLR,\n13–18 Jul 2020. URL https://proceedings.mlr.\npress/v119/kurtz20a.html .\nLaurent, B. and Massart, P. Adaptive estimation of a\nquadratic functional by model selection. Annals of\nStatistics , pp. 1302–1338, 2000.\nLeCun, Y ., Denker, J., and Solla, S. Optimal brain damage.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Advances in neural information processing systems , 2,\n1989.\nLee, M., He, X., Yih, W.-t., Gao, J., Deng, L., and Smolensky,\nP. Reasoning in vector space: An exploratory study of\nquestion answering. In ICLR , 2016.\nLee, N., Ajanthan, T., and Torr, P. H. Snip: Single-shot\nnetwork pruning based on connection sensitivity. arXiv\npreprint arXiv:1810.02340 , 2018.Lee, Y . T., Song, Z., and Zhang, Q. Solving empirical risk\nminimization in the current matrix multiplication time. In' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Conference on Learning Theory , pp. 2140–2157. PMLR,\n2019.\nLi, P., Li, X., and Zhang, C.-H. Re-randomized densification\nfor one permutation hashing and bin-wise consistent\nweighted sampling. Advances in Neural Information\nProcessing Systems , 32, 2019.\nLi, S., Song, Z., Xia, Y ., Yu, T., and Zhou, T. The closeness\nof in-context learning and weight shifting for softmax\nregression. arXiv preprint , 2023a.\nLi, X. and Li, P. C-MinHash: Improving minwise hashing' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='with circulant permutation. In Chaudhuri, K., Jegelka,\nS., Song, L., Szepesvari, C., Niu, G., and Sabato, S.\n(eds.), Proceedings of the 39th International Conference\non Machine Learning , volume 162 of Proceedings of\nMachine Learning Research , pp. 12857–12887. PMLR,\n17–23 Jul 2022. URL https://proceedings.mlr.\npress/v162/li22m.html .\nLi, Z., You, C., Bhojanapalli, S., Li, D., Rawat, A. S.,\nReddi, S. J., Ye, K., Chern, F., Yu, F., Guo, R., and\nKumar, S. Large models are parsimonious learners:' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Activation sparsity in trained transformers, 2022. URL\nhttps://arxiv.org/abs/2210.06313 .\nLi, Z., Song, Z., and Zhou, T. Solving regularized exp, cosh\nand sinh regression problems. arXiv preprint, 2303.15725 ,\n2023b.\nLiang, P., Bommasani, R., Lee, T., Tsipras, D., Soylu, D.,\nYasunaga, M., Zhang, Y ., Narayanan, D., Wu, Y ., Kumar,\nA., et al. Holistic evaluation of language models. arXiv\npreprint arXiv:2211.09110 , 2022.\nLiu, Z., Sun, M., Zhou, T., Huang, G., and Darrell, T.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Rethinking the value of network pruning. arXiv preprint\narXiv:1810.05270 , 2018.\nLiu, Z., Xu, Z., Ji, A., Zhang, J., Li, J., Chen, B., and\nShrivastava, A. Halos: Hashing large output space for\ncheap inference. Proceedings of Machine Learning and\nSystems , 4:110–125, 2022.\nLu, Y ., Dhillon, P., Foster, D. P., and Ungar, L. Faster ridge\nregression via the subsampled randomized hadamard\ntransform. In Advances in neural information processing\nsystems (NIPS) , pp. 369–377, 2013.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Malkov, Y ., Ponomarenko, A., Logvinov, A., and Krylov,\nV . Approximate nearest neighbor algorithm based on\nnavigable small world graphs. Information Systems , 45:\n61–68, 2014.\n13' metadata={'source': 'pdfs/paper_3.pdf', 'page': 12}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nMalkov, Y . A. and Yashunin, D. A. Efficient and robust ap-\nproximate nearest neighbor search using hierarchical nav-\nigable small world graphs. IEEE transactions on pattern\nanalysis and machine intelligence , 42(4):824–836, 2018.\nMeng, X. and Mahoney, M. W. Low-distortion subspace\nembeddings in input-sparsity time and applications to\nrobust linear regression. In Proceedings of the forty-fifth' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='annual ACM symposium on Theory of computing , pp.\n91–100, 2013.\nMerity, S., Xiong, C., Bradbury, J., and Socher, R. Pointer\nsentinel mixture models, 2016.\nMichel, P., Levy, O., and Neubig, G. Are sixteen heads\nreally better than one? Advances in neural information\nprocessing systems , 32, 2019.\nMihaylov, T., Clark, P., Khot, T., and Sabharwal, A. Can a\nsuit of armor conduct electricity? a new dataset for open\nbook question answering. In EMNLP , 2018.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='Min, S., Lyu, X., Holtzman, A., Artetxe, M., Lewis, M.,\nHajishirzi, H., and Zettlemoyer, L. Rethinking the role\nof demonstrations: What makes in-context learning work?\narXiv preprint arXiv:2202.12837 , 2022.\nMolchanov, P., Tyree, S., Karras, T., Aila, T., and Kautz, J.\nPruning convolutional neural networks for resource ef-\nficient inference. arXiv preprint arXiv:1611.06440 , 2016.\nNagel, M., Baalen, M. v., Blankevoort, T., and Welling,\nM. Data-free quantization through weight equalization' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='and bias correction. In Proceedings of the IEEE/CVF\nInternational Conference on Computer Vision , pp.\n1325–1334, 2019.\nNelson, J. and Nguyên, H. L. Osnap: Faster numerical linear\nalgebra algorithms via sparser subspace embeddings.\nIn2013 ieee 54th annual symposium on foundations of\ncomputer science , pp. 117–126. IEEE, 2013.\nNeyshabur, B. and Srebro, N. On symmetric and asymmetric\nlshs for inner product search. In International Conference\non Machine Learning (ICML) , pp. 1926–1934. PMLR,\n2015.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='NVIDIA. Fastertransformer. https://github.com/\nNVIDIA/FasterTransformer .\nNVIDIA. Gpu performance background user’s\nguide, 2022. URL https://docs.nvidia.\ncom/deeplearning/performance/\ndl-performance-gpu-background/index.\nhtml .\nPark, G., Park, B., Kwon, S. J., Kim, B., Lee, Y ., and Lee,\nD. nuqmm: Quantized matmul for efficient inference of\nlarge-scale generative language models. arXiv preprint\narXiv:2206.09557 , 2022.Pope, R., Douglas, S., Chowdhery, A., Devlin, J., Bradbury,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='J., Levskaya, A., Heek, J., Xiao, K., Agrawal, S., and\nDean, J. Efficiently scaling transformer inference. arXiv\npreprint arXiv:2211.05102 , 2022.\nQin, L., Song, Z., and Wang, Y . Fast submodular function\nmaximization. CoRR , abs/2305.08367, 2023a.\nQin, L., Song, Z., Zhang, L., and Zhuo, D. An online and\nunified algorithm for projection matrix vector multipli-\ncation with application to empirical risk minimization. In\nAISTATS , 2023b.\nRadford, A., Wu, J., Child, R., Luan, D., Amodei, D., and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='Sutskever, I. Language models are unsupervised multitask\nlearners. 2019.\nRaffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S.,\nMatena, M., Zhou, Y ., Li, W., and Liu, P. J. Exploring\nthe limits of transfer learning with a unified text-to-text\ntransformer. arXiv e-prints , 2019.\nRazenshteyn, I., Song, Z., and Woodruff, D. P. Weighted\nlow rank approximations with provable guarantees. In\nProceedings of the forty-eighth annual ACM symposium\non Theory of Computing , pp. 250–263, 2016.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='Sarlos, T. Improved approximation algorithms for large\nmatrices via random projections. In 2006 47th annual\nIEEE symposium on foundations of computer science\n(FOCS) , pp. 143–152. IEEE, 2006.\nSeo, M., Lee, J., Kwiatkowski, T., Parikh, A. P., Farhadi,\nA., and Hajishirzi, H. Real-time open-domain question\nanswering with dense-sparse phrase index. In ACL, pp.\n4430–4441, 2019.\nShrivastava, A., Song, Z., and Xu, Z. Sublinear least-squares\nvalue iteration via locality sensitive hashing. arXiv' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='preprint arXiv:2105.08285 , 2021.\nSmith, J. E. A study of branch prediction strategies. In\n25 years of the international symposia on Computer\narchitecture (selected papers) , pp. 202–215, 1998.\nSohler, C. and Woodruff, D. P. Subspace embeddings\nfor the l1-norm with applications. In Proceedings of\nthe forty-third annual ACM symposium on Theory of\ncomputing , pp. 755–764, 2011.\nSong, Z. and Ye, M. Efficient asynchronize stochas-\ntic gradient algorithm with structured data. CoRR ,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='abs/2305.08001, 2023.\nSong, Z. and Yu, Z. Oblivious sketching-based central path\nmethod for linear programming. In International Confer-\nence on Machine Learning , pp. 9835–9847. PMLR, 2021.\n14' metadata={'source': 'pdfs/paper_3.pdf', 'page': 13}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nSong, Z., Woodruff, D. P., and Zhong, P. Low rank approx-\nimation with entrywise l1-norm error. In Proceedings of\nthe 49th Annual ACM SIGACT Symposium on Theory of\nComputing , pp. 688–701, 2017.\nSong, Z., Woodruff, D. P., and Zhong, P. Relative error\ntensor low rank approximation. In Proceedings of the\nThirtieth Annual ACM-SIAM Symposium on Discrete\nAlgorithms (SODA) , pp. 2772–2789. SIAM, 2019.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='Song, Z., Zhang, L., and Zhang, R. Training multi-layer\nover-parametrized neural network in subquadratic time.\narXiv preprint arXiv:2112.07628 , 2021.\nSong, Z., Wang, W., and Yin, C. Fast and efficient\nmatching algorithm with deadline instances. CoRR ,\nabs/2305.08353, 2023a.\nSong, Z., Yang, X., Yang, Y ., and Zhang, L. Sketching meets\ndifferential privacy: fast algorithm for dynamic kronecker\nprojection maintenance. In International Conference on\nMachine Learning (ICML) , 2023b.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='Tang, R., Lu, Y ., Liu, L., Mou, L., Vechtomova, O., and Lin,\nJ. Distilling task-specific knowledge from bert into simple\nneural networks. arXiv preprint arXiv:1903.12136 , 2019.\nTillet, P., Kung, H.-T., and Cox, D. Triton: an interme-\ndiate language and compiler for tiled neural network\ncomputations. In Proceedings of the 3rd ACM SIGPLAN\nInternational Workshop on Machine Learning and\nProgramming Languages , pp. 10–19, 2019.\nTouvron, H., Cord, M., Douze, M., Massa, F., Sablayrolles,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='A., and Jégou, H. Training data-efficient image trans-\nformers & distillation through attention. In International\nConference on Machine Learning , pp. 10347–10357.\nPMLR, 2021.\nVeit, A., Wilber, M. J., and Belongie, S. Residual networks\nbehave like ensembles of relatively shallow networks. Ad-\nvances in neural information processing systems , 29, 2016.\nViterbi, A. Error bounds for convolutional codes and an\nasymptotically optimum decoding algorithm. IEEE trans-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='actions on Information Theory , 13(2):260–269, 1967.\nWang, B. and Komatsuzaki, A. GPT-J-6B: A\n6 billion parameter autoregressive language\nmodel. https://github.com/kingoflolz/\nmesh-transformer-jax , May 2021.\nWang, R. and Woodruff, D. P. Tight bounds for lp oblivious\nsubspace embeddings. 2018.\nWang, X., Xiong, Y ., Wei, Y ., Wang, M., and Li, L. Lightseq:\nA high performance inference library for transformers.\nInProceedings of the 2021 Conference of the North' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='American Chapter of the Association for ComputationalLinguistics: Human Language Technologies: Industry\nPapers , pp. 113–120, 2021.\nWoodruff, D. P. Sketching as a tool for numerical linear\nalgebra. Foundations and Trends ®in Theoretical\nComputer Science , 10(1–2):1–157, 2014.\nXiao, G., Lin, J., Seznec, M., Demouth, J., and Han, S.\nSmoothquant: Accurate and efficient post-training\nquantization for large language models. arXiv preprint\narXiv:2211.10438 , 2022.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='Xie, S. M., Raghunathan, A., Liang, P., and Ma, T.\nAn explanation of in-context learning as implicit\nbayesian inference. In International Conference\non Learning Representations , 2022. URL https:\n//openreview.net/forum?id=RdJVFCHjUMI .\nXue, H.-J., Dai, X., Zhang, J., Huang, S., and Chen, J. Deep\nmatrix factorization models for recommender systems.\nInIJCAI , pp. 3203–3209, 2017.\nYao, Z., Aminabadi, R. Y ., Zhang, M., Wu, X., Li, C., and\nHe, Y . Zeroquant: Efficient and affordable post-training' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='quantization for large-scale transformers. arXiv preprint\narXiv:2206.01861 , 2022.\nYu, G.-I., Jeong, J. S., Kim, G.-W., Kim, S., and Chun, B.-G.\nOrca: A distributed serving system for {Transformer-\nBased}generative models. In 16th USENIX Symposium\non Operating Systems Design and Implementation (OSDI\n22), pp. 521–538, 2022.\nZandieh, A., Han, I., Daliri, M., and Karbasi, A. Kdeformer:\nAccelerating transformers via kernel density estimation.\narXiv preprint arXiv:2302.02451 , 2023.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='Zhang, L. Speeding up optimizations via data structures:\nFaster search, sample and maintenance. Master’s thesis,\nCarnegie Mellon University, 2022.\nZhang, M., Wang, W., Liu, X., Gao, J., and He, Y . Navi-\ngating with graph representations for fast and scalable\ndecoding of neural language models. Advances in neural\ninformation processing systems , 31, 2018.\nZhao, R., Hu, Y ., Dotzel, J., De Sa, C., and Zhang, Z.\nImproving neural network quantization without retraining' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='using outlier channel splitting. In International conference\non machine learning , pp. 7543–7552. PMLR, 2019.\n15' metadata={'source': 'pdfs/paper_3.pdf', 'page': 14}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nContents: In Section A, we present an extended discussion on LLM inference and related works. In Section B, we provide\nmore observation plots for slowly changing activation and further observation on the possibility of sparsifying LLMs via layer\nskipping. In Section C, we provide experiment details. In Section D, we demonstrate implementation details. In Section E,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='we provide detailed benchmarks regarding our implementation. In Section F, we define some basic notations and definitions.\nIn Section G, we define subspace embedding and show the norm preserving. In Section H, we introduce distances, angles, and\ninner product. In Section I, we provide the distance between different functions. In Section J, we provide the Near-neighbor\nSearch data structure. In Section K, we discuss self-attention as a clustering algorithm in depth.\nA Related Work' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='Generative LLM inference. Taking OPT-175B as an example, assume 6 A100 80GB PCIe, based on the hardware\nspecifications, we compare two main phases of inference time LLM, namely prompting and token generation in Table 1, and\ntwo major components, namely Multi-Head-Attention block and MLP block in Table 2. In practice, the token generation\nphase usually dominates the end-to-end test latency due to IO latency. Generating only two tokens is about the same latency as' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='prompting. Further, during token generation, the MLP block is 2 ×more expensive in both FLOPs and IO access. The hardware\nis often at low utilization because memory reads and writes are more limited on modern hardware than tensor core computation.\nGiven the rapid development of LLM, there is an emergence of systems that are specialized for LLM inference, such as\nFaster Transformer (NVIDIA), Orca (Yu et al., 2022), LightSeq (Wang et al., 2021), PaLM inference (Pope et al., 2022),' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='TurboTransformers (Fang et al., 2021), and Deepspeed-Inference (Aminabadi et al., 2022). In practice, the token generation\nphase usually dominates the end-to-end inference time. Although the state-of-the-art systems introduce some helpful system\noptimizations for speedup, there is a lack of careful algorithm and system co-design to unleash the full potential of hardware\nefficiency during the LLM inference computation.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='Near-neighbor Search for Efficient Deep Neural Networks. Near-neighbor Search is a well-studied problem with wide\napplications in recommendation system (Xue et al., 2017; Hall & Attenberg, 2015), question answering (Boytsov et al., 2016;\nSeo et al., 2019; Chang et al., 2020) and natural language processing (Bengio et al., 2003; Lee et al., 2016). There has been a\nline of work using Near-neighbor Search techniques such as Locality-sensitive hashing (Gionis et al., 1999) and Graph-based' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='indexing (Malkov et al., 2014) for efficient deep neural network training or inference (Zhang et al., 2018; Chen et al., 2019;\n2020a; Kitaev et al., 2020; Chen et al., 2021b;a; Liu et al., 2022).\nQuantization, pruning, distillation for LLM inference. Various system relaxations have been studied for decades for\nmodel inference in machine learning. For example, quantization (Han et al., 2015; Jacob et al., 2018; Nagel et al., 2019; Zhao' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='et al., 2019), pruning (Molchanov et al., 2016; Liu et al., 2018; He et al., 2019; Hoefler et al., 2021), and distillation (Hinton\net al., 2015; Cho & Hariharan, 2019; Tang et al., 2019; Touvron et al., 2021) have been applied to speed up the inference of\nthe machine learning model. Active research has recently attempted to apply such techniques in LLM inference. For example,\nzeroQuant (Yao et al., 2022) and nuQmm (Park et al., 2022) implement customized CUDA kernels to support tenor-wise' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='or group-wise quantization for LLM inference; LLM.int8 (Dettmers et al., 2022) adopts a mixed INT8/FP16 computation to\ndiminish the influence of activation outliers; SmoothQuant (Xiao et al., 2022) enables efficient 8-bit weight and activation for\nLLM inference; GPTQ (Frantar et al., 2022) adopts a one-shot weight quantization method based on approximate second-order\ninformation for accuracy and efficiency; SparseGPT (Frantar & Alistarh, 2023) introduces an approximate sparse regression' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='solver to enable the sparsity in LLM inference; (Bansal et al., 2022) has reported that a small set of attention heads can perform\nprimitive induction operations associated with in-context learning, and use this property to prune LLM for acceleration.\nResidual connections in neural networks. Residual connection shows great advantages for neural network generalization,\nit provides additional paths for activations to reach the latter parts of the neural network by skipping some layers (He et al.,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='2016). The advancement of residual connections can be viewed as ensembles of multiple shallow neural networks (Veit\net al., 2016). Plenty of active research has discussed the effectiveness of residual connections (Balduzzi et al., 2017; Bello\net al., 2021; Allen-Zhu & Li, 2019; Frei et al., 2019). However, as far as we know, there is no former work that leverages\nthe property of residual connections to improve the efficiency of LLM inference.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='B Additional Observation on Slowly Changing Observation\nFirst, we present more plots on the cosine similarity between representations. Figure 9 plots the cosine similarity between\nactivation across layers on OPT family. It is evident that similarity is high for the larger models.\nThere are two residual connections inside a transformer layer, one around the attention block, and the other one around the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='MLP block. The residual connection can be written as X+F(X), where Fis either the Multi-Head Attention or two MLP\nLayer. Figure 10 plots the cosine similarity between XandX+F(X), which is close to 1.0, and the cosine similarity between\n16' metadata={'source': 'pdfs/paper_3.pdf', 'page': 15}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0 5 10 15 20\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8\n(a) OPT-1.3B\n0 510 15 20 25 30\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8 (b) OPT-6.7B\n05101520253035\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8 (c) OPT-13B\n0 10 20 30 40\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8\n(d) OPT-30B\n010 20 30 40 50 60' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='Transformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8 (e) OPT-66B\n0 20 40 60 80\nTransformer Layer0.00.20.40.60.81.0Cosine Similarityn = 1\nn = 2\nn = 4\nn = 8 (f) OPT-175B\nFigure 9. Cosine similarity between layer land layer l+1for various model.\nXandF(X), which is close to 0.0. This happens because ∥X∥is significantly greater than ∥F(X)∥, shown in the purple.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='In the first layer, ∥F(X)∥is larger, which explains the low cosine similarity. The magnitude of the L2norm is different across\nmodels, however, we observe a similar trend with models of different sizes. There exists a normalization layer before F(X)\nand the layer normalization scale ∥X∥to a consistent magnitude across layers (e.g. 85 for OPT-30B, 110 for OPT175B),\nbut not necessarily scale down ∥X∥.\nC Additional Experiment Detail\nC.1 Large Batch Size' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='To help understand where the speed-up comes from when batch size is greater than 1, we present the Union Contextual Sparsity\n(fraction of neurons/heads that are not used by any of the inputs in the batch) of different batches sizes for MLP and Attention\nblocks, respectively, in Figure 11. Union Contextual Sparsity is calculated as 1.0 - the union of activated MLP neurons or\nAttention heads in the batch / total neurons or heads. The union operation is essential to realize a fast sparse GEMM.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='Surprisingly the number of MLP neurons/Attention heads that DEJAVU activated does not grow linearly with the batch size.\nThis suggests a power law distribution rather than a uniform distribution of parameter access from all input examples. Further,\na larger batch size can easily lead to out-of-memory for long sequence settings due to the limited GPU memory, the giant\nlarge model size, and the stored KV cache. For example, the total GPU memory of 8 80GB A100 is 640GB. Model parameters' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='are around 350GB for OPT175B. The KV cache for a batch size 32 with a sequence longer than 1920 tokens has already\nfilled up the GPU memory.\nC.2 Near Neighbor classifier\nIn the DEJAVU framework, any near-neighbor search method under the inner product metric would be sufficient to predict\na sparsity pattern. ""Training predictor"" is to reduce the cost of on-the-fly prediction, rather than training the model itself.\n17' metadata={'source': 'pdfs/paper_3.pdf', 'page': 16}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0.00.20.40.60.81.0Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 5 10 15 20\nTransformer Layer0.00.20.40.60.81.0Cosine SimilarityResidual At MLP\nCos(X, X+F(X))\nCos(X, F(X))020406080100120\nNorm||X||\n||F(X)||\n||LN(X)||\n020406080100120\nL2 Norm||X||\n||F(X)||\n||LN(X)||\n(a) OPT-1.3b\n0.00.20.40.60.81.01.2Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 5 10 15 20 25 30' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='Transformer Layer0.00.20.40.60.81.0Cosine SimilarityResidual At MLP\nCos(X, X+F(X))\nCos(X, F(X))050100150200250\nNorm||X||\n||F(X)||\n||LN(X)||\n050100150200250\nL2 Norm||X||\n||F(X)||\n||LN(X)|| (b) OPT-6.7b\n0.2\n0.00.20.40.60.81.0Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 10 20 30 40\nTransformer Layer0.00.20.40.60.81.0Cosine SimilarityResidual At MLP\nCos(X, X+F(X))\nCos(X, F(X))050100150200\nNorm||X||\n||F(X)||\n||LN(X)||\n050100150200\nL2 Norm||X||\n||F(X)||\n||LN(X)|| (c) OPT-13B' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='0.000.250.500.751.00Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 10 20 30 40\nTransformer Layer0.00.20.40.60.81.0Cosine SimilarityResidual At MLP\nCos(X, X+F(X))\nCos(X, F(X))0100200300\nNorm||X||\n||F(X)||\n||LN(X)||\n0100200300\nL2 Norm||X||\n||F(X)||\n||LN(X)||\n(d) OPT-30B\n0.25\n0.000.250.500.751.00Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 10 20 30 40 50 60\nTransformer Layer0.2\n0.00.20.40.60.81.0Cosine SimilarityResidual At MLP\nCos(X, X+F(X))' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='Cos(X, F(X))0100200300400\nNorm||X||\n||F(X)||\n||LN(X)||\n0100200300400\nL2 Norm||X||\n||F(X)||\n||LN(X)|| (e) OPT-66B\n0.50\n0.25\n0.000.250.500.751.00Cosine SimilarityResidual At Attention\nCos(X, X+F(X))\nCos(X, F(X))\n0 20 40 60 80\nTransformer Layer0.25\n0.000.250.500.751.00Cosine SimilarityResidual At MLP\nCos(X, X+F(X))\nCos(X, F(X))0500100015002000\nNorm||X||\n||F(X)||\n||LN(X)||\n05001000150020002500\nL2 Norm||X||\n||F(X)||\n||LN(X)|| (f) OPT-175B' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='Figure 10. Cosine similarity between XandF(X), and the cosine similarity between XandX′in orange color. L2norm of XandF(X)\nandXafter layer normalization in purple on the right. Except on the first layer, ∥X∥is significantly higher than ∥F(X)∥.∥F(X)∥is\nhigher at the first layer, which corresponds to the low cosine similarity at the first layer.\nFor example, in our exploration stage mentioned in Section 4.1, we adopt HNSW, a state-of-art near-neighbor search method,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='to predict MLP sparse pattern, and we can see from the following table there is no drop in the perplexity at 90 % sparsity\nratio. However, due to the high dimensionality of embedding and HNSW’s reliance on CPU, the time HNSW took to identify\nthe sparsity pattern is 10ms, which is longer than the MLP computation.\nIn our paper, we choose a neural network classifier as our near neighbor search method to take advantage of the fast matrix' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='multiplication on GPU. And training such classifiers to predict sparsity patterns is not only cheaper in terms of training cost\nbut also inherently different from the method concept.\n18' metadata={'source': 'pdfs/paper_3.pdf', 'page': 17}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0 20 40 60 80 96\nTransformer Layer0.50.60.70.80.91.0Union Contextual Sparsity\nBatch size\n2\n4\n8\n16\n32\n(a) MLP\n0 20 40 60 80 96\nTransformer Layer0.40.50.60.70.80.9Union Contextual Sparsity\nBatch size\n2\n4\n8\n16\n32 (b) Attention\nFigure 11. Union contextual sparsity with larger batch size.\nOPT-1.3B OPT-1.3B + HNSW\nHellaswag 0.4154 0.4314\nC4 14.2 14.4' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='Table 8. Sparsify from the Depth: Skipping or parallel entire transformer blocks may not lead to catastrophic drop in accuracy at test time.\nModel COPA Hellaswag Lambada OpenBookQA PIQA Winogrande\nOPT-175B 0.8600 0.7814 0.7584 0.4460 0.8096 0.7261\n- Parallel 2 0.8300 0.7737 0.7762 0.4520 0.8030 0.7096\n- Parallel 4 0.5200 0.2519 0 0.2720 0.5092 0.4870\n- Skip 2/8 0.8000 0.7112 0.6387 0.4220 0.7840 0.6630\n- Skip 2/4 0.6900 0.4409 0.0240 0.3400 0.6882 0.5383' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='Bloom 0.8000 0.7460 0.6771 0.4480 0.7949 0.7040\n- Parallel 2 0.8100 0.7404 0.6992 0.4360 0.7813 0.7048\n- Parallel 4 0.6200 0.3176 0.1325 0.2720 0.5593 0.5217\n- Skip 2/8 0.7900 0.6829 0.5936 0.4120 0.7699 0.6614\n- Skip 2/4 0.6600 0.5538 0.3023 0.3580 0.7046 0.5549\nC.3 Future Possibility: Skipping Layer\nDeja Vu currently sparsifies from the perspective of model width. Here, we explore the possibility of sparsification from' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='model depth. As observed in Section 3, we show that the activation of large language models changes slowly across blocks.\nThis property can be leveraged to increase the efficiency of a trained model by parallelizing, reordering, or skipping certain\nintermediate sub-blocks without significantly impacting the overall accuracy.\nSetting Wiki(ppl) C4(ppl)\nBaseline 11.57 10.17\nSkip every 2 layers 21.16 16.58\nSkip every 4 layers 13.45 11.37' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='Improving the inference efficiency of Transformer models is a challenging task due to their sequential execution of\nTransformer layers. Each sub-block depends on the output of the previous one, leading to low hardware efficiency, particularly\nduring the token generation phase where each forward pass is computed for only one token. However, the sequential execution\nof blocks and sub-blocks yields computation bubbles, and the latter involves a large amount of communication overhead.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='Here, we present an interesting observation that can potentially alleviate these challenges. We found that the activation of\nthe model changes slowly across blocks. Specifically, the cosine similarity of activations between adjacent blocks is often\nabove 0.99. This suggests that the blocks might take the previous activation as input – parallelize or reorder the blocks –' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='without significantly affecting the output. Slowly changing activations suggest that it may be possible to parallelize, reorder,\nor even skip blocks while maintaining a similar output. Some existing models, such as GPT-J (Wang & Komatsuzaki, 2021),\nGPT-NeoX (Black et al., 2022), and PaLM (Chowdhery et al., 2022) already placed the Attention block and MLP block\n19' metadata={'source': 'pdfs/paper_3.pdf', 'page': 18}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nin parallel in training to facilitate parallel computation and reduce the communication overhead.\nHere we investigate the possibility at inference time. And surprisingly, we found parallelizing those blocks for models that\nare trained in a sequence manner will not hurt the performance of downstream tasks significantly. And surprisingly, we found' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='parallelizing those blocks for models that are trained in a sequence manner will not hurt the performance of downstream\ntasks significantly. TableC.3 presents some preliminary results of OPT-175B and Bloom\nGiven the activation yand Transformer layer l, we have:\neyl←yl+MHAl(yl)\nbyl←eyl+MLPl(eyl)\nParallelizing two blocks refers to placing the Attention and MLP blocks in parallel, i.e.:\nbyl←y+MHAl(yl)+MLPl(yl)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='Parallelizing four blocks then parallelize the blocks of two Transformer layers, defined as follows:\nbyl+1←yl+MHAl(yl)+MLPl(yl)+MHAl+1(yl)+MLPl+1(yl)\nSkipping layers is straightforward, which drops an entire Transformer layer for every nlayers.\nWe are surprised to find that parallel two layers preserve accuracy on a series of tasks across models. Besides, randomly\nskipping 25% layers doesn’t lead to catastrophic quality. Our findings suggest from the downstream task perspective, the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='activation patterns within the model are relatively consistent across different blocks, providing a potential avenue for future\nresearch on model compression and optimization.\nD Implementation Details\nFigure 12 presents a more detailed workflow of DEJAVU . The left diagram shows how an input yperforms the sparse MHA\nwith selected indices 0,3, predicted by the head predictor. Similarly, the right diagram shows how an input yperforms the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='sparse MLP with selected indices 0,2, predicted by the neuron predictor of that layer.\nSelected Head Index!! = {0,3}Attention with """"!#,""""!$,""""!%$%&($)""""!\'\tOutput Projection)%*""!($)%(($)"""""")\t!*= {0,2}$σ($""""""+)""""""+ MLP""""($)\nSelected Neurons IndexSparsified AttentionSparsified MLP\nFigure 12. Detailed diagram on the sparsified computation process of MLP and Attention. Notation refers to Section 2.3' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='Next, we will present a general explanation of two optimizations we used in DEJAVU implementation. Kernel fusion: A\nstandard implementation of sparse matrix-vector multiply (e.g., Wx in PyTorch) that separately indexes a subset of the\nmatrix W[idx,:]before multiplying with input xwould incur 3 ×the amount of memory IOs: one to load a subset of Wfrom\nGPU memory, one to write that subset to a different contiguous region in memory, and one to load that (now contiguous)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='subset in again to multiply with x. Similarly, to use sparse matrix multiply routines (e.g., cuSparse), we would first need\nto convert W[idx,:]to sparse format, again incurring more memory IOs. We instead fuse the indexing and the multiplication\nstep: we load a subset of W[idx,:]to memory, along with x, perform the multiply, then write down the result. This fused\nimplementation (in Triton (Tillet et al., 2019)) yields up to 4 ×speedup compared to a standard PyTorch implementation' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='(Section E). Memory coalescing: the weight matrices are conventionally stored in row-major format. This allows us to load\n20' metadata={'source': 'pdfs/paper_3.pdf', 'page': 19}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nW[idx,:]optimally (as the second dimension is contiguous in memory). However, for cases where we need to load W[:,idx]\n(attention output projection and the 2nd weight matrix in the MLP) this format significantly slows down memory loading,\nasidxcould contain indices pointing to non-contiguous memory. A simple solution is to store these matrices in column-major' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='format (i.e., storing W⊤in contiguous row-major format), then use the same fused kernel above. This transposition is done\nonce when loading the model, and incurs no added cost during generation.\nE Benchmarking Sparse MLP and Sparse Attention\n0.2 0.4 0.6 0.8\nMLP density100200300400Time (us)Sparse MLP benchmark\nDense MLP\nSparse MLP baseline (PyTorch)\nSparse MLP - Deja vu\nFigure 13. Speed benchmarking of the MLP layer of OPT-175B on 8xA100s. Our sparse implementation is up to 4.5 ×faster than the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='baseline implementation in PyTorch. Our sparse MLP implementation remains faster than dense MLP for density up to 0.8.\nWe validate that our hardware-aware implementation of sparse MLP and sparse attention (Section 4.4) yields wall-clock\nspeed up compared to both dense MLP/attention and compared to the standard implementation in PyTorch.\nRecall that our implementation fuses the sparse indexing and the multiplication (W1\nSM)⊤yfor weight matrices (W1)⊤and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='input vector y. In cases where we need to index W2\nSM, we store the transpose of W2to ensure memory coalescing. For the\nbaseline implementation in PyTorch, we index (W1\nSM)⊤as a separate operation before multiplying with y, which incurs\nmore memory reads/writes.\nSimilarly, we fuse the sparse indexing and the multiplication (WQ\nSA)⊤y,(WK\nSA)⊤y,(WV\nSA)⊤yfor weight matrices (WQ)⊤,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='(WK)⊤,(WV)⊤and input vector y. Note we usually concatenate all three matrices in the standard implementation, but we sep-\narate them here for clarity. In cases where we need to index WO\nSA, we store the transpose of WOto ensure memory coalescing.\nIn Figure 13 and Figure 14, our sparse MLP and attention implementations are 4-5 ×faster than the baseline implementation\nin Pytorch, and remains faster than the dense version for density up to 0.8.\nF Notations and Basic Definitions' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='For a positive integer n, let[n]:={1,2,···,n}. For a matrix A∈Rn×n, letAi,:andA:,jbe two column vectors corresponding\nto the i-th row and the j-th column of Arespectively, and Ai,jbe the entry at the i-th row and the j-th column. For a vector\nx∈Rn, let√x∈Rndenote the vector with the i-th entry being√xianddiag( x)∈Rn×ndenote the diagonal matrix with\nthei-th diagonal entry being xi. For two matrices A,W∈Rn×n, let∥A∥W:=(Pn\ni=1Pn\nj=1Wi,jA2\ni,j)1/2andW◦Adenote' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='the matrix where (W◦A)i,j=Wi,jAi,j. For matrix W∈Rn×n, letDWi:=diag( Wi,:)withi∈[n].\nFor two vectors x∈Rnandw∈Rn\n≥0, let∥x∥w:= (Pn\ni=1wix2\ni)1/2. For a vector x, we denote ∥x∥2:= (Pn\ni=1x2\ni)1/2as its\n21' metadata={'source': 'pdfs/paper_3.pdf', 'page': 20}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n0.2 0.4 0.6 0.8\nAttention density100200300Time (us)Sparse attention benchmark\nDense attention\nSparse attention baseline (PyTorch)\nSparse attention - Deja vu\nFigure 14. Speed benchmarking of the attention layer of OPT-175B on 8xA100s. Our sparse implementation is up to 5 ×faster than the\nbaseline implementation in PyTorch. Our sparse attention implementation remains faster than dense MLP for density up to 0.8.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='ℓ2norm. We denote ∥x∥p:=(Pn\ni=1|xi|p)1/pas itsℓpnorm. For a square matrix A, we denote tr[A]as the trace of matrix A.\nFor a matrix A∈Rn×k(suppose n≥k), we use ∥A∥to denote its spectral norm, i.e., ∥A∥=supx∥Ax∥2/∥x∥2. We use ∥A∥F\nto denote its Frobenius norm ∥A∥F:=(Pn\ni=1Pk\nj=1A2\ni,j)1/2.\nSuppose matrix A∈Rn×khas SVD decomposition UΣV⊤where U∈Rn×k(this matrix has orthonormal columns),\nΣ∈Rk×kis a diagonal matrix, and V∈Rk×k. We call columns of Uare singular vectors. We use A†∈Rk×nto denote the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='Moore-Penrose pseudoinverse, then A†=VΣ−1U⊤. Suppose Σ∈Rk×kis sorted diagonal matrix, let σ1,···,σkdenote the\ndiagonal entries of Σ. Then we call σithei-th singular value of matrix, and we write it as σi(A).\nFor any symmetric matrix B∈Rk×k, we define its eigenvalue decomposition as UΛU⊤, where Λis a diagonal matrix. Let\nλ1,···,λkdenote the entries on diagonal of Λ∈Rk×k. We say λiis the i-th eigenvalue. Usually we write it as λi(B).\nThe connection between eigenvalues and singular values is\nσ2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='i(A)=λi(A⊤A)\nWe use notation A⪰0to denote that matrix Ais positive semidefinite (psd). Mathematically, A⪰0means for all vectors\nx, we have x⊤Ax≥0.\nSimilarly, for two squarer matrices AandB, we use A⪰Bto denote the case where for all vectors x,x⊤Ax≥x⊤Bx.\nWe use Pr[]andE[]for probability and expectation. We denote max{a,b}as the maximum between aandb. We denote\nmin{a,b}(resp. max{a,b}) as the minimum (reps. maximum) between aandb.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='Throughout, for non-negative real numbers a and b, we use the notation a=(1±ϵ)bifa∈[(1−ϵ)b,(1+ϵ)b].\nG Subspace Embeddings and Norm Preserving\nIn Section G.1, we show the norm preserving of the soft-max functions. In Section G.2, we show the norm preserving of the\nReLU function. In Section G.3, we introduce the folded Guassian distribution. In Section G.4, we introduce the ℓ2subspace' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='embedding. In Section G.5, we introduce the ℓ1subspace embedding. In Section G.6, we introduce different sketching\nmatrices for subspace embedding.\n22' metadata={'source': 'pdfs/paper_3.pdf', 'page': 21}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nG.1 Soft-Max Functions\nLetK∈Rs×dandV∈Rd×s. Inspired by the softmax unit in the attention scheme of large language models. The softmax\nrelated regression has been studied in many settings (Zandieh et al., 2023; Alman & Song, 2023; Brand et al., 2023; Li et al.,\n2023b; Deng et al., 2023b;a; Gao et al., 2023a; Li et al., 2023a; Gao et al., 2023b). In this work, we follow the standard' metadata={'source': 'pdfs/paper_3.pdf', 'page': 22}"
"page_content='softmax definition. We define σ1:Rs→Rsto be a softmax function, i.e., for any vector y∈Rs, theσ(y)can be written as\nσ1(y)i=exp(yi)\nPd\nj=1exp(yj),∀i∈[d]\nThe standard softmax is ℓ1version. In this work, we also consider the ℓ2generalization. We define σ2:Rs→Rsto be a\nsoftmax function ( ℓ2version), i.e., for any vector y∈Rs, theσ(y)can be written as\nσ2(y)i=exp(yi)\n(Pd\nj=1exp(2 yj))1/2,∀i∈[d]\nWe define function f:Rd→Rd\nf(x)=V·(σ(K·x)) (3)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 22}"
"page_content='Definition G.1. We say X ⊂Rdis a rank- ksubspace, if there is an orthonormal basis U∈Rd×k, for any x∈ X, there is\ny∈Rksuch that\nx=Uy.\nWe can have\nLemma G.2. Letτ∈(0,1). LetX ⊂Rddenote a subspace with rank k. Letfbe defined based on σ2function. Let Vis\na random Gaussian matrices with d≥Ω(ϵ−2(k+log(1 /δ)))rows. Let V=τV, then we have with probability 1−δ\n(1−ϵ)τ∥x∥2≤∥f(x)∥≤(1+ϵ)τ∥x∥2.\nfor all unit vectors x∈X.\nFurther, if d=O(k+log(1 /δ)), then we have\n0.5τ∥x∥2≤∥f(x)∥≤2τ∥x∥2.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 22}"
"page_content='Remark G.3.The above condition implies that fis a shrinking operator but also not shrinking arbitrarily small.\nProof. Given d≥Ω(ϵ−2(k+log(1 /δ))), by using Lemma G.11 , we have\n(1−ϵ)∥y∥2≤∥V y∥2≤(1+ϵ)∥y∥2\nAs the input of the function fhere is the output of a softmax function ( ℓ2version), we know that ∥y∥2=1.\nThus, we have\n(1−ϵ)≤∥V y∥2≤(1+ϵ)\nBy rescaling V, we have\n(1−ϵ)∥x∥2≤∥V y∥2≤(1+ϵ)∥x∥2.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 22}"
"page_content='Lemma G.4. Letτ∈(0,1). LetX ⊂Rddenote a subspace with rank k. Letfbe defined based on σ1function. Suppose\nVis a random Gaussian matrix with d≥Ω((k+log(1 /δ)))rows. Let V=1\n2τV.\nThen we have\n1\n4√sτ·∥x∥2≤∥f(x)∥2≤τ·∥x∥2\nfor all unit vectors x.\nProof. By property of subspace embedding, we know that if d≥Ω(ϵ−2(s+log(1 /δ))),\n(1−ϵ)∥y∥2≤∥V y∥2≤(1+ϵ)∥y∥2\nBy property of function of f, we know we only need to care ∥y∥1=1, this implies that\n1√s∥y∥1≤∥y∥2≤∥y∥1\n23' metadata={'source': 'pdfs/paper_3.pdf', 'page': 22}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nOn one hand, we have\n∥V y∥2≤(1+ϵ)·∥y∥2\n≤(1+ϵ)·∥y∥1\n= (1+ ϵ), (4)\nwhere the first step follows from ∥V y∥2≤(1+ϵ)∥y∥2, the second step follows from ∥y∥2≤∥y∥1and the last step follows\nfrom∥y∥1=1.\nOn the other hand, we have\n∥V y∥2≥(1−ϵ)∥y∥2\n≥1√s(1−ϵ)∥y∥1\n=1√s(1−ϵ), (5)\nwhere the first step follows from (1−ϵ)∥y∥2≤∥V y∥2, the second step follows from1√s∥y∥1≤∥y∥2and the last step follows\nfrom∥y∥1=1.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 23}"
"page_content='Combining Eq. (5)and Eq. (4) together, we have\n(1−ϵ)1√s≤∥V y∥2≤(1+ϵ)\nChoosing ϵ=1/2, we have\n1\n2√s≤∥V y∥2≤2.\nByV=1\n2τVand∥x∥2=1, we have\n1\n4√sτ∥x∥2≤∥V y∥2≤τ∥x∥2.\nG.2 ReLU Functions\nWe use ϕ:R→Rto denote ReLU function, i.e., ϕ(z)=max {z,0}.\nWe define function g:Rd→Rd\ng(x)=V·(ϕ(K·x)) (6)\nLetK∈Rs×dandV∈Rd×s.\nLemma G.5. LetX ⊂Rddenote a rank- ksubspace. Let Kdenote a random Gaussian matrix. Let Vdenote a random' metadata={'source': 'pdfs/paper_3.pdf', 'page': 23}"
"page_content='Gaussian matrix. Let s≥Ω(ϵ−2klog(1/(δϵ))). Letd≥Ω(ϵ−2(k+log(1 /δ))). Then we know with high probability 1−δ,\nfor all unit vector x∈X\n(1−ϵ)∥x∥2≤∥f(x)∥2≤(1+ϵ)∥x∥2\nProof. Suppose s≥Ω(ϵ−2log(1/δ)).\nUsing Lemma G.6, Fact G.7, we can show that for each fixed\n(1−ϵ)∥x∥2≤∥ϕ(Kx)∥2≤(1+ϵ)∥x∥2\nholds with probability 1−δ.\nBy a standard ϵ-net argument (Lemma G.9), the net points in Xis at most (10/ϵ)O(k).\nTaking a union bound over all the net points, we can show that for all x∈X\n(1−ϵ)∥x∥2≤∥ϕ(Kx)∥2≤(1+ϵ)∥x∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 23}"
"page_content='holds with probability 1−δ/2ands≥Ω(ϵ−2klog(1/(δϵ))).\nFurther, we using Lemma G.11, we can show that\n(1−ϵ)∥ϕ(Kx)∥2≤∥f(x)∥2≤(1+ϵ)∥ϕ(Kx)∥2\nholds with probability 1−δ/2.\nCombining together,\n(1−ϵ)2∥x∥2≤∥f(x)∥2≤(1+ϵ)2∥x∥2\n24' metadata={'source': 'pdfs/paper_3.pdf', 'page': 23}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nholds with probability 1−δ.\nRescaling the ϵ, we complete the proof.\nG.3 Folded Gaussian Distribution\nWe state a standard tool from literature,\nLemma G.6 (Lemma 1 on page 1325 of Laurent and Massart (Laurent & Massart, 2000)) .LetX∼X2\nkbe a chi-squared\ndistributed random variable with kdegrees of freedom. Each one has zero means and σ2variance.\nThen,\nPr[X−kσ2≥(2√\nkt+2t)σ2]≤exp(−t)\nPr[kσ2−X≥2√\nktσ2]≤exp(−t)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='Further if k≥Ω(ϵ−2t)andt≥Ω(log(1 /δ)), then we have\nPr[|X−kσ2|≤ϵkσ2]≤δ.\nWe prove the following property,\nFact G.7. Leth,q∈Rpbe fixed vectors and h̸= 0,W∈Rm×pbe random matrix with i.i.d. entries Wi,j∼ N(0,2\nm), and\nvector v∈Rmdefined as vi=ϕ((Wh)i)=1(W(h+q))i≥0(Wh)i. Then,\n•|vi|follows i.i.d. from the following distribution: with half probability |vi|=0, and with the other half probability |vi|\nfollows from folded Gaussian distributions |N(0,2∥h∥2\nm)|.\n•m∥v∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='2∥h∥2is in distribution identical to χ2\nω(chi-square distribution of order ω) where ωfollows from binomial distribution\nB(m,1/2).\nProof. We assume each vector Wiis generated by first generating a gaussian vector g∼N(0,2I\nm)and then setting Wi=±g\nwhere the sign is chosen with half-half probability. Now, |⟨Wi,h⟩|=|⟨g,h⟩|only depends on g, and is in distribution identical\nto|N(0,2∥h∥2\nm)|. Next, after the sign is determined, the indicator 1⟨Wi,h+q⟩≥0is1with half probability and 0with another' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='half. Therefore, |vi|satisfies the aforementioned distribution. As for ∥v∥2, letting ω∈{0,1,...,m}be the variable indicator\nhow many indicators are 1, then ω∼B(m,1/2)andm∥v∥2\n2∥h∥2∼χ2\nω.\nG.4 ℓ2subspace embedding\nWe define a standard notion in sketching technique.3\nDefinition G.8 (ℓ2subspace embedding (Sarlos, 2006)) .A(ϵ,δ,ℓ 2)-subspace embedding for the column space of an n×d\nmatrix Ais a matrix Sfor which\nPr[∀x∈Rd,∥SAx∥2\n2=(1±ϵ)∥Ax∥2\n2]≥1−δ.\nThe above condition is equivalent to' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='Pr[∥U⊤U−U⊤S⊤SU∥≤ϵ]≥1−δ.\nwhere the Uis the orthonormal basis of A.\nFor the reason of above conditions are equivalent, we refer the readers to the survey (Woodruff, 2014).\nWe state a standard tool in literature,\nLemma G.9 (Lemma 5 in (Woodruff, 2014)) .LetX ⊂Rdbe rank k. For any γ∈(0,1), there is a γ-netNofXfor which\n|N|≤(1+4 /γ)k.\n3We remark that sketching technique has widely applied to many applications such as linear regression, low-rank approximation (Clarkson' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='& Woodruff, 2013; Nelson & Nguyên, 2013; Lu et al., 2013; Boutsidis et al., 2016; Cohen, 2016; Razenshteyn et al., 2016; Song et al., 2017;\n2019), linear programming (Song & Yu, 2021; Dong et al., 2021; Jiang et al., 2021; Gu & Song, 2022), semi-definite programming (Gu &\nSong, 2022; Song et al., 2023b), empirical risk minimization(Lee et al., 2019; Qin et al., 2023b), training over-parameterized neural network' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='(Brand et al., 2021; Song et al., 2021; Alman et al., 2022; Hu et al., 2022; Zhang, 2022).\n25' metadata={'source': 'pdfs/paper_3.pdf', 'page': 24}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nG.5 ℓ1subspace embedding\nWhen p=1, using Cauchy random variables, Sohler and Woodruff (Sohler & Woodruff, 2011) showed there exist ℓ1oblivious\nsubspace embeddings with O(dlogd)rows and κ=O(dlogd). This approach was generalized by using p-stable random\nvariables in work of Meng and Mahoney (Meng & Mahoney, 2013) to ℓp-norms when 1<p< 2, where they showed there' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='existℓpoblivious subspace embeddings with O(dlogd)rows and κ=O((dlogd)1/p). Unlike the case when p= 2, due to\nthe large distortion\nIn (Wang & Woodruff, 2018), they show for every 1≤p<2, any oblivious subspace embedding with dimension rhas distortion\nκ=Ω(1\n(1\nd)1/p·log2/pr+(r\nn)1/p−1/2). They also give sparse oblivious subspace embeddings for every 1≤p<2which are optimal\nin dimension and distortion, up to poly (logd)factors. Importantly for p=1, they achieve r=O(dlogd),κ=O(dlogd)and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='s=O(logd)non-zero entries per column.\nDefinition G.10 (ℓ1subspace embedding) .Let0<α<β be parameters. We will say a matrix Sis anℓ1subspace embedding\nfor an n×dmatrix Aif there are constants c1,c2>0so that for all x∈Rd,\n∥Ax∥≤∥SAx∥1≤dc1∥Ax∥1,\nandShas at most dc2rows.\nG.6 Random Matrices\nMatrices b Time for R·A Reference\nRandom Gaussian ϵ−2(d+log(1 /δ)) Tmat(b,n,d) Thm. 6 of (Woodruff, 2014)\nSRHT ϵ−2(√\nd+√logn)2log(d/δ)ndlog(ϵ−1d(logn)) Thm. 7 of (Woodruff, 2014)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='AMS ϵ−2(d+log(1 /δ)) Tmat(b,n,d) Follow from JL guarantee\nCount-sketch ϵ−2δ−1d2nnz(A) Thm. 9 of (Woodruff, 2014)\nSparse embedding ϵ−2d·polylog(d/(ϵδ)) ϵ−1nnz(A)polylog(d/(ϵδ)) Thm. 10 (2) of (Woodruff, 2014)\nSparse embedding ϵ−2d1+γϵ−1nnz(A)poly(1 /γ) Thm. 10 (1) of (Woodruff, 2014)\nTable 9. Summary for different sketching matrices for subspace embedding. The sketching matrix Rhas size b×n. The vectors are from the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='column subspace of matrix Awith size n×d.ϵ∈(0,1)is the error parameter, and δ∈(0,1)is the probability parameter. Tmat(a,b,c)denotes\nthe running time of fast matrix multiplication of two matrices with size a×bandb×c.In the first sparse embedding matrix, each column has\ns≥ϵ−1polylog(d/(ϵδ))non-zero entries; In the second sparse embedding matrix, each column has s≥ϵ−1poly(1/γ)non-zero entries,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='γ >0is a tunable parameter that gives different trade-offs, and δcan be as small as 1/poly(d).For count-sketch matrices, the subspace\nembedding guarantee is proved from JL moment property, instead of directly from JL guarantee.\nLemma G.11 (Theorem 6 of (Woodruff, 2014)) .Let0< ϵ,δ < 1andS=1√\nkR∈Rk×nwhere the entries Ri,jofRare\nindependent standard normal random variables. Then if k= Θ(ϵ−2(d+log(1 /δ))), then for any fixed n×dmatrix A, with' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='probability 1−δ,Sis a(1±ϵ)ℓ2-subspace embedding for A, that is, simultaneously for all x∈Rd,∥SAx∥2=(1±ϵ)∥Ax∥2.\nHereC >0is an absolute constant.\nWe consider several standard sketching matrices:\n1. Random Gaussian matrices.\n2. Subsampled randomized Hadamard/Fourier transform (SRHT) matrices (Lu et al., 2013).\n3. AMS sketch matrices (Alon et al., 1996), random {−1,+1}per entry.\n4.Count-Sketch matrices (Charikar et al., 2002), each column only has one non-zero entry, and is −1,+1half probability' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='each.\n5.Sparse embedding matrices (Nelson & Nguyên, 2013), each column only has snon-zero entries, and each entry is\n−1√s,+1√shalf probability each.\n6. Uniform sampling matrices.\nDefinition G.12 (Random Gaussian matrix) .We say R∈Rb×nis a random Gaussian matrix if all entries are sampled from\nN(0,1/b)independently.\n26' metadata={'source': 'pdfs/paper_3.pdf', 'page': 25}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nDefinition G.13 (Subsampled randomized Hadamard/Fourier transform matrix (Lu et al., 2013)) .We say R∈Rb×nis a\nsubsampled randomized Hadamard transform (SRHT) matrix4if it is of the form R=p\nn/bSHD , where S∈Rb×nis a random\nmatrix whose rows are b uniform samples (without replacement) from the standard basis of Rn,H∈Rn×nis a normalized' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='Walsh-Hadamard matrix, and D∈Rn×nis a diagonal matrix whose diagonal elements are i.i.d. Rademacher random variables.\nDefinition G.14 (AMS sketch matrix (Alon et al., 1996)) .Leth1,h2,···,hbbebrandom hash functions picking from a 4-wise\nindependent hash family H={h:[n]→{−1√\nb,+1√\nb}}. Then R∈Rb×nis a AMS sketch matrix if we set Ri,j=hi(j)\nDefinition G.15 (Count-sketch matrix (Charikar et al., 2002)) .Leth:[n]→[b]be a random 2-wise independent hash function' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='andσ: [n]→{− 1,+1}be a random 4-wise independent hash function. Then R∈Rb×nis a count-sketch matrix if we set\nRh(i),i=σ(i)for all i∈[n]and other entries to zero.\nDefinition G.16 (Sparse embedding matrix I (Nelson & Nguyên, 2013)) .We say R∈Rb×nis a sparse embedding matrix\nwith parameter sif each column has exactly snon-zero elements being ±1/√suniformly at random, whose locations are\npicked uniformly at random without replacement (and independent across columns)5.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='Definition G.17 (Sparse embedding matrix II (Nelson & Nguyên, 2013)) .Leth: [n]×[s]→[b/s]be a random 2-wise\nindependent hash function and σ:[n]×[s]→{− 1,1}be a 4-wise independent. Then R∈Rb×nis a sparse embedding matrix\nII with parameter sif we set R(j−1)b/s+h(i,j),i=σ(i,j)/√sfor all (i,j)∈[n]×[s]and all other entries to zero6.\nDefinition G.18 (Uniform sampling matrix) .We say R∈Rb×nis a uniform sampling matrix if it is of the form R=p\nn/bSD ,' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='where S∈Rb×nis a random matrix whose rows are b uniform samples (without replacement) from the standard basis of\nRn, andD∈Rn×nis a diagonal matrix whose diagonal elements are i.i.d. Rademacher random variables.\nH Distances, Angles, and Inner Product\nMost of the properties in this section are very standard in literature, e.g., see (Gu et al., 2023).\nLetU∈Rn×kdenote an orthonormal basis, we use U⊥∈Rn×(n−k)denote the matrix such that UU⊤+U⊥U⊤\n⊥=In.\nDefinition H.1. LetX∈Rn×kandY∈Rn×k.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='For any matrix X, and for orthogonal matrix Y(Y⊤Y=Ik) we define\n•tanθ(Y,X):=∥Y⊤\n⊥X(Y⊤X)−1∥\nFor orthogonal matrices YandX(Y⊤Y=IkandX⊤X=Ik), we define\n•cosθ(Y,X):=σmin(Y⊤X).\n–It is obvious that cos(Y,X)=1/∥(Y⊤X)−1∥andcos(Y,X)≤1.\n•sinθ(Y,X):=∥(I−Y Y⊤)X∥.\n–It is obvious that sinθ(Y,X)=∥Y⊥Y⊤\n⊥X∥=∥Y⊤\n⊥X∥andsinθ(Y,X)≤1.\n•dist(Y,X):=min Q∈Ok∥Y Q−X∥\nwhere Okis the set of k×korthogonal matrices.\nLemma H.2 (Structural lemma for orthogonal matrices) .LetX,Y∈Rn×kbe orthogonal matrices. Then\n(Y⊤X)⊥=Y⊤\n⊥X.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='Proof. Let us first compute the Gram of Y⊤X, which is\nX⊤Y Y⊤X=X⊤(I−Y⊥Y⊤\n⊥)X\n=X⊤X−X⊤Y⊥Y⊤\n⊥X\n=Ik−X⊤Y⊥Y⊤\n⊥X,\n4In this case, we require logno be an integer.\n5For our purposes the signs need only be O(logd)-wise independent, and each column can be specified by a O(logd)-wise independent\npermutation, and the seeds specifying the permutations in different columns need only be O(logd)-wise independent.\n6This definition has the same behavior as sparse embedding matrix I for our purpose\n27' metadata={'source': 'pdfs/paper_3.pdf', 'page': 26}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nwhere the first step follows from Y⊥Y⊤\n⊥+Y Y⊤=I, the second step follows from simple algebra, and the last step follows\nfromXis an orthogonal matrix, so X⊤=X−1.\nThis means that (Y⊤X)⊥=Y⊤\n⊥X.\nLemma H.3 (Orthogonal and inverse share singular vectors) .LetA∈Rk×kbe non-singular, then A⊥andA−1have the\nsame set of singular vectors. Consequently, ∥A⊥A−1∥=∥A⊥∥∥A−1∥.\nProof. LetA∈Rk×kandA⊤A+A⊤' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='⊥A⊥=Ik, we will show that ∥A⊥A−1∥=∥A⊥∥∥A−1∥. Let x∈Rkbe the unit\neigenvector of Athat realizes the spectral norm, note that\n∥A⊥x∥2\n2= 1−∥A∥2,\nwe argue that xcorresponds to the smallest singular value of A⊥via contradiction. Suppose there exists some unit vector\nywith∥A⊥y∥2<∥A⊥x∥2, by definition, we know that ∥A⊥y∥2\n2+∥Ay∥2\n2= 1, this means that ∥Ay∥2>∥Ax∥2=∥A∥,\ncontradicts the definition of spectral norm. Similarly, if zis the unit vector that realizes the spectral norm of A⊥, then it is' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='also singular vector corresponds to the smallest singular value of A, or equivalently, the spectral norm of A−1. Our above\nargument essentially implies that A⊥andA−1have the same set of singular vectors. The proof is then straightforward:\nsuppose A⊥z=λzandA−1z=µz, then\nA⊥A−1z=A⊥µz\n=µ(A⊥z)\n=λµz,\nwhere the first step follows from our assumption, the second step follows from µis a real number and a real number multiplying' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='a matrix is commutative and follows from the associative property, and the third step follows from our assumption.\nThus, we have ∥A⊥A−1∥=∥A⊥∥∥A−1∥, and we have proved the assertion.\nLemma H.4. LetX,Y∈Rn×kbe orthogonal matrices, then\ntanθ(Y,X)=sinθ(Y,X)\ncosθ(Y,X).\nProof. Due to Lemma H.2, we have (Y⊤X)⊥=Y⊤\n⊥X. Thus, tanθ(Y,X)=∥(Y⊤X)⊥(Y⊤X)−1∥. The proof then follows\nstraightforwardly from Lemma H.3.\nLemma H.5. LetX,Y∈Rn×kbe orthogonal matrices, then sin2θ(Y,X)+cos2θ(Y,X)=1 .' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='Proof. Recall that cosθ(Y,X) =1\n∥(Y⊤X)−1∥andsinθ(Y,X) =∥Y⊤\n⊥X∥, by Lemma H.2, we know that (Y⊤X)⊥=Y⊤\n⊥X,\ntherefore sinθ(Y,X) =∥(Y⊤X)⊥∥. Let A:=Y⊤X, by Lemma H.3, we know that A⊥andA−1have the same singular\nvectors, or equivalently, the singular vector realizing ∥A⊥∥corresponds to the smallest singular value of A. Letz∈Rkbe\nthe unit singular vector with singular value ∥A⊥∥, then\nz⊤A⊤Az+z⊤A⊤\n⊥A⊥z= 1,\n∥A⊥∥2+σ2\nmin(A)= 1,\ncos2θ(Y,X)+sin2θ(Y,X)= 1.\nThis completes the proof.\nH.1 Angle is close' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='Lemma H.6. Letϵ∈(0,0.1)Letxdenote a unit vector, i.e., ∥x∥2=1.\nLetz=(x+y)/∥x+y∥2.\nIf∥y∥2≤ϵ·∥x∥2, thenp\n1−⟨x,z⟩2≤2√ϵ\nProof. We have\n∥x+y∥2≥ ∥x∥2−∥y∥2\n≥1−ϵ\nwhere the first step follows from triangle inequality.\n28' metadata={'source': 'pdfs/paper_3.pdf', 'page': 27}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nWe also have\n∥x+y∥2≤ ∥x∥2+∥y∥2\n≤1+ϵ (7)\nWe have\n(1−ϵ)2≥1−2ϵ (8)\nWe also have1\n(1+ϵ)2≥1−3ϵ (9)\nwhere ϵ∈(0,0.1).\nCombining Eq. (8) and Eq. (9), we have\n1\n(1+ϵ)2·(1−ϵ)2≥(1−2ϵ)·(1−3ϵ)\n= 1−5ϵ+6ϵ2\n≥1−5ϵ+ϵ\n= 1−4ϵ (10)\nwhere the first step follows from Eq. (8) and Eq. (9) and the rest of them follow from simple algebra.\nFinally, we have\n1−⟨x,z⟩2= 1−⟨x,x+y\n∥x+y∥2⟩2\n= 1−1\n∥x+y∥2\n2⟨x,x+y⟩2\n= 1−1\n∥x+y∥2\n2·(∥x∥2\n2+⟨x,y⟩)2\n= 1−1\n∥x+y∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 28}"
"page_content='2·(1+⟨x,y⟩)2\n≤1−1\n(1+ϵ)2·(1+⟨x,y⟩)2\n≤1−1\n(1+ϵ)2·(1−ϵ)2\n≤1−(1−4ϵ)\n= 4ϵ,\nwhere the first step follow the definition of z, the second step follows from the reorganization, the third step follows from\nthe definition of inner product, the fourth step follows from ∥x∥2=1, the fifth step follows from Eq. (7), the sixth step follows\nfrom 1+⟨x,y⟩≥1−|⟨x,y⟩|≥1−∥x∥2·∥y∥2≥1−ϵ, the seventh step follows from Eq. (10) and the last step follows from\nsimple algebra.\nI Function Approximations' metadata={'source': 'pdfs/paper_3.pdf', 'page': 28}"
"page_content='We first we show the function approximation for two operators in Section I.1, which means that there are two functions. Then\nwe show the function approximations for four operators in Section I.2.\nI.1 Function Approximations for Two Operators\nLemma I.1. Letf1:Rd→Rdand let f2:Rd→Rd.\nAssume the the following conditions\n•Condition 1a. f1is a linear function\n•Condition 1b. ∥f1(x)∥2≤ϵ1∥x∥2(f1is shrinking)\n•Condition 1c. ∥f1(x)−f1(y)∥2≤L1∥x−y∥2(f1is Lipschitz)\n29' metadata={'source': 'pdfs/paper_3.pdf', 'page': 28}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\n•Condition 2a. f2is a linear function\n•Condition 2b. ∥f2(x)∥2≤ϵ2∥x∥2(f2is shrinking)\n•Condition 2c. ∥f2(x)−f2(y)∥2≤L2∥x−y∥2(f2is Lipschitz)\nWe define three functions\n•\ng1(x)=: (I+f1)·(I+f2)(x)\n=x+f2(x)+f1(x+f2(x))\n•\ng2(x)=: (I+f2)·(I+f1)(x)\n=x+f1(x)+f2(x+f1(x))\n•\ng3(x)=: (I+f1+f2)(x)\n=x+f1(x)+f2(x)\nThen we can show that\n•Part 1. ∥g1(x)−g2(x)∥2≤2ϵ1ϵ2∥x∥2(iff1andf2are linear functions)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 29}"
"page_content='•Part 2. ∥g1(x)−g2(x)∥2≤(ϵ2·L1+ϵ1·L2)∥x∥2(iff1andf2are Lipschitz functions)\n•Part 3. ∥g1(x)−g3(x)∥2≤ϵ1ϵ2∥x∥2(iff1is a linear function)\n•Part 4. ∥g1(x)−g3(x)∥2≤ϵ2·L1∥x∥2(iff1is a Lipschitz function)\n•Part 5. ∥g2(x)−g3(x)∥2≤ϵ1ϵ2∥x∥2(iff2is a linear function)\n•Part 6. ∥g2(x)−g3(x)∥2≤ϵ1·L2∥x∥2(iff2is a Lipschitz function)\nProof. Part 1.\nWe have\n∥g1(x)−g2(x)∥2≤ ∥g1(x)−g3(x)∥2+∥g3(x)−g2(x)∥2\n≤ϵ1ϵ2∥x∥2+ϵ1ϵ2∥x∥2\n= 2ϵ1ϵ2∥x∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 29}"
"page_content='where the first step follows from triangular inequality, the second step follows from Part 3 and Part 5 and the last step follows\nfrom simple algebra.\nPart 2.\nWe have\n∥g1(x)−g2(x)∥2≤ ∥g1(x)−g3(x)∥2+∥g3(x)−g2(x)∥2\n≤ϵ2·L1∥x∥2+ϵ1·L2∥x∥2\n= (ϵ2·L1+ϵ1·L2)∥x∥2\nwhere the first step follows from triangular inequality, the second step follows from Part 4 and Part 6 and the last step follows\nfrom simple algebra.\nPart 3.\n30' metadata={'source': 'pdfs/paper_3.pdf', 'page': 29}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nWe have\n∥g1(x)−g3(x)∥2=∥f1(x+f2(x))−f1(x)∥2\n=∥f1(x+f2(x)−x)∥2\n=∥f1(f2(x))∥2\n≤ϵ1·∥f2(x)∥2\n≤ϵ1·ϵ2·∥x∥2,\nwhere the first step follows from the definition of g1andg3, the second step follows from the fact that f1is a linear function, the\nthird step follows from simple algebra, the fourth step follows from Condition 1b and the last step follows from Condition 2b.\nPart 4.\n∥g1(x)−g3(x)∥2=∥f1(x+f2(x))−f1(x)∥2\n≤L1·∥x+f2(x)−x∥2\n=L1·∥f2(x)∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 30}"
"page_content='≤L1·ϵ2∥x∥2,\nwhere the first step follows from definition of g1andg3, the second step follows from Condition 1c, the third step follows\nfrom simple algebra and the last step follows from Condition 2b.\nPart 5.\nWe have\n∥g2(x)−g3(x)∥2=∥f2(x+f1(x))−f2(x)∥2\n=∥f2(x+f1(x)−x)∥2\n=∥f2(f1(x))∥2\n≤ϵ2·∥f1(x)∥2\n≤ϵ2·ϵ1·∥x∥2,\nwhere the first step follows from the definition of g2andg3, the second step follows from the fact that f2is a linear function, the' metadata={'source': 'pdfs/paper_3.pdf', 'page': 30}"
"page_content='third step follows from simple algebra, the fourth step follows from Condition 2b and the last step follows from Condition 1b.\nPart 6.\n∥g2(x)−g3(x)∥2=∥f2(x+f1(x))−f2(x)∥2\n≤L2·∥x+f1(x)−x∥2\n=L2·∥f1(x)∥2\n≤L2·ϵ1∥x∥2,\nwhere the first step follows from definition of g1andg3, the second step follows from Condition 2c, the third step follows from\nsimple algebra and the last step follows from Condition 1b.\nI.2 Function Approximations for Four Operators' metadata={'source': 'pdfs/paper_3.pdf', 'page': 30}"
"page_content='Lemma I.2. For each i∈[4], we assume the following conditions\n•i(a)fiis a linear function\n•i(b)∥fi(x)∥2≤ϵi∥x∥2(fiis shriking)\n•i(c)∥fi(x)−fi(y)∥2≤Li∥x−y∥2(fiis Lipschitz)\nWe define three functions\n•g1(x):=(I+f1)·(I+f2)·(I+f3)·(I+f4)(x)\n•g2(x):=(I+f1)·(I+f3)·(I+f2)·(I+f4)(x)\n•g3(x):=(I+f1+f2+f3+f4)(x)\n31' metadata={'source': 'pdfs/paper_3.pdf', 'page': 30}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nThen, we can show that\n•Part 1. ∥g1(x)−g2(x)∥2≤2(ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2\n(iffi,∀i∈[4]are linear functions)\n•Part 2. ∥g1(x)−g2(x)∥2≤(2L1ϵ2+ 2L1ϵ3+ 2L1ϵ4+L2ϵ3+ 2L2ϵ4+ 2L3ϵ4+ 2L1ϵ2ϵ3+ 2L1ϵ2ϵ4+ 2L1ϵ3ϵ4+\nL2ϵ3ϵ4+2L1ϵ2ϵ3ϵ4+L3ϵ2+L3ϵ2ϵ4)∥x∥2(iffi,∀i∈[4]are Lipschitz functions)\n•Part 3. ∥g1(x)−g3(x)∥2≤(ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='(iffi,∀i∈[4]are linear functions)\n•Part 4. ∥g1(x)−g3(x)∥2≤(L1ϵ2+L1ϵ3+L1ϵ4+L2ϵ3+L2ϵ4+L3ϵ4+L1ϵ2ϵ3+L1ϵ2ϵ4+L1ϵ3ϵ4+L2ϵ3ϵ4+\nL1ϵ2ϵ3ϵ4)∥x∥2(iffi,∀i∈[4]are Lipschitz functions)\n•Part 5. ∥g2(x)−g3(x)∥2≤(ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2\n(iffi,∀i∈[4]are linear functions)\n•Part 6. ∥g2(x)−g3(x)∥2≤(L1ϵ2+L1ϵ3+L1ϵ4+L2ϵ4+L3ϵ2+L3ϵ4+L1ϵ2ϵ3+L1ϵ2ϵ4+L1ϵ3ϵ4+L3ϵ2ϵ4+\nL1ϵ2ϵ3ϵ4)∥x∥2\n(iffi,∀i∈[4]are Lipschitz functions)\nProof. Part 1.\nWe have' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='∥g1(x)−g2(x)∥2≤ ∥g1(x)−g3(x)∥2+∥g3(x)−g2(x)∥2\n≤2(ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2\nwhere the first step follows from triangular inequality and the last step follows from Part 3 and Part 5.\nPart 2.\nWe have\n∥g1(x)−g2(x)∥2≤ ∥g1(x)−g3(x)∥2+∥g3(x)−g2(x)∥2\n≤(2L1ϵ2+2L1ϵ3+2L1ϵ4+L2ϵ3+2L2ϵ4+2L3ϵ4+2L1ϵ2ϵ3+2L1ϵ2ϵ4+2L1ϵ3ϵ4+\nL2ϵ3ϵ4+2L1ϵ2ϵ3ϵ4+L3ϵ2+L3ϵ2ϵ4)∥x∥2\nwhere the first step follows from triangular inequality and the last step follows from Part 4 and Part 6.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='Part 3. We have\n∥g1(x)−g3(x)∥2=∥(I+f1)·(I+f2)·(I+f3)·(x+f4(x))−(I+f1+f2+f3+f4)(x))∥2\n=∥(x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+\nf3(x+f4(x)))+f1(x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x))))\n−(I+f1+f2+f3+f4)(x)))∥2\n=∥f3(f4(x))+f2(f4(x)+f3(x+f4(x)))+f1(f4(x)+\nf3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x)))))∥2\n=∥f3(f4(x))+f2(f4(x))+f2(f3(x))+f2(f3(f4(x)))+\nf1(f4(x))+f1(f3(x))+f1(f3(f4(x)))+f1(f2(x))+f1(f2(f4(x)))\n+f1(f2(f3(x)))+f1(f2(f3(f4(x)))))∥2\n≤ ∥f3(f4(x))∥2+∥f2(f4(x))∥2+∥f2(f3(x))∥2+∥f2(f3(f4(x)))∥2+' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='∥f1(f4(x))∥2+∥f1(f3(x))∥2+∥f1(f3(f4(x)))∥2+∥f1(f2(x))∥2+∥f1(f2(f4(x)))∥2+\n∥f1(f2(f3(x)))∥2+∥f1(f2(f3(f4(x))))∥2\n≤(ϵ3ϵ4+ϵ2ϵ4+ϵ2ϵ3+ϵ2ϵ3ϵ4+ϵ1ϵ4+ϵ1ϵ3+ϵ1ϵ3ϵ4+ϵ1ϵ2+ϵ1ϵ2ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ3ϵ4)∥x∥2\n= (ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2,\nwhere the first step follows from the definition of g1andg3, the second step follows from simple algebra, the third step follows' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='from reorganization, the fourth step follows from the fact that all fi,∀i∈[4]are linear function, the fifth step follows from\ntriangular inequality, the sixth step follows from i(b)and the last step follows from reorganization.\n32' metadata={'source': 'pdfs/paper_3.pdf', 'page': 31}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nPart 4. We have\n∥g1(x)−g3(x)∥2=∥(I+f1)·(I+f2)·(I+f3)·(x+f4(x))−(I+f1+f2+f3+f4)(x))∥2\n=∥x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+\nf3(x+f4(x)))+f1(x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x))))\n−(I+f1+f2+f3+f4)(x))∥2\n=∥f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x)))\n+f1(x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x))))\n−f1(x)−f2(x)−f3(x))∥2\n=∥f3(x+f4(x))−f3(x)+f2(x+f4(x)+f3(x+f4(x)))−f2(x)\n+f1(x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x))))−f1(x)∥' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='≤L3∥x+f4(x)−x∥2+L2∥x+f4(x)+f3(x+f4(x))−x∥2\n+L1∥x+f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x)))−x∥2\n≤L3∥f4(x)∥2+L2∥f4(x)+f3(x+f4(x))∥2\n+L1∥f4(x)+f3(x+f4(x))+f2(x+f4(x)+f3(x+f4(x)))∥2\n≤L3ϵ4∥x∥2+L2ϵ4∥x∥2+L2ϵ3∥x+f4(x)∥2\n+L1ϵ4∥x∥2+L1ϵ3∥x+f4(x)∥2+L1ϵ2∥x+f4(x)+f3(x+f4(x))∥2\n≤L3ϵ4∥x∥2+L2ϵ4∥x∥2+L2ϵ3∥x∥+L2ϵ3ϵ4∥x∥2\n+L1ϵ4∥x∥2+L1ϵ3∥x∥2+L1ϵ3ϵ4∥x∥2+L1ϵ2∥x∥2+L1ϵ2ϵ4∥x∥2+L1ϵ2ϵ3∥x+f4(x)∥2\n≤L3ϵ4∥x∥2+L2ϵ4∥x∥2+L2ϵ3∥x∥+L2ϵ3ϵ4∥x∥2\n+L1ϵ4∥x∥2+L1ϵ3∥x∥2+L1ϵ3ϵ4∥x∥2+L1ϵ2∥x∥2+L1ϵ2ϵ4∥x∥2+L1ϵ2ϵ4∥x∥2+L1ϵ2ϵ3ϵ4∥x∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='= (L3ϵ4+L2ϵ4+L2ϵ3+L2ϵ3ϵ4+L1ϵ4+L1ϵ3+L1ϵ3ϵ4+L1ϵ2+L1ϵ2ϵ4+L1ϵ2ϵ3+L1ϵ2ϵ3ϵ4)∥x∥2\n= (L1ϵ2+L1ϵ3+L1ϵ4+L2ϵ3+L2ϵ4+L3ϵ4+L1ϵ2ϵ3+L1ϵ2ϵ4+L1ϵ3ϵ4+L2ϵ3ϵ4+L1ϵ2ϵ3ϵ4)∥x∥2\nwhere the first step follows from the definition of g1andg3, the second step follows from simple algebra, the third step follows\nfrom simple algebra, the fourth step follows from reorganization, the fifth step follows from the fact that all fi,∀i∈[4]are Lips-' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='chitz functions, the sixth step follows from simple algebra, the seventh step follows from i(b), the eighth step follows from trian-\ngular inequality, the ninth step follows from i(b), the tenth step follows from i(b)and the last step follows from reorganization.\nPart 5. We have\n∥g2(x)−g3(x)∥2=∥(I+f1)·(I+f3)·(I+f2)·(x+f4(x))−(I+f1+f2+f3+f4)(x))∥2\n=∥(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+\nf2(x+f4(x)))+f1(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+f2(x+f4(x))))\n−(I+f1+f2+f3+f4)(x)))∥2\n=∥f2(f4(x))+f3(f4(x))+' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='f3(f2(x+f4(x)))+f1(f4(x))+f1(f2(x+f4(x)))+f1(f3(x+f4(x)+f2(x+f4(x)))))∥2\n≤(ϵ2ϵ4+ϵ3ϵ4+ϵ3ϵ2+ϵ3ϵ2ϵ4+ϵ1ϵ4+ϵ1ϵ2+ϵ1ϵ2ϵ4+ϵ1ϵ3+ϵ1ϵ3ϵ4+ϵ1ϵ3ϵ2+ϵ1ϵ3ϵ2ϵ4)∥x∥2\n= (ϵ1ϵ2+ϵ1ϵ3+ϵ1ϵ4+ϵ2ϵ3+ϵ2ϵ4+ϵ3ϵ4+ϵ1ϵ2ϵ3+ϵ1ϵ2ϵ4+ϵ1ϵ3ϵ4+ϵ2ϵ3ϵ4+ϵ1ϵ2ϵ3ϵ4)∥x∥2,\nwhere the first step follows from the definition of g2andg3, the second step follows from simple algebra, the third step follows\nfrom the fact that all fi,∀i∈[4]are linear function, the fourth step follows from triangular inequality and i(b), and the last' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='step follows from reorganization.\n33' metadata={'source': 'pdfs/paper_3.pdf', 'page': 32}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nPart 6. We have\n∥g2(x)−g3(x)∥2=∥(I+f1)·(I+f3)·(I+f2)·(x+f4(x))−(I+f1+f2+f3+f4)(x))∥2\n=∥(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+f2(x+f4(x)))\n+f1(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+f2(x+f4(x))))\n−(I+f1+f2+f3+f4)(x)))∥2\n=∥f2(x+f4(x))−f2(x)+f3(x+f4(x)+f2(x+f4(x)))−f3(x)\n+f1(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+f2(x+f4(x))))−f1(x)∥2\n≤ ∥f2(x+f4(x))−f2(x)∥2+∥f3(x+f4(x)+f2(x+f4(x)))−f3(x)∥2\n+∥f1(x+f4(x)+f2(x+f4(x))+f3(x+f4(x)+f2(x+f4(x))))−f1(x)∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='≤L2ϵ4∥x∥2+L3ϵ4∥x∥2+L3ϵ2∥x+f4(x)∥2\n+L1ϵ4∥x∥2+L1ϵ2∥x+f4(x)∥2+L1ϵ3∥x+f4(x)+f2(x+f4(x))∥2\n≤L2ϵ4∥x∥2+L3ϵ4∥x∥2+L3ϵ2∥x∥2+L3ϵ2ϵ4∥x∥2\n+L1ϵ4∥x∥2+L1ϵ2∥x∥2+L1ϵ2ϵ4∥x∥2+L1ϵ3∥x∥+L1ϵ3ϵ4∥x∥2+L1ϵ3ϵ2∥x∥2+L1ϵ3ϵ2ϵ4∥x∥2\n= (L1ϵ2+L1ϵ3+L1ϵ4+L2ϵ4+L3ϵ2+L3ϵ4+L1ϵ2ϵ3+L1ϵ2ϵ4+L1ϵ3ϵ4+L3ϵ2ϵ4+L1ϵ2ϵ3ϵ4)∥x∥2\nwhere the first step follows from the definition of g2andg3, the second step follows from simple algebra, the third step follows' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='from reorganization, the fourth step follows from triangular inequality, the fifth step follows from the fact that all fi,∀i∈[4]are\nLipschitz functions and i(b), the sixth step follows from triangular inequality, and the last step follows from reorganization.\nJ Nearest Neighbor Search Data Structure\nWe use the reduction-based approximate MaxIP method with LSH data-structure to achieve sublinear iteration cost. Note' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='that we choose this method due to its clear theoretical guarantee on the retrieval results. It is well-known that an LSH\ndata-structures is used for approximate nearest neighbor problem. The following definition of approximate nearest neighbor\nsearch is very standard in literature (Arya & Mount, 1993; Indyk & Motwani, 1998a; Datar et al., 2004; Andoni et al., 2014;\n2015; Andoni & Razenshteyn, 2015; Indyk & Wagner, 2018; Andoni et al., 2017; 2018; Dong et al., 2019; Chen et al., 2020b;' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='Li & Li, 2022; Li et al., 2019).\nJ.1 LSH andMaxIP\nWe start with the defining the Approximate Nearest Neighbor ( ANN ) problem (Arya & Mount, 1993; Indyk & Motwani,\n1998a; Datar et al., 2004; Andoni et al., 2014; 2015; Andoni & Razenshteyn, 2015; Indyk & Wagner, 2018; Andoni et al.,\n2017; 2018; Dong et al., 2019; Chen et al., 2020b) as:\nDefinition J.1 (Approximate Nearest Neighbor ( ANN )).Letc>1andr∈(0,2)denote two parameters. Given an n-vector' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='setY⊂Sd−1on a unit sphere, the objective of the (c,r)-Approximate Nearest Neighbor ( ANN ) is to construct a data structure\nthat, for any query x∈Sd−1such that miny∈Y∥y−x∥2≤r, it returns a vector zfromYthat satisfies ∥z−x∥2≤c·r.\nTheANN problem can be solved via locality sensitive hashing ( LSH) (Indyk & Motwani, 1998a; Datar et al., 2004; Indyk\n& Wagner, 2018). In this paper, we use the standard definitions of LSH (see Indyk and Motwani (Indyk & Motwani, 1998a)).' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='Definition J.2 (Locality Sensitive Hashing) .Letc >1denote a parameter. Let p1,p2∈(0,1)denote two parameters and\np1> p2. We say a function family His(r,c·r,p1,p2)-sensitive if and only if, for any vectors x,y∈Rd, for any hchosen\nuniformly at random from H, we have:\n• if∥x−y∥2≤r, then Prh∼H[h(x)=h(y)]≥p1,\n• if∥x−y∥2≥c·r, then Prh∼H[h(x)=h(y)]≤p2.\nNext, we show that LSH solves ANN problem with sublinear query time complexity.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='Theorem J.3 (Andoni, Laarhoven, Razenshteyn and Waingarten (Andoni et al., 2017)) .Letc>1andr∈(0,2)denote two\nparameters. One can solve (c,r)-ANN on a unit sphere in query time O(d·nρ)using preprocessing time O(dn1+o(1))and\nspace O(n1+o(1)+dn), where ρ=2\nc2−1\nc4+o(1).\n34' metadata={'source': 'pdfs/paper_3.pdf', 'page': 33}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nHere we write o(1)is equivalent to O(1/√logn). Note that we could reduce dtono(1)with Johnson–Lindenstrauss\nLemma (Johnson & Lindenstrauss, 1984). Besides, we could achieve better ρusing LSH in (Andoni & Razenshteyn, 2015)\nif we allowed to have more proprocessing time.\nIn this work, we focus on a well-known problem in computational complexity: approximate MaxIP . In this work, we follow' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='the standard notation in (Chen, 2018) and define the approximate MaxIP problem as follows:\nDefinition J.4 (Approximate MaxIP ).Letc∈(0,1)andτ∈(0,1)denote two parameters. Given an n-vector dataset Y⊂Sd−1\non a unit sphere, the objective of the (c,τ)-MaxIP is to construct a data structure that, given a query x∈Sd−1such that\nmax y∈Y⟨x,y⟩≥τ, it retrieves a vector zfromYthat satisfies ⟨x,z⟩≥c·max y∈Y⟨x,y⟩.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='In many applications, it is more convenient to doing inner product search in a transformed/projected space compared to doing\ninner product search in the original space. Thus, we propose the following definitions (Definition J.5 and Definition J.6)\nDefinition J.5 (Projected MaxIP ).Letϕ,ψ:Rd→Rkdenote two transforms. Given a data set Y⊆Rdand a point x∈Rd,\nwe define (ϕ,ψ)-MaxIP as follows:\n(ϕ,ψ)-MaxIP (x,Y):=max\ny∈Y⟨ϕ(x),ψ(y)⟩' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='Definition J.6 (Projected approximate MaxIP ).Letϕ,ψ:Rd→Rkdenote two transforms. Given an n-vector dataset\nY⊂Rdso that ψ(Y)⊂Sk−1, the goal of the (c,ϕ,ψ,τ )-MaxIP is to construct a data structure that, given a query x∈Rdand\nϕ(x)∈Sk−1such that max y∈Y⟨ϕ(x),ψ(y)⟩≥τ, it retrieves a vector z∈Ythat satisfies ⟨ϕ(x),ψ(z)⟩≥c·(ϕ,ψ)-MaxIP (x,Y).\nJ.2 Connections\nFact J.7. Letexdenote the vector that ⟨ex,x⟩≥1−1\n2ϵ2, where both exandxare unit vectors. We have\n∥ex−x∥2≤ϵ\nProof.\n∥ex−x∥2= (∥ex∥2\n2+∥x∥2' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='2−2⟨x,ex⟩)1/2\n= (2−2⟨x,ex⟩)1/2\n≤(2−2(1−1\n2ϵ2))1/2\n=ϵ\nNow, we complete the proof.\nLemma J.8. Letexdenote the vector that ⟨ex,x⟩≥1−1\n2ϵ2, where both exandxare unit vectors. Let 0.01c·τ >ϵ .\nSuppose there is a z∈Y, where ∥z∥2=1, such that\n⟨x,z⟩≥c·max\ny∈Y⟨x,y⟩\nNote that max y∈Y⟨x,y⟩≥τ. Then, we can find a z∈Ysuch that\n⟨ex,z⟩≥1\n2c·max\ny∈Y⟨x,y⟩\nProof. We have\n⟨ex,z⟩=⟨x,z⟩+⟨ex−x,z⟩\n≥ ⟨x,z⟩−|⟨ex−x,z⟩|\n≥ ⟨x,z⟩−∥ex−x∥2·∥z∥2\n≥ ⟨x,z⟩−ϵ\n≥c·max\ny∈Y⟨x,y⟩−ϵ\n≥0.99·c·max\ny∈Y⟨x,y⟩' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='where the first step follows from simple algebra, the second step follows from the fact that ⟨x, y⟩ ≥ −|⟨ x, y⟩|, the\nthird step follows from the property of inner product, the fourth step follows from Fact J.7, the fifth step follows from\n⟨x,z⟩≥c·max y∈Y⟨x,y⟩and the final step follows from the fact that 0.01c·τ >ϵ .\n35' metadata={'source': 'pdfs/paper_3.pdf', 'page': 34}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nJ.3 Efficient Transformations\nWe have learned from that (c,τ)-MaxIP on a unit sphere Sd−1using LSH forANN . Therefore, the next step is to transform\nthe direction search procedure in iterative optimization algorithm into a MaxIP on a unit sphere. To achieve this, we formulate\nthe direction search as a projected approximate MaxIP (see Definition J.5). We start with presenting a pair of transformation' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='ϕ0,ψ0:Rd→Rd+1such that, given a function g:Rd→R, for any x,yin a convex set K, we have\nϕ0(x):=[∇g(x)⊤,x⊤∇g(x)]⊤, ψ0(y):=[−y⊤,1]⊤. (11)\nIn this way, we show that\n⟨y−x,∇g(x)⟩=−⟨ϕ0(x),ψ0(y)⟩,\nargmin\ny∈Y⟨y−x,∇g(x)⟩= argmax\ny∈Y⟨ϕ0(x),ψ0(y)⟩ (12)\nTherefore, we could transform the direction search problem into a MaxIP problem.\nNext, we present a standard transformations (Neyshabur & Srebro, 2015) that connects the MaxIP toANN in unit sphere.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='For any x,y∈Rd, we propose transformation ϕ1,ψ1:Rd→Rd+2such that\nϕ1(x)=h\n(D−1\nxx)⊤0q\n1−∥xD−1x∥2\n2i⊤\nψ1(y)=h\n(D−1\nyy)⊤q\n1−∥yD−1y∥2\n20i⊤\n(13)\nHereDx,Dyare some constant that make sure both x/D xandy/D yhave norms less than 1. Under these transformations,\nbothϕ1(x)andψ1(y)have norm 1andargmax y∈Y⟨ϕ1(x),ψ1(y)⟩=argmax y∈Y⟨x,y⟩.\nCombining transformations in Eq. (11) and Eq. (13), we obtain query transform ϕ:Rd→Rd+3with form ϕ(x)=ϕ1(ϕ0(x))' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='and data transform ϕ:Rd→Rd+3with form ψ(y)=ψ1(ψ0(y)). Using ϕandψ, we transform the direction search problem in\noptimization into a MaxIP in unit sphere. Moreover, given a set Y⊂Rdand a query x∈Rd, the solution zof(c,ϕ,ψ,τ )-MaxIP\nover(x,Y)has the propriety that ⟨z−x,∇g(x)⟩≤c·miny∈Y⟨y−x,∇g(x)⟩. Thus, we could approximate the direction search\nwithLSH based MaxIP data-structure.\nNote that only MaxIP problem with positive inner product values could be solved by LSH. We found the direction search' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='problem naturally satisfies this condition. We show that if gis convex, given a set S⊂Rd, we have mins∈S⟨∇g(x),s−x⟩≤0\nfor any x∈B(S), where Bis the convex hull of S. Thus, max y∈Y⟨ϕ0(x),ψ0(y)⟩is non-negative following Eq. (12).\nJ.4 Data Structures\nIn this section, we present a formal statement that solves (c,τ)-MaxIP problem on unit sphere using LSH for(c,r)-ANN .\nTheorem J.9. Letc∈(0,1)andτ∈(0,1). Given a set of n-vector set Y⊂ Sd−1on the unit sphere, there exists a data' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='structure with O(dn1+o(1))preprocessing time and O(n1+o(1)+dn)space so that for any query x∈Sd−1, we take O(d·nρ)\nquery time to retrieve the (c,τ)-MaxIP ofxinYwith probability at least 0.97, where ρ:=2(1−τ)2\n(1−cτ)2−(1−τ)4\n(1−cτ)4+o(1)\nProof. We know that ∥x−y∥2\n2=2−2⟨x,y⟩for all x,y∈Sd−1. In this way, if we have a LSH data-structure for (c,r)-ANN .\nIt could be used to solve (c,τ)-MaxIP withτ=1−0.5r2andc=1−0.5c2r2\n1−0.5r2. Next, we write c2as\nc2=1−c(1−0.5r2)\n0.5r2=1−cτ\n1−τ.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='Next, we show that if the LSH is initialized following Theorem J.3, it takes query time O(d·nρ), space O(n1+o(1)+dn)and\npreprocessing time O(dn1+o(1))to solve (c,τ)-MaxIP through solving (c,r)-ANN , where\nρ=2\nc2−1\nc4+o(1)=2(1−τ)2\n(1−cτ)2−(1−τ)4\n(1−cτ)4+o(1).\nIn practice, cis increasing as we set parameter τclose to MaxIP (x,Y). There is also another LSH data structure (Andoni\n& Razenshteyn, 2015) with longer preprocessing time and larger space that could solve the (c,τ)-MaxIP with similar query' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='7It is obvious to boost probability from constant to δby repeating the data structure log(1/δ)times.\n36' metadata={'source': 'pdfs/paper_3.pdf', 'page': 35}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\ntime complexity. We refer readers to Section 8.2 in (Shrivastava et al., 2021) for more details8. Moreover, Corollary J.9 could\nbe applied to projected MaxIP problem.\nTheorem J.10. Letc∈(0,1)andτ∈(0,1). Letϕ,ψ:Rd→Rkdenote two transforms. Let Tϕdenote the time to compute\nϕ(x)andTψdenote the time to compute ψ(y). Given a set of n-points Y∈Rdwithψ(Y)⊂ Sk−1on the sphere, one can' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='construct a data structure with O(dn1+o(1)+Tψn)preprocessing time and O(n1+o(1)+dn)space so that for any query\nx∈Rdwithϕ(x)∈Sk−1, we take query time complexity O(d·nρ+Tϕ)to solve (c,ϕ,ψ,τ )-MaxIP with respect to (x,Y)with\nprobability at least 0.9, where ρ:=2(1−τ)2\n(1−cτ)2−(1−τ)4\n(1−cτ)4+o(1).\nProof. The preprocessing phase can be decomposed in two parts.\n• It takes O(Tψn)time to transform every y∈Yintoψ(y).' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='• It takes O(O(dn1+o(1))time and O(dn1+o(1)+dn)to index every ψ(y)intoLSH using Theorem J.9.\nThe query phase can be decomposed in two parts.\n• It takes O(Tϕ)time to transform every x∈Rdintoϕ(x).\n• It takes O(d·nρ)time perform query for ϕ(x)inLSH using Theorem J.9.\nK Self-attention layer as a clustering algorithm\nThe self-attention layer in the Transformer looks like mean-shift clustering. Suppose {(xj,vj)}are a bunch of key and value' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='pairs and qis the query. Note that q=Wqx,k=Wkxandv=Wvxare computed by three projection matrices Wk,Wq\nandWvfrom a common x. Then from self-attention we have:\nv=X\njpjvj=P\njexp(x⊺W⊺\nqWkxj)WvxjP\njexp(x⊺W⊺\nqWkxj)=WvP\njexp(x⊺W⊺\nqWkxj)xjP\njexp(x⊺W⊺\nqWkxj)(14)\nwhere∼(q,kj):=exp( q⊺kj)=exp( x⊺W⊺\nqWkxj)andpj=∼(q,kj)/P\nj∼(q,kj).\nOn the other hand, mean-shift clustering looks like the following:\nm(x)=P\njK(xj,x)xjP\njK(xj,x)(15)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='where K(xj,x)is a kernel matrix that measure the similarity between xjandx. According to the mean-shift algorithm,\nin the next iteration, we will simply replace xwithm(x).\nSo in some sense, self-attention is just to do some kind of clustering for the input embedding qandk, plus a transformation of the\nembedding to another place. The term “projection” is due to the fact that there is a projection matrix Wvonxfor the next level.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='Residue connection and LayerNorm . Compared to mean-shift, Transformer layer has residue connection. Therefore, for\nsingle-headed attention, what you actually get is v+x, followed by a LayerNorm. For the residue connection, the mean-shift\nanalog already shows the output m(x)contains x+part. The reason why we need residue connection is that the self-attention\npart might only model the “change” of xin the mean-shift picture, rather than the full update of x.\nL The role of self-attention' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='Consider we have a vocabulary of size mandddimensional embedding space. In practice, many papers in NLP have reported\nclustering behaviors of word embeddings: such a clustering of word embedding naturally occurs after training.\nAn explanation for the above phenomenon is that, by grouping these word embedding together, we might generalize better, since\nsimilarity in word now can transfer (e.g., A linked to B, B linked to C, then A might link to C as well) and generalization follows.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='Let’s treat it as a fact and focus on how this is achieved and how self-attention plays a role here.\n8Recently, there a line of work that use fast MaxIP data structure to speedup the iterative-type optimization algorithms (Shrivastava et al.,\n2021; Song & Ye, 2023; Qin et al., 2023a; Song et al., 2023a).\n37' metadata={'source': 'pdfs/paper_3.pdf', 'page': 36}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nL.1 The capacity of embedding layer\nFirst let us take a look at the following pairwise distance constraints between word embedding (e.g., some words should\nbe close to each other, some should be far away from each other) as the following:\n∥xi−xj∥=D(i,j) (16)\nwhere D(i,j)is large for iandjthat should be far apart and D(i,j)is small for iandjthat are close to each other. In' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='visualization, this is called Multidimensional Scaling (MDS) (Cox & Cox, 2008).\nNote that in neural network training, the constraint (Eqn. 16) is not directly enforced during training, but the clustering\nnaturally happens. Since we talk about capacity, how we achieve Eqn. 16 doesn’t matter for now.\nIn general we cannot find a fixed low-dimensional embedding ( d≪m) to satisfy these constraints, since we only have md' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='parameters ( mvectors, each has dentries), but m2constraint. So two vectors that are supposed to be close may not be close\nenough (but hopefully they remain close to each other).\nL.2 The role of self-attention\nFor this, the self-attention mechanism comes to the rescue, trading model-size with additional computation. It fulfills what\n(static) embedding cannot achieve: to further group the embedding vectors together in a multi-layer structure.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='Note that one sentence never covers all dvocabularies. Once the words in the sentence are picked, they are grouped together\nvia self-attention layers to collectively represent a concept that can be useful for the task.\nL.3 How the clustering happens through self-attention?\nNow one fundamental questions arise: How the static clustering of embedding happens during end-to-end training? In\npractice, no one explicitly enforces the MDS constraint (Eqn. 16).' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='Let’s start with a simple example. we have two unit embedding: xandywith the normalization condition that ∥x∥2= 1\nand∥y∥2=1, and a simple self-attention layer (without projection) which output z:\nz=(1−p)x+py (17)\nWhere the attention map is:\np=ex⊺y\nex⊺x+ex⊺y=1\n1+e1−x⊺y(18)\nNote that here we attend to xso0<p< 1/2always. The last two is due to normalization condition.\nNow we consider a loss function L=−1\n2∥z∥2\n2. The intuition behind is that “for some reason, we found that zis a good' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='representation for our task, and want to make sure its length is as long as possible”.\nUnder this context, what would be the gradient rule for xandy? Will they cluster together?\nThe answer is yes! We could compute\n∂z\n∂x= (1 −p)I+∂p\n∂x(y−x)⊺(19)\n∂z\n∂y=pI+∂p\n∂y(y−x)⊺(20)\nLett:=1−x⊺yand define the following function with respect to t:\nf(t):=(x−y)⊺z=(1−2p)(1−x⊺y)>0 (21)\nTherefore, we can compute the gradient for xand gradient for y:\n−gx:=−∂L\n∂x=−∂z\n∂x∂L\n∂z=(1−p)2x+p(1−p)(1−f(t))y (22)\n−gy:=−∂L' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='∂y=−∂z\n∂y∂L\n∂z=p2y+p(1−p)(1−f(t))x (23)\nNote that since xandyare kept to be normalized, the term (1−p)2xin∂L/∂ xis gone (and similarly p2yforgy). So how\nxandymove depends on the sign of 1−f(t).\nWith some computation, we could see 0< f(t)<1when t <1.5424 . In summary, if x⊺y>−0.4576 , then the (negative)\ngradient of xpushes it towards yand pushes xtowards y, and the clustering of static embedding happens during training.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='Note that since both xandyare normalized, −1≤x⊺y≤1, so this is a quite loose condition and can be easily satisfied.\n38' metadata={'source': 'pdfs/paper_3.pdf', 'page': 37}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nL.4 Multiple embeddings\nPeople might wonder what happen to multiple unit embeddings x,y1,y2,...,yK? In this case, we can similarly define\nself-attention probability pi(note that here we consider the case that every embedding attends to x):\npi:=ex⊺yi\nex⊺x+P\njex⊺yj=ex⊺yi\n1+P\njex⊺yj(24)\nDefine pS:=PK\ni=1pi=1−1\n1+P\njex⊺yj<1and we have:\nz=(1−pS)x+X\nipiyi (25)\nLet˜pi:=pi/pSbe the (normalized) probability on yiand¯y:=1\npSP\nipiyi=P' metadata={'source': 'pdfs/paper_3.pdf', 'page': 38}"
"page_content='i˜piyibe the weighted mean of {yi}other\nthanx, then we have:\nz=(1−pS)x+pS¯y (26)\nNow we can still compute the partial derivative:\n∂pj\n∂x=pj[−pS¯y+yj] (27)\n∂pj\n∂yi=pi[−pj+I(i=j)]x (28)\nwhich gives\n∂z\n∂x= (1 −pS)I+X\nj∂pj\n∂x(yj−x)⊺(29)\n∂z\n∂yi=piI+X\nj∂pj\n∂yi(yj−x)⊺(30)\nAfter some manipulation, we have:\n∂z\n∂x=(1−pS)[I+pS¯y(¯y−x)⊺]+pSQ (31)\nwhere Q:=P\nj˜pj(yj−¯y)(yj−¯y)⊺is the weighted covariance matrix of data points {yj}.' metadata={'source': 'pdfs/paper_3.pdf', 'page': 38}"
"page_content='Similar to the two unit case, we want to check −gxto see how the embedding xchanges over time.\n−gx=−∂L\n∂x=−∂z\n∂x∂L\n∂z(32)\n= (1 −pS)2x+pS\x02\n(1−2pS)x⊺¯y−(1−pS)+pS∥¯y∥2\x03¯y+pSQz\nIf things are already quite clustered, then ∥¯y∥≈1(usually ∥¯y∥2<1since sphere is a convex set), Qz≈0(since Qspans\non the tangent space of zat the sphere and zis perpendicular to it), and we have:\n−gx≈(1−pS)2x+pS(1−2pS)(x⊺¯y−1)¯y (33)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 38}"
"page_content='It is clear that x⊺¯y<1. When pS>1/2, which is high likely for large K, then−gxhas positive component of ¯yandxwill\nmove towards ¯y.\nOn the other hand, we could also check\n∂z\n∂yi=pi[I+(1−pS)x(¯y−x)⊺]+pix(yi−¯y)⊺(34)\nwhich gives an expression of −gy:\n· (35)\nWith the same argument, it moves towards ¯y(so all yiwill cluster together) and towards x.\nWhen there is a WkandWqbefore the embedding, following the same logic, only the column subspace of Wk(orWq) will' metadata={'source': 'pdfs/paper_3.pdf', 'page': 38}"
"page_content='be clustered together. On the other hand, the value part will be different in order to enable encoding of more complicated\nconcepts based on co-occurrence of multiple tokens.\nM Link self-attention with generative models.\nConsider the following self-attention structure. Consider an embedding matrix X∈Rn×dand for embedding xiandxj, let\nyij=ϕ(xi;xj):=(1−βij)xi+βijxj, β ij:=ex⊺\nixj\nex⊺\nixi+ex⊺\nixj(36)\n39' metadata={'source': 'pdfs/paper_3.pdf', 'page': 38}"
"page_content='Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time\nHere ϕ(xi;xj) :=xi+βij(xj−xi)is the self-attention operation. More properties of this operator ϕneed to be explored.\nThen we want to maximize the following objective:\nmax\nX,∥xi∥2=1X\nijkP(k|i,j)y⊺\nijxk (37)\nor more formally, using a softmax to avoid trivial solution xi≡x, we have:\nmax\nX,∥xi∥2=1J:= max\nX,∥xi∥2=1X\nijkP(k|i,j)logδijk, δijk:=ey⊺\nijxk\nP\nkey⊺\nijxk(38)\nwhich is:\nmax\nX,∥xi∥2=1X\nijkP(k|i,j)""\ny⊺\nijxk−logX\nkey⊺\nijxk#\n(39)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 39}"
"page_content='We can compute its gradient update. Here we assume the index knever appears in index iandj(encoding and decoding\nmatrices are decoupled), then by gradient rule, we have:\n˙xk=∂L\n∂xk=P⊥\nxkX\nijP(k|i,j)(1−δijk)yij (40)\nwhere P⊥\nxkis the projection matrix that projects a vector to the orthogonal complement space of xk. The projection is due\nto the constraint ∥xk∥2=1. If the training converges ( ˙xk=0), then we know thatX\nijP(k|i,j)(1−δijk)yij=γxk (41)' metadata={'source': 'pdfs/paper_3.pdf', 'page': 39}"
"page_content='for some γ >0(note that γ <0will be an unstable stationary point).\nDepending on different structure of the generative model specified by P(k|i,j), we might end up learning different embedding\nmatrix X.\nThe first thing we want to check is independency. Assume that for some specific token kandi, we have P(k|i,j) =P(k|i)\nfor any j, which means that the frequency of token khas nothing to do with the second entry j. Furthermore, token kis not' metadata={'source': 'pdfs/paper_3.pdf', 'page': 39}"
"page_content='connected with other token i′̸=i, i.e,P(k|i′,j)≡0. If we just let δijk=δ >0, then we have:\nP(k|i)X\njyij=γ′xk (42)\nwhich yields\nP(k|i)nxi+X\njβij(xj−xi)=γ′xk (43)\nAnd we could possibly show thatP\njβij(xj−xi)≈0since βij= 1/(1+e1−x⊺\nixj)applies equal weights for embeddings\naround xiand they cancel out. Therefore, xkis aligned with xi.\nAnother thing we might want to check is identification of two tokens. Assume that there exists two tokens j1andj2and' metadata={'source': 'pdfs/paper_3.pdf', 'page': 39}"
"page_content='specific kandi, so that P(k|i,j1)=P(k|i,j2). For other k,i,j combination P(k|i,j)≡0, then we have:\nP(k|i,j1)yij1=γ1xk (44)\n(not sure how to continue).\nIf we have Wq,WkandWv, then the formulation doesn’t change that much. The only difference here is that now\nβij:=ex⊺\niWpqxj\nex⊺\niWpqxi+ex⊺\niWpqxj(45)\nandy⊺\nijxknow becomes y⊺\nijWvxk.\n40' metadata={'source': 'pdfs/paper_3.pdf', 'page': 39}"
"page_content='Giraffe: Adventures in Expanding Context Lengths in LLMs\nArka Pal∗, Deep Karkhanis, Manley Roberts,\nSamuel Dooley, Arvind Sundararajan, Siddartha Naidu\nAbacus.AI\nAbstract\nModern large language models (LLMs) that rely on attention mechanisms are typically trained with\nfixed context lengths which enforce upper limits on the length of input sequences that they can handle\nat evaluation time. To use these models on sequences longer than the train-time context length, one' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='might employ techniques from the growing family of context length extrapolation methods — most of\nwhich focus on modifying the system of positional encodings used in the attention mechanism to indicate\nwhere tokens or activations are located in the input sequence. We conduct a wide survey of existing\nmethods of context length extrapolation on a base LLaMA or LLaMA 2 model, and introduce some of' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='our own design as well — in particular, a new truncation strategy for modifying the basis for the position\nencoding.\nWe test these methods using three new evaluation tasks (FreeFormQA, AlteredNumericQA, and\nLongChat-Lines) as well as perplexity, which we find to be less fine-grained as a measure of long context\nperformance of LLMs. We release the three tasks publicly as datasets on HuggingFace. We discover' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='that linear scaling is the best method for extending context length, and show that further gains can be\nachieved by using longer scales at evaluation time. We also discover promising extrapolation capabilities\nin the truncated basis. To support further research in this area, we release three new 13B parameter\nlong-context models which we call Giraffe : 4k and 16k context models trained from base LLaMA-13B,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='and a 32k context model trained from base LLaMA2-13B. We also release the code to replicate our\nresults.1\n1 Introduction\nIn recent years, transformers [1] have become the dominant neural network architecture in a variety of\nnatural language modelling tasks [2, 3], by dint of their flexibility and their amenability to being trained\non extremely large datasets [4, 5]. Subsequently, a popular term that has been adopted for these neural' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='networks is ‘Large Language Models’ (LLMs) — with the ‘Large’ referring both to the training dataset size\nas well as their parameter count (and indeed, the associated training and environmental cost).\nA key element of the standard transformer architecture is its inherent insensitivity to the ordering of\nthe input elements. Attention is naturally a set-like operation in which the position of the elements does' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='not matter [1]. However, the order of elements is crucial for many important tasks such as parsing natural\nlanguage, coding, forecasting, etc. Thus it is necessary to inject positional information into the inputs of the\nLLM, typically in the form of positional encodings.\nOne possible desideratum of a positional encoding scheme is context length extrapolation : the ability\nto use the LLM for inference on input lengths longer than those it was trained on. Due to the quadratic' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='complexity growth of the attention mechanism in transformers, it is often infeasible to train on large context\nlengths. The benefit of increased context length is diverse - allowing reading longer documents and papers,\n∗Correspondence to arka@abacus.ai .\n1Github repo at: https://github.com/abacusai/Long-Context .\n1arXiv:2308.10882v1  [cs.AI]  21 Aug 2023' metadata={'source': 'pdfs/paper_1.pdf', 'page': 0}"
"page_content='more internal consistency in long conversations with users in LLM-powered chatbots, working on bigger\ncodebases, and so on. We can break context length extrapolation down into two main paradigms. First,\nthere is finetuned extrapolation where a model previously pretrained on shorter contexts is allowed to finetune,\nor update model weights based on the longer context length. Additionally, there is zero-shot extrapolation' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='where a model previously pretrained on short contexts is immediately evaluated on longer context lengths\nwith the same weights as the shorter context model.\nIn this paper, we focus primarily on zero-shot extrapolation and make the following key contributions :\nBenchmark of different context extrapolation schemes We conduct a survey of methods for context\nlength extrapolation with a pretrained base model, and try a few of our own inventions as well. In particular,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='we present a new truncated basis for position encodings. The focus in this paper on pretrained models is\nalso different from other work in the literature [6, 7], which tend to instead train from scratch with a\nchosen positional encoding scheme. As mentioned above, although LLMs have been successful, training\nthem is a costly enterprise. Well known closed-source model include GPT-4 [8] and Claude [9]. Recently the' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='open-source LLaMA [10] has been released by a team at Meta AI, and this was followed by the improved\nLLaMA2 [11]. In our view, the resources required to train competitive base models of this nature will remain\nconstrained to a few large players. Therefore, it is imperative to be able to modify the models as desired for\nthe end user—ideally, with a fraction of the compute power applied.\nOur main findings are that:\n•Linear interpolation is the best as a context length extrapolation method.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='•All context length extrapolation methods show degradation on task accuracy, even for lengths where\nthey provide otherwise coherent output (and perplexity scores are still reasonable).\n•Further context length increase can be achieved by utilising a higher scale factor at evaluation time\nthan finetune time, but seemingly only up to a factor of 2x.\nPublic release of LLM weights and evaluation datasets We release the weights of two new 13B' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='models trained from base LLaMA with an extended context length of 16k2and a context length of 4k3\non HuggingFace. We also release a 13B model trained to a length of 32k from base LLaMA 24. We call\nthis family of models Giraffe . In addition, we release three datasets (LongChat-Lines5, FreeFormQA6and\nAlteredNumericQA7) to evaluate long context performance of these, and other, models. LongChat-Lines is' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='a key-value fine-grained retrieval task. FreeFormQA and AlteredQA are question-answering datasets based\non the Natural Questions Dataset [12]. Some existing work [6, 7] focuses only on perplexity on a document\ncorpus evaluation set as their measure of extrapolation performance. We find that perplexity scores are not\nas sensitive a measure of long context performance as our introduced tasks.\n2 Related Work' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='RoPE In this work, we examine the efficacy of the positional encoding choice of LLaMA [10] for context\nlengths longer than the base model was trained on. The positional encoding used by LLaMA is RoPE (Rotary\nPosition Embedding) [13]. RoPE works by rotating slices of the query and key projection matrices at different\nspeeds. Thus for example even if the query and key are projected to the same encoding, they will be rotated' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='by different amounts depending on their position in the sequence. If they are subsequently unaligned, their\ndot product will be smaller relative to what it would be if they were not rotated at all. Conversely, they\n2https://huggingface.co/abacusai/Giraffe-v1-delta-13b-scaled-16\n3https://huggingface.co/abacusai/Giraffe-v1-delta-13b-scaled-4\n4https://huggingface.co/abacusai/Giraffe-v2-13b-32k\n5https://huggingface.co/datasets/abacusai/LongChat-Lines' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='6https://huggingface.co/datasets/abacusai/WikiQA-Free_Form_QA\n7https://huggingface.co/datasets/abacusai/WikiQA-Altered_Numeric_QA\n2' metadata={'source': 'pdfs/paper_1.pdf', 'page': 1}"
"page_content='could become more aligned, leading to a larger dot product and attention score. In RoPE, this rotation is\nhappening at different speeds on all 2-slices of the query and key in the embedding dimension, allowing the\nmodel to build a complex function of attention scores over distances. One of the main appeals of utilising\nthe RoPE method is that it ensures mathematically that the attention score function is dependent only on' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='therelative distance between a query and a key, rather than their absolute positions. This is considered to\nbe a desirable property of LLMs [13, 14].\nALiBi Although RoPE was successful in this aim, the work on ALiBi [6] demonstrated that RoPE was not\nable to perform zero-shot context length extrapolation. The ALiBi paper showed that RoPE quickly degraded\nas it was tested on context lengths longer than the model had seen during training; it also introduced its' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='own proposed alternative that showed superior extrapolation ability on their benchmarks. However, ALiBi\nhas its own shortcomings; its use of simple linear functions for modulating the attention scores over distance\nmeans that it cannot represent as complex distance-attention-functions as the Fourier basis of RoPE. In\naddition, ALiBi uses a single such function per head, further reducing expressive power. This may explain' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='why, although ALiBi does extrapolate, models which utilize it have worse performance than RoPE-based\nmodels on benchmarks such as MMLU [2] and the LMSys arena which measures human preferences [15].\nxPos Sun et al. [7] examine why RoPE fails to extrapolate successfully and determines that this is due to\nthe effect of the high frequency components causing residual noise in the attention score even when tokens' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='are long distances apart. They attempt to address this by adding an exponentially decaying amplitude\nterm to RoPE. This new method, called xPos, decays these noisy high frequency components faster than\nlow frequency components. This method shows good results on from-scratch training of LLMs [7], and the\nintuition driving it aligns with our own hypotheses on the deficiency of RoPE. However, Sun et al. do not' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='experiment in our setting of interest: taking a model pretrained with RoPE and seeing if it can be coaxed\n(via limited finetuning) to learn the xPos encoding instead. Furthermore, their experiments demonstrate\nthat Blockwise Causal Attention is necessary for them to achieve extrapolation.\nLinear Scaling/Positional Interpolation This simple but effective context length extrapolation tech-\nnique was concurrently reported by kaiokendev [16] and by a team at Meta [17]. The method that is used' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='here is to simply divide the position vector by a scaling factor which fits the input within the context length\nof the original model. The intuition of this technique is to utilize the LLM’s interpolation capability, rather\nthan relying on extrapolation. It is a well known phenomenon that neural networks tend to interpolate\nwithin a range of previously seen values better than they extrapolate outside that range (e.g. [18]). In' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='the specific case of positional encodings, [17] claim that positional interpolation avoids the risk of massive\nnumerical explosion in attention values associated with extrapolation. We perform many experiments on\nthis scheme and variations of it and report the results in this paper.\nRandomized Positional Encodings Ruoss et al present this method in [19]. During training they\nrandomly generate their position vector by drawing N many samples uniformly without replacement from' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='the range [1, L], where N is the training context length and L is a large value that is greater than the\n(assumed to be known prior) maximum evaluation context length. These sampled positions are then sorted\nin increasing order and act as the position inputs that the model sees at evaluation time. During evaluation,\nthe position inputs [1, ..., M] are given to the model. The authors claim improved performance on context' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='length extrapolation. We independently arrived at a similar idea to this paper but instead randomized by\ndrawing from sub-integer positions approximately in the range [1, N]; see Section 4 for further details. We\nalso note that Ruoss et al investigate the use of such a scheme for training LLMs from scratch, whereas\nwe are primarily interested in post-hoc finetuning a pretrained LLM with randomization to enable context\nlength extrapolation.\n3' metadata={'source': 'pdfs/paper_1.pdf', 'page': 2}"
"page_content='3 Assessing Long Context Extrapolation\nThe main question posed in this paper revolves around extending the context length capacity of LLMs. To\nevaluate this, a commononly used metric in the literature is perplexity [6, 7, 13]. However, as we show in\nSection 5.3, perplexity is somewhat coarse-grained for evaluating how well the model can use longer context\nwindows. Our intuition is that — in many natural language datasets — a reasonable perplexity score can' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='be achieved even if the model is only attending to information in a limited range (the final 512 tokens, say)\nof the context window. For example, a positional encoding scheme which simply masks out any elements\nof the context (and inner key and query activations in the attention heads) that are greater in length than\nwhat the model was trained on should succeed in achieving a reasonable perplexity score, but would fare\npoorly on the tasks we describe below.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='We expand upon existing work to look at the accuracy of a model when presented with problems which\nhave verifiable answers. In using this metric, we can evaluate how the model is using the additional contextual\ninformation in order to respond to prompts. We rely on two types of evaluation tasks to assess models’ ability\nto extract and use information from long input contexts: the first is key-value retrieval tasks and the other' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='is question answering tasks. By using these two types of tasks, we enforce the requirement of the model to\nattend to the full context in order to obtain high accuracies. We consider the retrieval task to be a more\npure test of information retrieval free of many natural language biases. However, the retrieval task is a\nsomewhat artificial construct which the LLM will likely not have seen during training, so we also include\nquestion answering to replicate more real-world tasks.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='LongChat-Lines We start with a synthetic fine-grained key-value retrieval task first proposed in [20] and\nalso used by [21]. While these works are excellent given the standard contexts of LLMs, they lack the longer\ncontext lengths that are needed to evaluate our experiments. Thus, we utilize the same task as [20, 21], but\ngenerate additional samples of longer context lengths. This task gives the model a prompt with lines of the\nform:\n•line grotesque-classmate: REGISTER CONTENT is <42527 >' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='•line imperfect-bull: REGISTER CONTENT is <3119 >\n•line supreme-inversion: REGISTER CONTENT is <13960 >\n•...\nThe model is asked to memorize the value corresponding to the REGISTER CONTENT for each line and\nis asked at the end to retrieve the value for a specific line. By varying the number of lines in the prompt,\nwe can control the context length. We release longer length versions of this task than in [20], and we also\nrelease the generation script for this task.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='WikiQA We also create two new datasets from the Natural Questions [12] dataset with longer context\nevaluations specifically in mind which we collectively term WikiQA. In this evaluation, the prompt given\nto the LLM is in the format of a Wikipedia document followed by a question pertaining to that document;\nthe model is asked to answer the question. We ensure the answer to the question is a short answer which is' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='either a single word or a small sentence that has an exact string match in the document given to the LLM\nas input. We call this task Free Form QA (FFQA) .\nA potential issue in a Wikipedia based dataset however is that the model could perhaps correctly answer\nfrom its pretrained corpus and not specifically using the information in the context. To resolve this, we have\ncreated another “altered” dataset, which we call Altered Numeric QA (AltQA) . This dataset consists' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content='only of questions which have numerical answers. Here, we change the answer and every occurrence of the\nanswer in the document to a different number, thus ensuring that the LLM must attend to the context, and\nonly the context, in order to give a correct answer. The modification is made as follows:\n4' metadata={'source': 'pdfs/paper_1.pdf', 'page': 3}"
"page_content=""... is the third and final part of Dante 's Divine Comedy , following\nthe Inferno  and the Purgatorio . It is an allegory telling of Dante 's\njourney through Heaven , guided by Beatrice  , who symbolises\ntheology . In the poem , Paradise is depicted as a series of\nconcentric spheres surrounding the Earth , consisting of the Moon ,\nMercury , Venus , the Sun , Mars , Jupiter , Saturn , the Fixed\nStars , the Primum Mobile and finally , the Empyrean . It was\nwritten in the early 14th century ..."" metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content=""Question: \nWho serv es as dante's guide through par adise\nReference Answer:\nBeatriceDocument:FreeForm WIkiQA\n... Greece has hosted the Summer Olympic Games on two\noccasions , the inaug ural modern Olympics in 1896 and again in\n2004 2009  . Both were held in Athens , which along with Paris and\nLos Angeles are the cities that have hosted the Olympic Games\ntwice , with London being the only city to have hosted them three\ntimes . The Greek capital also hosted the 1906 Intercalated Games"" metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content=', which at the time were considered to be Olympic Games by the\nInternational Olympic Committee ...\nQuestion: \nWhen w as the last time the olympics were in Greece?\nReference Answer:\n2009* \n*and not 2004Document:AlteredNumeric WIkiQAFigure 1: Example QA snippets from our WikiQA dataset.\n•If the answer is a year, which is quite frequent, (i.e. it is between 1000-2100), we change it to a different' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='random value within +/- 10 of the original value. We treat years as a special case so as not to disrupt\nthe overall coherence of the document by having highly anachronistic date values.\n•If the answer is any other number, we change it to a different random number which has the same\nnumber of digits.\nFigure 1 highlights examples from our WikiQA dataset. Since the contexts in our application are long,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='the location of the answer within the context could play a significant role in the model’s ability to answer\nthe question. Therefore, we utilize both of the WikiQA tasks to conduct analysis on the performance of\nthe LLM as both the answer location moves within the document (in the beginning 10%, the last 10%, or\nrandomly anywhere else), as well as with the question given at the beginning or the end of the prompt — in\na bid to replicate the analyses of [22].' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='4 Context Length Extrapolation Techniques\nWe examine several context length extrapolation techniques, including existing approaches (or slight varia-\ntions on them) as well as our own newly proposed approaches.\n4.1 Existing Context Length Extrapolation Techniques\nSeveral methods exist to adapt RoPE positional encodings to longer context lengths. We evaluated the\nfollowing techniques.\nLinear Scaling/Positional Interpolation Here, the position vector is divided by a scaling factor. Hence' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='if the original model was trained on a range of positions [0 ,1, ...,2048], say, then the new model will see instead\n[0\nx,1\nx, ...,2048\nx] where xis the scaling factor.\nxPos We wanted to examine whether a checkpoint trained with the base model’s RoPE encoding scheme\ncould be finetuned to the xPos [7] scheme. On top of the programming hurdle of patching the entire attention\nmodule to handle xPos’ unique transformation of keys and queries, the major issue presented by this sort of' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='adaptation is xPos’ sensitivity to floating point precision. The method relies on scaling the key by numeric\nvalues with large (absolute) exponents; these later cancel in the dot product with the query. For long\ncontexts, however, the large values can actually exceed the magnitude supported by float16. We chose to\n5' metadata={'source': 'pdfs/paper_1.pdf', 'page': 4}"
"page_content='work around this by performing the core attention operation in float32 at the cost of a 2X training slow\ndown.\nRandomized Position Encodings Here we randomize the distances between the position values uni-\nformly in the range [ ϵ,2] for 0 < ϵ≪1, rather than using the typical [0 ,1, ..., n ] which has fixed intervals\nof size 1. The intuition behind this approach is that by showing the model many different intra-position' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='distances at finetuning time, the model will be able to generalize to any choice of fine-grained positions at\nevaluation time, thereby allowing for an effective increase in context length by choosing smaller divisions.\nThis has some similarity to the procedure described in Ruoss et al. [19]. We set an upper bound of 2 so that\nthe model will in expectation see a final position of n (as E[X]≈1 for X∼U(ϵ,2)). We also set a positive,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='non-zero lower bound of ϵin order to avoid issues with position aliasing due to limited numerical precision.\n4.2 Newly Proposed Context Length Extrapolation Techniques\nPower Scaling In the original RoPE, the basis that is used is given by:\nΘ ={θi= 10000−2(i−1)\nd|i∈ {1,2, . . . ,d\n2}} (1)\nwhere dis the embedding dimension. We use instead the basis given by:\nΘ∗=(\nθ∗\ni=θi\x12\n1−2i\nd\x13k\n|i∈ {1,2, . . . ,d\n2})\n(2)' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='where kis a parameter to be set. By applying this transformation, the high frequency (short distance)\nelements of the basis are less affected than the low frequency (long distance) elements, which are made even\nlower in frequency – see Figure 2. By doing so, our hope was that the model would have to perform less\ncomplex extrapolation for the low frequencies where it has not seen the full range of the periodic function' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='during train time, and thereby extrapolate better. A potential issue however is that the model relies on\nspecific relationships across frequencies that a linear transform preserves but a non-linear transformation\ndestroys.\nTruncated Basis Beginning from Equation 1, we instead use the basis given by applying:\nθ∗\ni=\uf8f1\n\uf8f4\uf8f2\n\uf8f4\uf8f3θiforθi≥b,\nρfora < θ i< b,\n0 for θ∗\ni≤a.(3)\nWhere ρis a fixed value that is relatively small, and aandbare chosen cutoff values. The idea here is that we' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='wish to preserve the high frequency components of the basis but set the low frequency elements to a constant\nvalue—in this case, 0. By doing so with a judicious choice of cutoff a, the model will have experienced all\nvalues of the basis in the context length used during finetuning (due to the periodic nature of the sine and\ncosine functions) and should therefore extrapolate better to larger context lengths for evaluation. However,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='the model still needs to be able to distinguish between distances that span the entire context it was trained\non, so we include the ρfixed frequency as well. In summary, we hope that with this basis the model can avoid\nthe issue of having to learn complicated coefficients in the entire RoPE basis by instead learning smooth\nfunctions at longer distances (as demonstrated in the paper [17]).\nIn Figure 2, we visually compare the frequencies produced by the standard RoPE basis, power scaling,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='and truncation.\n6' metadata={'source': 'pdfs/paper_1.pdf', 'page': 5}"
"page_content='Figure 2: Comparison of the standard RoPE basis vs the power basis and the truncated basis. The x-axis\nspans over the embedding dimension, and the y-axis is the frequency value of the sine-cosine basis.\n5 Results & Discussion\nIn the following experiments, we finetuned a base LLaMA-13B model on a portion of the RedPajama dataset\n[5] which has been modified so that each data sample has a size of exactly 4096 tokens. We trained with' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='each positional encoding approach until the evaluation loss roughly plateaued. Loss curves can be found in\nAppendix B.\nWe then further applied instruction finetuning (IFT) with the Vicuna dataset [23] and using LoRA [24]\non the base model. However, we discovered that although IFT did boost accuracies on LongChat-Lines, it\ndid not significantly change the range of contexts that the base model was able to deal with (see Figure 5 in' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='Appendix C). This we found to be a marked contrast with the WikiQA variants; there, IFT was necessary\nfor the model to produce any meaningful results at all. Hence for LongChat-Lines, we used non-IFT models;\nfor WikiQA, we performed evaluation on a subset of the more promising models with additional IFT.\n5.1 Finetuned Context Length Extrapolation\nLongChat-Lines We conducted evaluations on LongChat-Lines with the techniques described in Section' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='4 and report the results in Table 1. We expected all models to be able to perform with a non-zero accuracy\nuntil at least 4200 given that the model is finetuned on context lengths of 4096 and convergence is achieved\nin all cases. However, this turned out not to be the case for xPos, which was not able to perform the task\nat all. We suspect this may be because xPos is too different from the RoPE basis for the model to be able' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='to adapt in finetuning; as we see in Appendix B, the training and evaluation loss for xPos was not able to\nreach the same values as the other methods. This may also be a product of the numerical precision issues\nthat are encountered in the implementation of xPos.\nLinear scaling is able to achieve successful context length extrapolation. It is worth mentioning here that\nwe would expect scaling with a factor of xto achieve non-zero accuracies up to 2048 ·x, due to the base' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='model being trained on a context length of 2048. Although this is observed with linear scaling with a factor\nof 4, we see in Table 1 much quicker degradation as context length increases with a scaling factor of 16. By\ncontext length 17 500 it is already recording 0% accuracy even though we naively would expect reasonable\n7' metadata={'source': 'pdfs/paper_1.pdf', 'page': 6}"
"page_content='Context\nLengthLinear\nScaling\n(Factor=4)Linear\nScaling\n(Factor=16)Power basis Truncated\nbasisRandomized\npositionxPos\n2500 0.7 0.64 0.96 0.42 0.54 0\n3600 0.64 0.42 0.64 0.26 0.54 0\n4200 0.56 0.56 0.1 0.18 0.3 0\n4800 0.66 0.62 0 0.14 0.14 0\n7100 0.36 0.4 0 0.04 0.16 0\n9400 0 0.22 0 0 0 0\n11800 0 0.14 0 0 0 0\n14000 0 0.12 0 0 0 0\n16000 0 0.1 0 0 0 0\n17500 0 0 0 0 0 0\n20000 0 0 0 0 0 0\n22000 0 0 0 0 0 0' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='Table 1: After finetuning LLaMA-13B with a base context length of 4096, this table represents evaluations\nwith different context extrapolation methods on LongChat-Lines. An accuracy of 1.0 would indicate perfect\nperformance and 0.0 indicates getting every evaluation wrong. The power basis uses a parameter of k= 0.5.\nThe truncated basis uses the following parameters: a=1\n82π\n2048, b=2π\n2048, ρ=1\n162π\n2048. Randomization uses a\nlower bound parameter of ϵ=1' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='16. Evaluations are all performed without additional instruction finetuning.\nperformance up to roughly 32 000 context length. We believe that this indicates that there are limits to the\ninterpolation methodology and are interested in examining this further in future work.\nThe power basis, although it performs best at the shortest context, also decays fastest and is unable to\nshow any extrapolation performance beyond 4200 context at all.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='The randomized position approach may appear to be extrapolating based on the results in the table.\nHowever, this is likely due to how we evaluated the model. At train time, the model samples distances\nuniformly in [ ϵ,2] as described in Section 4. At evaluation time, it is unclear a priori what the best choice\nof positions is. We tried a range of different approaches: fixed distances of size 1, uniform random in [ ϵ,2]' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='and uniform random in [ ϵ,1]. We found best results for extrapolation with the latter, so we report this. We\nhoped that by reducing the upper bound further, we could coax the desired context length extrapolation\nfrom the model. However, going to [ ϵ,0.5] and below significantly degraded the performance of the model.\nOur conclusion from this is that the model cannot independently learn to represent each position without' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='knowing the other positions as well. An interesting avenue for future work would be to condition the Q-proj\nand K-proj matrices on the sampled positions during training (and evaluation).\nThe truncated basis does seem to offer true context length extrapolation, as it is able to achieve non-zero\naccuracies on context lengths outside any values it has seen before. Although the performance does degrade' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='as the length increases and the current manifestation of this is inferior in performance to linear scaling,\nwe believe that this may be a direction of investigation that can lead to better extrapolation performance.\nTruncation can also be combined with linear scaling, as we discuss in Section 5.2.\nWikiQA Variants We further conducted evaluations of the linear scaling and truncated basis approaches' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='on the WikiQA variants described in Section 4. Unlike the retrieval task, we found that models were unable\nto perform this task successfully without any instruction finetuning, so we performed this analysis on only\na few approaches of interest. The results are shown in Tables 2 and 3. They largely match the pattern seen\nin LongChat-Lines—linear scaling with scale factor 4 is able to perform the task up to 7500 context but' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='not beyond, whilst scale factor 16 is able to surpass this cutoff but with a slope-off in accuracy. As with\nLongChat-Lines, we see that the models appear to show some degradation of accuracy as context length\nincreases. We see again that the truncated basis is able to extrapolate successfully to about the same context\nlength as in LongChat-Lines with comparable accuracies to linear scaling with scale 4, but again seemingly\ncannot go further than a context length of about 8k.\n8' metadata={'source': 'pdfs/paper_1.pdf', 'page': 7}"
"page_content='Context Length Linear scaling (Factor=4) Linear scaling\n(Factor=16)Truncated basis\n2000 0.72 0.69 0.74\n3800 0.72 0.73 0.73\n7500 0.62 0.70 0.46\n15000 0.00 0.68 0.00\n24000 0.00 0.56 0.00\n32000 0.00 0.18 0.00\nTable 2: Accuracy on the AlteredNumeric WikiQA variant of models finetuned with different context ex-\ntrapolation methods. Evaluations are all performed with instruction finetuning.\nContext Length Linear scaling (Factor=4) Linear scaling\n(Factor=16)Truncated basis\n1900 0.44 0.47 0.46' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='3800 0.49 0.44 0.55\n7600 0.46 0.45 0.36\n15000 0.00 0.48 0.00\n24000 0.00 0.42 0.00\n32000 0.00 0.20 0.00\nTable 3: Accuracy on the FreeForm WikiQA variant of models finetuned with different context extrapolation\nmethods. Evaluations are all performed with instruction finetuning.\n5.2 Zero-Shot Linear Scaling\nIn the previous section, we examined the performance of linear scaling using the same value at finetuning' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='time as at evaluation time. In this section, we instead investigate the effect of using a different scaling factor\nat evaluation time than what the model was trained on. In Table 4, we show results for this strategy as\nevaluated on LongChat-Lines. We find in general that if the model is trained with a scale factor of x, then\nthe model can successfully evaluate zero-shot with a scale factor of 2 x(with some reduction of accuracy' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='within the range of context lengths the model could previously handle). It also appears that at a scale factor\nof 16, the model is no longer able to increase its effective context length by using this approach. We also\nfind that evaluating with >2xleads to the model breaking and being unable to perform the task.\nWe show that zero-shot linear scaling can be applied successfully after finetuning with the truncated basis.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='Interestingly, whilst for linear scaling using a longer scale factor at evaluation time results in a deterioration\nof the accuracy on context lengths the model could previously handle, this does not appear to be the case\nfor the truncated basis—instead, the range of context lengths that the model achieves non-zero accuracy on\nincreases, and accuracy improves among context lengths the model was finetuned on.\n5.3 Comparing Perplexity To Tasks' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='In Section 3, we introduced two tasks which specifically require a long-context LLM to extract answers from\nthroughout the entire text, arguing that these tasks may assess long context performance better than raw\nperplexity. To analyze how perplexity fares as compared to these tasks, we report perplexities on a held\nout set of the RedPajama dataset for a subset of our trained models (see Table 5). Perplexity scores do' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='show a large increase when a context length is reached that the model is completely unable to deal with (for\nexample, beyond 2k on the base LLaMA model, or beyond 8k on the linear scale 4 model). However, they\nare appear less effective for showing the decrease in long-context capability within that effective range. In\nparticular, while we observe a steep slope-off in performance on LongChat-Lines and the WikiQA variants as' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='the context length increases for the linear scale 4 and truncated basis columns of Table 1), this degradation\nis not strongly reflected in the perplexity scores at those contexts. However, the linear scale 16 model\ndoes appear to have well-correlated perplexity and accuracy on our tasks. Perhaps most tellingly, we see\n9' metadata={'source': 'pdfs/paper_1.pdf', 'page': 8}"
"page_content='Train Scaling = 1 (base model) Train Scaling = 4 Train Scaling = 16 Truncated (Scaling = 1)\nContext Length Eval 1 Eval 2 Eval 4 Eval 4 Eval 8 Eval 16 Eval 16 Eval 32 Eval 64 Eval 1 Eval 2 Eval 4\n2500 0 0.32 0 0.88 0.64 0 0.64 0.24 0 0.42 0.58 0\n3600 0 0.3 0 0.8 0.58 0 0.42 0.26 0 0.26 0.44 0\n4200 0 0.18 0 0.86 0.62 0 0.56 0.12 0 0.18 0.28 0\n4800 0 0 0 0.86 0.62 0 0.62 0.22 0 0.14 0.26 0\n7100 0 0 0 0.64 0.38 0 0.4 0.12 0 0.04 0.04 0\n9400 0 0 0 0 0.32 0 0.22 0.12 0 0 0.04 0' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='11800 0 0 0 0 0.30 0 0.14 0.1 0 0 0.04 0\n14000 0 0 0 0 0.1 0 0.12 0.04 0 0 0 0\n16000 0 0 0 0 0.12 0 0.1 0.02 0 0 0 0\n17500 0 0 0 0 0 0 0 0 0 0 0 0\n20000 0 0 0 0 0 0 0 0 0 0 0 0\n22000 0 0 0 0 0 0 0 0 0 0 0 0\nTable 4: Accuracy on LongChat-Lines of models finetuned with different context extrapolation methods\nand then evaluated with additional linear scaling. Whenever the evaluation linear scale is greater than' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='the training linear scale, this produces zero-shot context length extrapolation. In general, increasing the\nevaluation context length 2x over train actually does double the usable context length, at an accuracy cost,\nfor train lengths 1 and 4 (for train length 16, it does not). The truncated basis accuracy improves with 2x\nscaling. A more aggressive scale up of 4x times the train length leads to apparent model failure.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='the shortcoming of perplexity for between-model comparisons. According to Table 5, the truncated basis\nperforms best at 8k and below; however, in Tables 1, 2 and 3 truncated is significantly lower in performance\ncompared to the linear scaled models at 8k context.\nPerplexity is commonly used in the literature to measure long context performance [13, 7], but we believe\nthese results show it is not in itself a sufficient measure of long context performance, but is best utilised with' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='other tasks which additionally probe the capabilities of the LLM.\nContext Length LLaMA Base Linear Scaling (Factor=4) Linear Scaling (Factor=16) Truncated Basis\n512 4.06 4.06 4.05 3.79\n1k 3.88 3.87 3.86 3.63\n2k 3.79 3.75 3.74 3.52\n4k 9022 3.66 3.66 3.46\n8k 7198 3.79 3.97 3.78\n16k 5141 14902 5.43 15793\n24k 4980 21236 8.73 13929\n32k 4408 55480 98.12 12534\nTable 5: Perplexity scores on a held out evaluation set of the RedPajama dataset at various context lengths' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='on different models. The evaluation length is 256 tokens, and the prompt given to the models is the previous\nN - 256 tokens of the document, where N is the context length we evaluated. When the context length is too\nlong for the model to handle effectively, the perplexity does blow up; however, within ranges the model can\nhandle, perplexity appears to be less sensitive to context-usage degradation than the LongChat-Lines task,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='and does not follow the same between-model ranking as that of LongChat-Lines or WikiQA.\n5.4 Analysis of question and answer positioning\nFor the WikiQA variants, we performed a stratified analysis of the effect of the position of the answer and\nthe question. As described in Section 4, we looked at the impact of placing the answer within the first 10%\nof the document, the last 10%, or elsewhere randomly. We also examined the effect of putting the question' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='at the beginning or the end of the prompt. The results are shown in Tables 6 and 7, performed on the model\nwith linear scaling with a factor of 16, with additional instruction finetuning.\n10' metadata={'source': 'pdfs/paper_1.pdf', 'page': 9}"
"page_content='Context Length Answer Location Question Location\nStart Middle End Start End\n2000 0.74 0.68 0.66 0.70 0.69\n3800 0.70 0.67 0.66 0.63 0.73\n7500 0.69 0.63 0.65 0.61 0.70\n15000 0.68 0.68 0.69 0.68 0.68\n24000 0.30 0.26 0.50 0.14 0.56\n32000 0.13 0.13 0.23 0.15 0.18\nTable 6: Stratified accuracy analysis on the AlteredNumericQA task. For answers, “Start” refers to the first\n10% of the document, “End” to the last 10%, and “Middle” to any other location. For questions, “Start”' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='refers to placing the question before the rest of the language prompt, and “End” refers to placing the question\nat the end. Results are reported on LLaMA-13B finetuned with a linear scale of 16, with IFT applied.\nContext Length Answer Location Question Location\nStart Middle End Start End\n1900 0.37 0.40 0.36 0.27 0.47\n3800 0.40 0.30 0.32 0.24 0.44\n7600 0.35 0.34 0.35 0.24 0.45\n15000 0.44 0.43 0.45 0.40 0.48\n24000 0.18 0.20 0.40 0.10 0.42\n32000 0.07 0.11 0.18 0.04 0.20' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='Table 7: Stratified accuracy analysis on the FreeFormQA task. For answers, “Start” refers to the first 10%\nof the document, “End” to the last 10%, and “Middle” to any other location. For questions, “Start” refers\nto placing the question before the rest of the language prompt, and “End” refers to placing the question at\nthe end. Results are reported on LLaMA-13B finetuned with a linear scale of 16, with IFT applied.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='We aimed to build on a similar analysis from [22]. However, we were not able to replicate the results shown\nin that paper on the LongChat-13B (16K) model (to which our modeling approach is most comparable). On\nboth FreeFormQA and AlteredNumericQA, we observed no clear trend with regards to the location of the\nanswer within the prompt and the model’s accuracy up to 15k context length. There also did not appear to be' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='a significant impact on the location of the question for AlteredNumericQA, but there is a noticeable impact\nobserved for FreeFormQA where having the question at the end appears to have a significant improvement\nin accuracy. However, at 24k and 32k context lengths we see a clear indication in both datasets for both the\nanswer at the end and question at the end returning superior accuracy to their placements elsewhere. These' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='results are a marked contrast to those in [22]. Our take away from this is that there is plausibly a great\ndeal of task-conditional variability in the performance of LLMs with regards to how well they can utilize all\nportions of the context; even small differences in task construction can lead to large differences in observed\ntrends.\n6 Conclusion and Limitations\nIn this paper we examined multiple approaches to finetuning a pretrained base LLaMA and LLaMA2 LLM' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='that has a limited context length such that it is capable of extrapolating zero-shot to new, longer context\nlengths. We compared the methods using perplexity, as well as two custom tasks that probe long context\nperformance; we find that the custom tasks offer a more fine-grained understanding of long context perfor-\nmance than perplexity. We showed that the method of linear interpolation performed best at context length' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='extrapolation, and found some promise in the potential of using a new basis which we termed the truncated\nbasis. We release three models which we call Giraffe that extend the context length of the base LLaMA and\nLLaMA 2 models using the method of linear interpolation.\nThere is significant room for building on the work presented in this paper. We note that all methods show\n11' metadata={'source': 'pdfs/paper_1.pdf', 'page': 10}"
"page_content='a degradation in accuracy on our evaluation tasks as context length increases, even though perplexity often\nremains reasonable and the model can still produce coherent outputs. This is a shortcoming that would be\nof interest to address, and in our view is necessary for claiming ‘true’ long context extrapolation ability of a\nmodel.\nThe limitations of this work are that we only conducted our perplexity analysis on a single document' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='dataset. Future work could look to replicate this analysis on other datasets. Additionally, we focused\nspecifically on context-length extrapolation from a pretrained base model, and in particular the LLaMA\nand LLaMA 2 models trained with RoPE positional encodings. Future work could investigate whether the\nanalysis herein extends to other positional encoding types and models. Future work could also address' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='the limitations of the linear interpolation method. We see some evidence on the LongChat-Lines task in\nparticular of accuracy degradation as the scale factor is increased. What is the limit of the size of scale\nfactor of this method? Is there a point beyond which it simply does not improve the range of contexts it\ncan handle? Furthermore, can the truncated basis approach which seems to show signs of true extrapolation' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='capability be modified in a manner to gain parity with or surpass the linear interpolation method? We\nbelieve these are some potential future directions of interest.\nReferences\n[1] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz\nKaiser, and Illia Polosukhin. Attention is all you need, 2023.\n[2] Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='Steinhardt. Measuring massive multitask language understanding, 2021.\n[3] Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan,\nHarri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger,\nMichael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder,\nMikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel\nHerbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin,\nSuchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh\nAchiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati,\nKatie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='Wojciech Zaremba. Evaluating large language models trained on code, 2021.\n[4] Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang,\nHorace He, Anish Thite, Noa Nabeshima, Shawn Presser, and Connor Leahy. The pile: An 800gb\ndataset of diverse text for language modeling, 2020.\n[5] Together Computer. Redpajama: An open source recipe to reproduce llama training dataset, April\n2023.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='[6] Ofir Press, Noah A. Smith, and Mike Lewis. Train short, test long: Attention with linear biases enables\ninput length extrapolation, 2022.\n[7] Yutao Sun, Li Dong, Barun Patra, Shuming Ma, Shaohan Huang, Alon Benhaim, Vishrav Chaudhary,\nXia Song, and Furu Wei. A length-extrapolatable transformer, 2022.\n[8] OpenAI. Gpt-4 technical report, 2023.\n[9] Anthropic. Introducing claude, 2023.\n[10] Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timoth´ ee' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='Lacroix, Baptiste Rozi` ere, Naman Goyal, Eric Hambro, Faisal Azhar, Aurelien Rodriguez, Armand\n12' metadata={'source': 'pdfs/paper_1.pdf', 'page': 11}"
"page_content='Joulin, Edouard Grave, and Guillaume Lample. Llama: Open and efficient foundation language models,\n2023.\n[11] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay\nBashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Can-\nton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu,\nBrian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev,\nPunit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu,\nYuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew\nPoulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael\nSmith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan\nNarang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. Llama 2: Open\nfoundation and fine-tuned chat models, 2023.\n[12] Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti,\nDanielle Epstein, Illia Polosukhin, Matthew Kelcey, Jacob Devlin, Kenton Lee, Kristina N. Toutanova,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='Llion Jones, Ming-Wei Chang, Andrew Dai, Jakob Uszkoreit, Quoc Le, and Slav Petrov. Natural ques-\ntions: a benchmark for question answering research. Transactions of the Association of Computational\nLinguistics , 2019.\n[13] Jianlin Su, Yu Lu, Shengfeng Pan, Ahmed Murtadha, Bo Wen, and Yunfeng Liu. Roformer: Enhanced\ntransformer with rotary position embedding, 2022.\n[14] Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. Self-attention with relative position representations,\n2018.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='[15] Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin,\nZhuohan Li, Dacheng Li, Eric. P Xing, Hao Zhang, Joseph E. Gonzalez, and Ion Stoica. Judging\nllm-as-a-judge with mt-bench and chatbot arena, 2023.\n[16] Things i’m learning while training superhot, 2023.\n[17] Shouyuan Chen, Sherman Wong, Liangjian Chen, and Yuandong Tian. Extending context window of\nlarge language models via positional interpolation, 2023.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='[18] Keyulu Xu, Mozhi Zhang, Jingling Li, Simon S. Du, Ken ichi Kawarabayashi, and Stefanie Jegelka.\nHow neural networks extrapolate: From feedforward to graph neural networks, 2021.\n[19] Anian Ruoss, Gr´ egoire Del´ etang, Tim Genewein, Jordi Grau-Moya, R´ obert Csord´ as, Mehdi Bennani,\nShane Legg, and Joel Veness. Randomized positional encodings boost length generalization of trans-\nformers, 2023.\n[20] anadim. A little retrieval test for large language models, May 2023.' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='[21] Dacheng Li*, Rulin Shao*, Anze Xie, Ying Sheng, Lianmin Zheng, Joseph E. Gonzalez, Ion Stoica,\nXuezhe Ma, and Hao Zhang. How long can open-source llms truly promise on context length?, June\n2023.\n[22] Nelson F. Liu, Kevin Lin, John Hewitt, Ashwin Paranjape, Michele Bevilacqua, Fabio Petroni, and\nPercy Liang. Lost in the middle: How language models use long contexts, 2023.\n[23] Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin,' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='Zhuohan Li, Dacheng Li, Eric. P Xing, Hao Zhang, Joseph E. Gonzalez, and Ion Stoica. Judging\nllm-as-a-judge with mt-bench and chatbot arena, 2023.\n13' metadata={'source': 'pdfs/paper_1.pdf', 'page': 12}"
"page_content='[24] Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and\nWeizhu Chen. Lora: Low-rank adaptation of large language models, 2021.\n14' metadata={'source': 'pdfs/paper_1.pdf', 'page': 13}"
"page_content='Context Length LLaMA 2 Linear (8)\n2500 0.48\n3600 0.42\n4200 0.32\n4800 0.74\n7100 0.56\n9400 0.50\n11800 0.50\n14000 0.42\n16000 0.38\n17500 0.14\n20000 0.14\n22000 0.08\n26000 0.00\n30000 0.00\nTable 8: LLaMA 2 performance on LongChat-Lines with a scale factor of 8.\nContext Length LLaMA 2 Linear (8)\nAltQA FFQA\n2k 0.72 0.56\n4k 0.76 0.55\n8k 0.71 0.56\n16k 0.59 0.44\n24k 0.36 0.28\n32k 0.15 0.10\nTable 9: LLaMA 2 performance on the WikiQA variants with a scale factor of 8.\nA LLaMA 2' metadata={'source': 'pdfs/paper_1.pdf', 'page': 14}"
"page_content='As we were finalising this paper, Meta released LLaMA 2 [11]. We verified that similar results of context\nlength extrapolation were achievable with LLaMA 2 by the linear interpolation method. We applied the\nsame method as described in Section 4, training LLaMA 2-13B on a portion of the RedPajama dataset\nmodified such that each data sample has a size of exactly 4096 tokens. We then also applied instruction' metadata={'source': 'pdfs/paper_1.pdf', 'page': 14}"
"page_content='finetuning with the Vicuna dataset. We used a scale of 8. Performance is shown in tables 8 and 9.\nWe see that the model is able to achieve non-zero accuracies on LongChat-Lines up to a context length\nof 22000, further than any of the models we tested in the main paper. The model is also able to achieve\nnon-zero performance on the WikiQA variants up to 32k context. However, we do see diminishing accuracy' metadata={'source': 'pdfs/paper_1.pdf', 'page': 14}"
"page_content='in both tasks as the context length increases. It is also worth noting that the accuracies on both tasks\nare slightly lower than LLaMA 1 with scale 16 on the context lengths which both models are capable of\nproducing non-zero results.\n15' metadata={'source': 'pdfs/paper_1.pdf', 'page': 14}"
"page_content='B Loss Curves\nFigure 3: Training loss curves of models during the initial fitting runs on 4096 token samples extracted from\nthe RedPajama dataset.\nFigure 4: Evaluation loss curves of models during the initial fitting runs on 4096 token samples extracted\nfrom the RedPajama dataset.\n16' metadata={'source': 'pdfs/paper_1.pdf', 'page': 15}"
"page_content='Figure 5: Comparison between IFT and non-IFT on LongChat-Lines with linear scaling of 4 applied. Al-\nthough IFT improves the accuracies, it does not extend the range of contexts on which the model obtains a\nnon-zero accuracy.\nC Impact of IFT\nAs mentioned in Section 5 of the main text, we found that instruction-fine-tuning with the Vicuna dataset\ndid improve accuracies on LongChat-Lines, but did not change the span of non-zero contexts for a given' metadata={'source': 'pdfs/paper_1.pdf', 'page': 16}"
"page_content='model. Figure 5 shows this on the model with linear interpolation with scale 4.\n17' metadata={'source': 'pdfs/paper_1.pdf', 'page': 16}"