Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.
On the other hand, working in the primal domain ensures a bound, but usually at the cost of enumerating a large number of trees. @cite_19 heuristically select a small number of trees to avoid being too inefficient, while @cite_0 focus on trying to speed up the updates on a given collection of trees. Another primal bound is weighted mini-bucket (WMB, @cite_5 ), which can represent a large collection of trees compactly and is easily applied to marginal MAP using the weighted log partition function viewpoint @cite_7 @cite_18 ; however, existing optimization algorithms for WMB are non-monotonic, and often fail to converge, especially on marginal MAP tasks.
@cite_18 @cite_7 @cite_0 @cite_19 @cite_5
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We consider requests for capacity in a given tree network T e (V, E) where each edge e of the tree has some integer capacity ue. Each request f is a node pair with an integer demand df and a profit wf which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b-matching problems. When all demands are 1, we have the integer multicommodity flow problem. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand dmax is at most the minimum edge capacity umin, we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed. We propose new succinct representations of ordinal trees and match various space time lower bounds. It is known that any n-node static tree can be represented in 2n p o(n) bits so that a number of operations on the tree can be supported in constant time under the word-RAM model. However, the data structures are complicated and difficult to dynamize. We propose a simple and flexible data structure, called the range min-max tree, that reduces the large number of relevant tree operations considered in the literature to a few primitives that are carried out in constant time on polylog-sized trees. The result is extended to trees of arbitrary size, retaining constant time and reaching 2n p O(n polylog(n)) bits of space. This space is optimal for a core subset of the operations supported and significantly lower than in any previous proposal. For the dynamic case, where insertion deletion (indels) of nodes is allowed, the existing data structures support a very limited set of operations. Our data structure builds on the range min-max tree to achieve 2n p O(n log n) bits of space and O(log n) time for all operations supported in the static scenario, plus indels. We also propose an improved data structure using 2n p O(nlog log n log n) bits and improving the time to the optimal O(log n log log n) for most operations. We extend our support to forests, where whole subtrees can be attached to or detached from others, in time O(log1pe n) for any e > 0. Such operations had not been considered before. Our techniques are of independent interest. An immediate derivation yields an improved solution to range minimum maximum queries where consecutive elements differ by ± 1, achieving n p O(n polylog(n)) bits of space. A second one stores an array of numbers supporting operations sum and search and limited updates, in optimal time O(log n log log n). A third one allows representing dynamic bitmaps and sequences over alphabets of size σ, supporting rank select and indels, within zero-order entropy bounds and time O(log n log σ (log log n)2) for all operations. This time is the optimal O(log n log log n) on bitmaps and polylog-sized alphabets. This improves upon the best existing bounds for entropy-bounded storage of dynamic sequences, compressed full-text self-indexes, and compressed-space construction of the Burrows-Wheeler transform. In a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1 e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1 e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1 e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1 e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of , removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good "strength" to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis. We study resource-limited online learning, motivated by the problem of conditional-branch outcome prediction in computer architecture. In particular, we consider (parallel) time and space-efficient ensemble learners for online settings, empirically demonstrating benefits similar to those shown previously for offline ensembles. Our learning algorithms are inspired by the previously published “boosting by filtering” framework as well as the offline Arc-x4 boosting-style algorithm. We train ensembles of online decision trees using a novel variant of the ID4 online decision-tree algorithm as the base learner, and show empirical results for both boosting and bagging-style online ensemble methods. Our results evaluate these methods on both our branch prediction domain and online variants of three familiar machine-learning benchmarks. Our data justifies three key claims. First, we show empirically that our extensions to ID4 significantly improve performance for single trees and additionally are critical to achieving performance gains in tree ensembles. Second, our results indicate significant improvements in predictive accuracy with ensemble size for the boosting-style algorithm. The bagging algorithms we tried showed poor performance relative to the boosting-style algorithm (but still improve upon individual base learners). Third, we show that ensembles of small trees are often able to outperform large single trees with the same number of nodes (and similarly outperform smaller ensembles of larger trees that use the same total number of nodes). This makes online boosting particularly useful in domains such as branch prediction with tight space restrictions (i.e., the available real-estate on a microprocessor chip). In mining data streams the most popular tool is the Hoeffding tree algorithm. It uses the Hoeffding's bound to determine the smallest number of examples needed at a node to select a splitting attribute. In the literature the same Hoeffding's bound was used for any evaluation function (heuristic measure), e.g., information gain or Gini index. In this paper, it is shown that the Hoeffding's inequality is not appropriate to solve the underlying problem. We prove two theorems presenting the McDiarmid's bound for both the information gain, used in ID3 algorithm, and for Gini index, used in Classification and Regression Trees (CART) algorithm. The results of the paper guarantee that a decision tree learning system, applied to data streams and based on the McDiarmid's bound, has the property that its output is nearly identical to that of a conventional learner. The results of the paper have a great impact on the state of the art of mining data streams and various developed so far methods and algorithms should be reconsidered. We consider the problem of online scheduling of jobs on unrelated machines with dynamic speed scaling to minimize the sum of energy and weighted flow time. We give an algorithm with an almost optimal competitive ratio for arbitrary power functions. (No earlier results handled arbitrary power functions for minimizing flow time plus energy with unrelated machines.) For power functions of the form f(s) = sα for some constant α > 1, we get a competitive ratio of O(α log α), improving upon a previous competitive ratio of O(α2) by [3], along with a matching lower bound of Ω(α log α). Further, in the resource augmentation model, with a 1 + e speed up, we give a 2(1 e + 1) competitive algorithm, with essentially the same techniques, improving the bound of 1 + O(1 e2) by [15] and matching the bound of [3] for the special case of fixed speed unrelated machines. Unlike the previous results most of which used an amortized local competitiveness argument or dual fitting methods, we use a primal-dual method, which is useful not only to analyze the algorithms but also to design the algorithm itself. We give a simple proof that the ranking algorithm of Karp, Vazirani and Vazirani [KVV90] is 1-1 e competitive for the online bipartite matching problem. The proof is via a randomized primal-dual argument. Primal-dual algorithms have been successfully used for many online algorithm problems, but the dual constraints are always satisfied deterministically. This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. The approach also generalizes easily to the vertex-weighted version considered by [AGKM11]. Further we show that the proof is very similar to the deterministic primal-dual argument for the online budgeted allocation problem with small bids (also called the AdWords problem) of [MSVV05]. Under certain conditions (known as the restricted isometry property, or RIP) on the mN matrix ˆ (where m<N ), vectors x 2 R N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y WDx even though ˆ � 1 .y is typically an .Nm -dimensional hyperplane; in addition, x is then equal to the element in ˆ � 1 .y of minimal 1-norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in ˆ � 1 .y with smallest 2.w -norm. If x .n is the solution at itera- tion step n, then the new weight w .n is defined by w .n WD Œjx .n j 2 C " 2� � 1=2 , i D 1; :::;N , for a decreasing sequence of adaptively defined "n; this updated weight is then used to obtain x .nC1 and the process is repeated. We prove that whensatisfies the RIP conditions, the sequence x .n converges for all y, re- gardless of whether ˆ � 1 .y contains a sparse vector. If there is a sparse vector in ˆ � 1 .y , then the limit is this sparse vector, and when x .n is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same al- gorithm with the "heavier" weight w .n D Œjx .n Very Fast Decision Tree (VFDT) in data stream mining has been widely studied for more than a decade. VFDT in essence can mine over a portion of an unbounded data stream at a time, and the structure of the decision tree gets updated whenever new data feed in; hence it can predict better upon the input of fresh data. Inherent from traditional decision trees that use information gains for tree induction, VFDT may suffer the same over-fitting problem where noise in the training data leads to excessive tree branches therefore decline in prediction accuracy. This problem is aggravated in stream mining because limited run-time memory (for storing the whole decision tree) and reasonable accuracy are often the criteria for implementing VFDT. Post-pruning that was a popular technique used in traditional decision tree to keep the tree size in check, however may not be applicable for VFDT in situ. In this paper a new model that extends from VFDT called Optimized VFDT is proposed, for controlling the tree size while sustaining good prediction accuracy. This is enabled by using an adaptive threshold tie and incremental pruning in tree induction. Experimental results show that an optimal ratio of tree size and accuracy can be achieved by OVFDT. Marginal MAP problems are known to be very difficult tasks for graphical models and are so far solved exactly by systematic search guided by a join-tree upper bound. In this paper, we develop new AND OR branch and bound algorithms for marginal MAP that use heuristics extracted from weighted mini-buckets enhanced with message-passing updates. We demonstrate the effectiveness of the resulting search algorithms against previous join-tree based approaches, which we also extend to accommodate high induced width models, through extensive empirical evaluations. Our results show not only orders-of-magnitude improvements over the state-of-the-art, but also the ability to solve problem instances well beyond the reach of previous approaches. Consider an ordered, static tree T where each node has a label from alphabet Σ. Tree T may be of arbitrary degree and shape. Our goal is designing a compressed storage scheme of T that supports basic navigational operations among the immediate neighbors of a node (i.e. parent, ith child, or any child with some label,…) as well as more sophisticated path-based search operations over its labeled structure. We present a novel approach to this problem by designing what we call the XBW-transform of the tree in the spirit of the well-known Burrows-Wheeler transform for strings [1994]. The XBW-transform uses path-sorting to linearize the labeled tree T into two coordinated arrays, one capturing the structure and the other the labels. For the first time, by using the properties of the XBW-transform, our compressed indexes go beyond the information-theoretic lower bound, and support navigational and path-search operations over labeled trees within (near-)optimal time bounds and entropy-bounded space. Our XBW-transform is simple and likely to spur new results in the theory of tree compression and indexing, as well as interesting application contexts. As an example, we use the XBW-transform to design and implement a compressed index for XML documents whose compression ratio is significantly better than the one achievable by state-of-the-art tools, and its query time performance is order of magnitudes faster. We consider succinct or space-efficient representations of trees that efficiently support a variety of navigation operations. We focus on static ordinal trees, that is, arbitrary static rooted trees where the children of each node are ordered. The set of operations is essentially the union of the sets of operations supported by previous succinct representations [Jacobson 1989; Munro and Raman 2001; 1999] to which we add the level-ancestor operation.Our representation takes 2n p o(n) bits to represent an n-node tree, which is within o(n) bits of the information-theoretic minimum, and supports all operations in O(1) time on the RAM model. These operations also provide a mapping from the n nodes of the tree onto the integers 1, …, n . In addition to the existing motivations for studying such data structures, we are motivated by the problem of representing XML documents compactly so that XPath queries can be supported efficiently. One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions The random assignment problem asks for the minimum-cost perfect matching in the complete n × n bipartite graph Knn with i.i.d. edge weights, say uniform on [0, 1]. In a remarkable work by Aldous [Aldous, D. 2001. The ζ(2) limit in the random assignment problem. RSA18 381--418], the optimal cost was shown to converge to ζ(2) as n → ∞, as conjectured by Mezard and Parisi [Mezard, M., G. Parisi. 1987. On the solution of the random link matching problem. J. Phys.48 1451--1459] through the so-called cavity method. The latter also suggested a nonrigorous decentralized strategy for finding the optimum, which turned out to be an instance of the belief propagation (BP) heuristic discussed by Pearl [Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco]. In this paper we use the objective method to analyze the performance of BP as the size of the underlying graph becomes large. Specifically, we establish that the dynamic of BP on Knn converges in distribution as n → ∞ to an appropriately defined dynamic on the Poisson weighted infinite tree, and we then prove correlation decay for this limiting dynamic. As a consequence, we obtain that BP finds an asymptotically correct assignment in O(n2) time only. This contrasts with both the worst-case upper bound for convergence of BP derived by [Bayati, M., D. Shah, M. Sharma. 2008. Max-product for maximum weight matching: Convergence, correctness, and LP duality. IEEE Trans. Inform. Theory54(3) 1241--1251.] and the best-known computational cost of Θ(n3) achieved by Edmonds and Karp's algorithm [Edmonds, J., R. Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM19 248--264]. The success of CNNs in various applications is accompanied by a significant increase in the computation and parameter storage costs. Recent efforts toward reducing these overheads involve pruning and compressing the weights of various layers without hurting original accuracy. However, magnitude-based pruning of weights reduces a significant number of parameters from the fully connected layers and may not adequately reduce the computation costs in the convolutional layers due to irregular sparsity in the pruned networks. We present an acceleration method for CNNs, where we prune filters from CNNs that are identified as having a small effect on the output accuracy. By removing whole filters in the network together with their connecting feature maps, the computation costs are reduced significantly. In contrast to pruning weights, this approach does not result in sparse connectivity patterns. Hence, it does not need the support of sparse convolution libraries and can work with existing efficient BLAS libraries for dense matrix multiplications. We show that even simple filter pruning techniques can reduce inference costs for VGG-16 by up to 34 and ResNet-110 by up to 38 on CIFAR10 while regaining close to the original accuracy by retraining the networks. The success of CNNs in various applications is accompanied by a significant increase in the computation and parameter storage costs. Recent efforts toward reducing these overheads involve pruning and compressing the weights of various layers without hurting original accuracy. However, magnitude-based pruning of weights reduces a significant number of parameters from the fully connected layers and may not adequately reduce the computation costs in the convolutional layers due to irregular sparsity in the pruned networks. We present an acceleration method for CNNs, where we prune filters from CNNs that are identified as having a small effect on the output accuracy. By removing whole filters in the network together with their connecting feature maps, the computation costs are reduced significantly. In contrast to pruning weights, this approach does not result in sparse connectivity patterns. Hence, it does not need the support of sparse convolution libraries and can work with existing efficient BLAS libraries for dense matrix multiplications. We show that even simple filter pruning techniques can reduce inference costs for VGG-16 by up to 34 and ResNet-110 by up to 38 on CIFAR10 while regaining close to the original accuracy by retraining the networks. We develop and analyze methods for computing provably optimal maximum a posteriori probability (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: a) a tree-relaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and b) a tree-reweighted max-product message-passing algorithm that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the mode-finding problem and a reweighted form of the max-product (min-sum) message-passing algorithm. We give a (ln n+ 1)-approximation for the decision tree (DT) problem. An instance of DT is a set of mbinary tests T= (T 1 , ..., T m ) and a set of nitems X= (X 1 , ..., X n ). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the total external path length of the tree is minimized. Total external path length is the sum of the depths of all the leaves in the tree. DT has a long history in computer science with applications ranging from medical diagnosis to experiment design. It also generalizes the problem of finding optimal average-case search strategies in partially ordered sets which includes several alphabetic tree problems. Our work decreases the previous upper bound on the approximation ratio by a constant factor. We provide a new analysis of the greedy algorithm that uses a simple accounting scheme to spread the cost of a tree among pairs of items split at a particular node. We conclude by showing that our upper bound also holds for the DT problem with weighted tests. Primal-dual algorithms have played an integral role in recent developments in approximation algorithms, and yet there has been little work on these algorithms in the context of LP relaxations that have been strengthened by the addition of more sophisticated valid inequalities. We introduce primal-dual schema based on the LP relaxations devised by for the minimum knapsack problem as well as for the single-demand capacitated facility location problem. Our primal-dual algorithms achieve the same performance guarantees as the LP-rounding algorithms of which rely on applying the ellipsoid algorithm to an exponentially-sized LP. Furthermore, we introduce new flow-cover inequalities to strengthen the LP relaxation of the more general capacitated single-item lot-sizing problem; using just these inequalities as the LP relaxation, we obtain a primal-dual algorithm that achieves a performance guarantee of 2. Computational experiments demonstrate the effectiveness of this algorithm on generated problem instances. A Euclidean approximate sparse recovery system consists of parameters k,N, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, the system approximates x by ^x=D(Φ x), which must satisfy ||x - x||2≤ C ||x - xk||2, where xk denotes the optimal k-term approximation to x. (The output ^x may have more than k terms). For each vector x, the system must succeed with probability at least 3 4. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, D. In this paper, we give a system with m=O(k log(N k)) measurements--matching a lower bound, up to a constant factor--and decoding time k log O(1) N, matching a lower bound up to log(N) factors. We also consider the encode time (i.e., the time to multiply Φ by x), the time to update measurements (i.e., the time to multiply Φ by a 1-sparse x), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to log(k) factors. The columns of Φ have at most O(log2(k)log(N k)) non-zeros, each of which can be found in constant time. Our full result, an FPRAS, is as follows. If x=xk+ν1, where ν1 and ν2 (below) are arbitrary vectors (regarded as noise), then, setting ^x = D(Φ x + ν2), and for properly normalized ν, we get [||^x - x||22 ≤ (1+e)||ν1||22 + e||ν2||22,] using O((k e)log(N k)) measurements and (k e)logO(1)(N) time for decoding.