This problem becomes severe when the observed time series are very noisy and this is often the case, rather than the exception, for multiple curve alignment.

the solution to DTW is extremely efficient and global optimality (with respect to the DTW objective function) is guaranteed, through the application of dynamic programming [11].


If we select L and R to have a very small number of columns, say r, then the number of real decision variables in the optimization problems (2) and ( 3) is reduced from mn to r(m + n), a dramatic improvement in the dimensionality of the problem.

the new formulation is nonconvex with respect to L and R so it might not be efficiently solvable.


The performances of both methods follow similar curves.

the performance of neural fingerprints with small random weights follows a different curve, and is substantially better.


[40], which is an inductive, embeddingbased approach to semi-supervised learning.

planetoid-I does not use any graph structural information during inference; instead, it uses the graph structure as a form of regularization during training.


For example, in the permutation equivalent subset [{(1, 1), (2, 2)}], the equal constraint is that the first and the second index number are equal and the third and fourth index are also equal.

the unequal constraint requires that the first two index numbers are different from those of the last two.


Objects under deformation often observed large variation globally.

their local structures are somewhat more invariant to the deformations.


Generalization responses (figure 10) show a persistent effect of stimuli that merely resemble a rewarded stimulus.

animals do not terminally confuse normally rewarded and normally non-rewarded stimuli, since if a reward is provided in the latter case, then it engenders OA activity (as an unexpected reward should), and if it is not provided, then there is no depression (as would be the case if an expected reward was not delivered) (Schultz, 1998).


Recurrent neural networks (RNNs) have attained state-of-the-art performance on a variety of tasks, including speech recognition [Graves et al., 2013], language modeling [Melis et al., 2017, Merity et al., 2017, and machine translation , Wu et al., 2016.


Another alternative method is to consider a randomized sampling of the input tensor in each iteration of tensor decomposition [27,2].

such methods can be expensive due to I/O calls and are sensitive to the sampling distribution.


Powerful visual QA architectures contain components, such as ResNets, which are highly capable visual processors capable of detecting complicated textures and forms.

as demonstrated by CLEVR, they lack an ability to reason about the features they detect.


While this method has shown some empirical success [21], its theoretical justification is lacking, since rewards received after a non-optimal action no longer relate to the hard-max Q-values Q • .

the algorithms we propose incorporate the log-probabilities of the actions on a multi-step rollout, which is crucial for the version of softmax consistency we consider.


Suppose T is derived from samples from distribution D and Q from another distribution D . If the two distributions are close (say in L 1 distance), or, more specifically, if the changes do not significantly change the probability that a query y has f (y) ≥ τ , then the empirical false positive rate on the test set may still be relatively accurate.

in practice it may be hard to provide such guarantees on the nature of future queries.


We note that there is a flurry of research on heuristics-based methods to address the multi-objective stochastic optimization problem (see e.g., [8] and [1] for a recent survey on existing methods).

in contrast to this study, most of these approaches do not have theoretical guarantees.


Both of these results suggest that the algorithms people use to sample mental representations are more complex than DS or RwM, and, like MC 3 , are instead adapted to sampling from multimodal distributions.

if people are adapted to multimodal distributions, their behavior appears not to change even when they are actually sampling from a unimodal distribution.


This suffices for many tasks, such as the SVM example above.

with recent successes in deep learning, neural networks are increasingly becoming the method of choice for many machine learning applications.


Understanding and modeling the interactions between factors of variation of data is an open problem.

supervised learning of the mapping of x to target (y) involves a relatively simpler (yet challenging) problem of finding those factors of variation (F y ) that contain all the information required for predicting y and discarding all the others (F y ).


Note that Equation (10) has been mentioned by Grünwald [14], based on an earlier observation of Zhang [36].

the PAC-Bayesian theorems proposed by the latter do not bound the generalization loss directly, as the "classical" PAC-Bayesian results [8,24,29] that we extend to regression in forthcoming Section 4 (see the corresponding remarks in Appendix A.1).


The following theorem is the main result in Abbeel and Ng [1].

some aspects of their analysis are not covered by this theorem, such as the complexity of each iteration of the projection algorithm, and the sensitivity of the algorithm to various approximations.


For example, a function composed of many piecewise linear segments might have a long description length under a typical continuous parametrization (e.g., the radial basis kernel described below), because it violates the smoothness assumptions of the prior.

conditional on the changepoints between segments, the function can be decomposed into independent parts each of which is well-described by a simple continuous parametrization.


This leads to high Order Accuracy, but low Sequence Accuracy.

adding Auto or Freq loss regularizers, as in ( 7) and ( 8), significantly improves the Seq.


Pruning the weights of the baseline models with a threshold of 1e − 4 resulted in 1.5M zeroed-out parameters for the L2 regularizer and 2.8M zeroed-out parameters for the L1 regularizer.

these zeroed out weights are sparsely located within units (neurons).


Our next hope would be to obtain a different good approximation algorithm.

we can show that most likely this is not possible: Theorem 1.


Algorithms for LDR-SD Generic and near-linear time algorithms for matrix-vector multiplication by LDR matrices with even more general operators, including both the LDR-TD and LDR-SD classes, were recently shown to exist [14].

complete algorithms were not provided, as they relied on theoretical results such as the transposition principle [6] that only imply the existence of algorithms.


where e k the k-th elementary symmetric polynomial.

one easily shows that the values r(µ, ν) we derive above for (k-) DPPs are loosely lower-bounded by κ |1−p| , where κ is the condition number of the kernel matrix L. it is possible to obtain a closer SR distribution to ν ∝ det(L) p than the baseline choice of the DPP with kernel L: indeed, as L is positive semi-definite, we can also consider a DPP parametrized by kernel L p . For the rest of this section, we define µ as the SR measure corresponding to the DPP with kernel L p : µ(S) = det(L p [S])/ det(I + L p ), and ν as the ESR measure such that µ(S) ∝ det(L[S]) p . Note that ν remains absolutely continuous with regard to µ.


In fact, they consider a similar setting, but treat the mean vector as random and generated independently from a known normal distribution.

we assume a fixed but unknown mean vector and establish a minimax lower bound as well as an adaptive source coding method that adapts to the fixed mean vector within the parameter space.


Therefore, while switching between Φ = W ΦD and Φ = Φ may influence the initialization and possibly the update rules of a particular Type II algorithm, it does not fundamentally alter the underlying cost function.

type I methods do not satisfy this invariance.


The relative running time for the QR-distribution was comparable to k-means and only slightly higher than uniform sampling.

it never exceeded a factor of two compared to its competitors and remained negligible compared to the full optimization task, see Figure 2   set, where the uniform and k-means based algorithms showed larger values.


MHE is originally defined on Euclidean distance, as indicated in Thomson problem.

we further consider minimizing hyperspherical energy defined with respect to angular distance, which we will refer to as angular-MHE (A-MHE) in the following paper.


Macro-MCVI's policy takes one macro-action, "moving northeast until reaching the border", to localize and another macro-action, "navigating to the goal", to reach the goal.

both MCVI and SARSOP fail to match the performance of Macro-MCVI even when they are run 100 times longer.


Note that complete move-making also requires us to solve the linear program (4).

since problem (4) is independent of the unary potentials and the edge weights, it only needs to be solved once beforehand in order to compute the approximate solution for any metric labeling problem defined using the distance function d. Theorem 1 implies that the approximation factor of the complete rounding procedure is very large for distance functions that are highly non-submodular.


The intuition behind ICA is that it find a linear transformation on the input that minimizes the multiinformation among the outputs (Y j ).

corEx searches for Y j 's so that multi-information among the X i 's is minimized after conditioning on Y . ICA assumes that the signals that give rise to the data are independent while CorEx does not.


Thus, it is critical for a practical system to have a time complexity sublinear in N . One naive approach is to select a subset of arms ahead of time, such as volumetric spanners [19].

this is specialized for an efficient exploration only and can rule out a large number of good arms.


Sections 2 and 3 provide theoretical insight into the fundamental limits of estimation of v 0 from quadratic observations of the form βv 0 v 0 T + Z.

as previously mentioned, the maximum likelihood estimator of Section 3 is NP-hard to compute, in general.


It is expected that the performance can be even more robust.

we have a remark about the basic assumption of the cavity method, namely that the addition or removal of a spin causes a small change in the system describable by a perturbative approach.


Then, for example, one can compute the likelihood of the survey data given parameters of the individual model by summing out the individual variables.

this is highly impractical: if our map has L grid cells, then the variable elimination algorithm run on this model would instantiate tabular potentials of size L M .  shows the CGM for this model, which we obtain by analytically marginalizing away the individual variables to get a new model on their sufficient statistics, which are the tables n t,t+1 with entries n t,t+1 (i, j) equaling the number of birds that fly from i to j from time t to t + 1.


Examples of uniformly convex sets include L p balls for any 1 < p < ∞ with modulus of convexity δ Lp ( ) = C p p for p ≥ 2 and a constant C p and δ Lp ( ) = (p − 1) 2 for 1 < p ≤ 2.

l 1 and L ∞ units balls are not uniformly convex.


A recent scheme called Federated Averaging SGD [5] attempts such a data parallelization in the context of deep learning with significant success; however, they still use a central parameter server.

deep learning with decentralized computation can be achieved via gossip SGD algorithms [6,7], where agents communicate probabilistically without the aid of a parameter server.


The counterfactual fairness principle however forbids us from using Y : using the twin network 4 of Pearl [28], we see in Figure 1(e) that Y a and Y a need not be identically distributed given the background variables.

any function of variables not descendants of A can be used a basis for fair decision making.


[2017] proposed the accelerated (stochastic) power method with optimal rate.

its empirical performance seems not as good as expected in our experiments.


Together with our bound on the log-partition function, this yields This expression is convex in each of (θ, φ) and a, but is not jointly convex.

it is straightforward to show that equation ( 6) and its unlinearized version have a unique point satisfying first-order stationarity conditions.


Our architecture is also related to the multi-dimensional RNN or LSTM [30,3,8].

both the standard RNN and LSTM contain multiple non-linear units and thus do not fit into our proposed affinity framework.


In Rocksample (5, 7), the reward performance of CC-POMCP is comparable to CALP when more than 2 seconds of search time is allowed while at the same time satisfying the cost constraint.

baseline agent basically exhibited random behavior since the Monte-Carlo return at early stage mostly violates cost constraints for all actions.


The loss based decoding approach [12] suggests decoding so as to minimize loss.

it does not provide significant guidance in the choice of encoding method, or the feedback between encoding and decoding which we analyze here.


Its main limitation is that much of the previous work requires additional assumptions such as a small number of seed images in the beginning [13], texts and HTML tags associated with images [19], and user-labeled images [6].

no additional meta-data are required in our approach.


In the generative process above, we mentioned that it is reminiscent of the well known stick breaking construction from Ishwaran & James [12], where you break a stick of length one but it is not the same.

we can effectively reparameterize the model, starting with Equation (2), due to two useful identities in distribution: 1´ř ăj P for j " 1, . . . , K. Indeed, using this reparameterization, we obtain the corresponding joint in terms of K p0, 1q-valued stick-breaking weights tV j u K j"1 which correspond to a stick-breaking representation.


The inter-trial interval (ITI) between subsequent trials was 15 sec.

a new trial could only start when during the last 3 sec before it there were no wrong (ITI) pokes, so as to penalize spontaneous poking.


Overall, this opens up the possibility to annotate action video datasets much more efficiently at a moderate cost.

as explained above, we observe a significant drop when removing the spatial supervision.


In addition, they considered a stochastic version of their algorithms but provided no second-order convergence guarantee.

we also develop a stochastic algorithm with provable second-order convergence guarantee.


Using this approach, let m be the number of hypotheses, we can get a bound of O(m).

this solution ignores the fact that many hypotheses can share the same arm so that choosing an arm yields information for many hypotheses.


Prior versus learned structure Before learning, the convolutional network has little structure specialized to natural images.

the GSM model uses a multi-scale wavelet representation that is known for its suitability in representing natural image statistics.


In Section 4.3, we showed that γ K (P m , Q n ) does not have a nice closed form expression unlike in the case of B being an RKHS.

in the following, we show that for K in Examples 1-3, γ K (P m , Q n ) has a closed form expression for certain choices of q.


Replacing the input vector cJlt +1 by its projection on the linear span of the BVs when updating the GP parameters induces changes in the GP2.

the replacement of the true feature vector by its approximation leaves the mean function unchanged at each B V i = 1, t. That is, the functions (a(x))t+1 from ( 6) and (a(x))t+1 = L~=I Qt+1(i)Ko(Xi,X) have the same value at all Xl.


( 10) is a Semi-Definite Programming (SDP) problem [3], and in general can be solved using SDP packages such as SeDuMi [16].

solving a SDP problem is computationally expensive and does not easily scale to a large number of training examples.


A deterministic ray-conjunction classifier is then specified by choosing the thresholds values It is tempting at this point to choose t i = (a i + b i )/2 ∀i ∈ i (i.e., in the middle of each interval).

we will see shortly that the PAC-Bayes theorem offers a better guarantee for another type of deterministic classifier.


For example, the instance of the person in the inner bounding box of the image in Figure 2(b) has comparably higher SC than the car instance in the outer bounding box when considering "person" class.

when it is in the super-bag of "car", its SC is lower than that of the car instance.


This is satisfied if η ≥ 1 2 + 1 2β in the worst-case, and under the assumption that node delays are i.i.d., it is satisfied in expectation if η ≥ 1 √ β .

this condition is only required for a worst-case analysis, and the algorithm may perform well in practice even when this condition is not satisfied.


A classical K-armed bandit strategy such as UCB, which only uses the highest fidelity, can obtain at best O(λ (2) K log(Λ/λ (2) )) regret [9].

this paper will present multi-fidelity strategies that achieve O (λ (1) K + λ (2) |K g |) log(Λ/λ (2) ) regret.


Comparatively, the densely connected networks are able to explore new information from previous outputs since the f k t (•) is not shared across steps.

different f k t (•) may extract the same type of features multiple times, leading to high redundancy.


We provide a rigorous statistical analysis of our estimator in the high-dimensional setting.

existing approaches for robust sparse GGMs estimation lack statistical guarantees.


Second, along the whole search process, greedy has a decreasing trend, likely due to over-exploitation in early stages.


When a state variable is introduced, we wish to solve Eq.

( 2) for a fixed query state s ∈ S. scenarios are not i.i.d.


Conventional vision approaches that build upon image classifiers work better than simple image features (such as GIST), which is sensible given that object categories provide relatively more semantics than simple features.

all the baselines still have performances far from human's, because they miss the key mechanism for inferring the "width" of the latent concept represented by a set of images (instead of a single image as conventional approaches assume).


In multidimensional scaling, for example, pairwise distance measurements are used to form a kernel matrix and PCA is performed on this matrix to embed the data in a low-dimensional subspace.

due to constraints, it may not be possible to measure pairwise distances for all variables, rendering the kernel matrix incomplete.


In turn, the region of synaptic depression tends to follow the bump.

if the time scale of synaptic depression is large, the recovery of the synaptic depressed region is slowed down, and cannot catch up with the bump motion.


The POMDP framework presents a reward function r(s, a) based on the state and action.

the belief MDP presents a reward function ρ(b, a) based on beliefs.


It is also clear that SGD with different step sizes can achieve similar generalization bounds.

the computational complexity to fulfill this statistical potential can be significantly different.


Most previous methods generally assume that the multiple tasks can be well correlated given the shared representation learned by the feature layers conv1-f c7 of deep networks [25,22,32,27].

it may be vulnerable if different tasks are not well correlated under deep features, which is common as higher layers are not safely transferable and tasks may be dissimilar.


(2017) leverage an adversarial training scheme where a binary CNN-based discriminator is used to evaluate whether a transferred sentence is real or fake, ensuring that transferred sentences match real sentences in terms of target style.

in practice, the error signal from a binary classifier is sometimes insufficient to train the generator to produce fluent language, and optimization can be unstable as a result of the adversarial training step.


It suggests that we might be able to embed data into 2 and then simply apply well-established methods for 2 nearest neighbor search.


If an optimization problem has a sparse solution, it is more suitable than its counterpart RCD.

the theoretical convergence rate is still O(1/ ).


We can see that as predicted by theory, the raw commute distance does not identify the cluster structure.

the cluster structure is still visible in the kernel corresponding to the commute distance, the pseudoinverse graph Laplacian L † .


the origin) satisfies the approximation condition.

the ordering (and hence the ranks) of those distances remains the same.


Secondly, [10] provides the up-to-date Rademacher complexity bounds of both L 1,∞ and L 2,2 norm-constrained fully connected DNNs without bias neurons.

it is not straightforward to extend their results to fully connected DNNs with a bias neuron in each hidden layer.


One simple solution is to add an extra image channel where each pixel holds the side information [10], which is equivalent to using 1st-layer filter bias terms that change with the side information.

this may not be the most effective solution when the side information is a high-level property with a complex relationship with the image appearance (e.g., the camera angle).


Hex requires complex strategy, making it challenging for deep Reinforcement Learning algorithms; its large action set and connection-based rules means it shares similar challenges for AI to Go.

games can be simulated efficiently because the win condition is mutually exclusive (e.g.


One concern might be that most observations of 2-state fluctuations in vivo have been when the animal is anesthetized, implying that this kind of neuronal dynamics is an artifact of the anesthetized state.

these fluctuations have been observed in several different kinds of anesthesia, including local anesthesia [16].


Figure 1 shows the value λ(a 0 , b 0 ) given by eq.

( 5) by numerical calculations, for the cases N = 2, 3, .., lambda/N a0||b0|| "Gener:N=2" "Gener:N=3" "Gener:N=4" "Gener:N=5" "Gener:N=6" "lambda=1" if N ≥ 4, the learning coefficient λ(a 0 , b 0 ) is always smaller than N , even if the true parameter is not contained in singularities.


synaptic weights, time constants, etc.).

the η-function assigns concrete nonlinear operations to the LGMD [7].


For example, a 2-hop enclosing subgraph will contain all the information needed to calculate any first and second-order heuristics.

although first and second-order heuristics are well covered by local enclosing subgraphs, an extremely large h seems to be still needed for learning high-order heuristics.


The standard assumption underpinning DBNs is that of stationarity: time-series data are assumed to have been generated from a homogeneous Markov process.

regulatory interactions and signal transduction processes in the cell are usually adaptive and change in response to external stimuli.


This can be done via L , where 1 is a vector of 1's.

with a user defined matrix multiplication, the eigensolver in rARPACK runs slightly slower.


The update w i,t → w i,t+1 is based on the gradients ∇ i,t of a loss function L(•) satisfying L (∆) = −g(∆).

the update A i,t−1 → A i,t uses the rank one matrix 4 x t x t . In both the update of w i,t and the one involving A i,t−1 , the reader should observe the role played by the signs s i,t .


Many recent works [8,9,4,10,11] followed the idea of [3] to construct feed-forward networks by unfolding and truncating iterative algorithms, as fast trainable regressors to approximate the solutions of sparse coding models.

progress has been slow towards understanding the efficient approximation from a theoretical perspective.


The last expression can thus be further bounded by where in the first inequality we use the fact for m ≥ m * , ui x (m) ≥ ui x (m * ) ≥ ui 2x (M ) ≥ 1, and that h(y) is positive and increasing when y ≥ 1.


The algorithm is based on optimal coloring of chordal graphs and requires at most log interventions to learn any causal graph where is the chromatic number of the chordal skeleton.

one important open problem appears when one also considers the size of the used interventions: Each intervention is an experiment where the scientist must force a set of variables to take random values.


The natural approach would be to try and generalize Algorithm 1 to work in multiple dimensions.

the obvious extension requires querying the function along every direction in a covering of the unit sphere so that we know the behavior of the function along every direction.


Once we have analytic forms for the posteriors in ( 16), the predictive density is approximated as which is a non-Gaussian integral that cannot be computed analytically.

following the same argument as in [9,13], we can calculate analytically its mean and covariance: where All expectations are taken w.r.t.


This might seem a pessimistic result, since under model (1), we essentially have Kn observations of the coefficient vector β * with the same design matrix but K independent noise realizations.

the thresholds as well as the rates of convergence in high-dimensional results such as Theorem 1 are not determined by the noise variance, but rather by the number of interfering variables (p − s).


Most previous work in the literature tackled the influence estimation and maximization problems for infinite time window [1,2,3,4,5,6].

in most cases, influence must be estimated or maximized up to a given time, i.e., a finite time window must be considered [7].


Simulating all the latent variables X L K jointly is useful since, in general, this will reduce the autocorrelation when compared to simulating the variables x j one at a time [26].

it is also possible to employ PMCMC to construct an algorithm in between these two extremes, a strategy that we believe will be particularly useful in the context of PGMs.


Designing a principled procedure for such risk-shaping is not trivial, and is beyond the scope of this paper.

we believe that there is much potential to risk shaping as it may be the key for handling model misspecification in dynamic decision making.


To classify documents represented as a set of vectors, we can use support measure machines (SMMs), which are a kernel-based discriminative learning method on distributions [8].

these low dimensional representations of documents or words might not be helpful for improving classification performance because the learning criteria for obtaining the representation and the classifiers are different.


when discretizing the gradient flow with step size h, the eigenvalues of the Jacobian at a fixed point are projected into the unit ball along rays from 1.

this is only possible if the eigenvalues lie in the left half plane and requires extremely small step sizes h if the eigenvalues are close to the imaginary axis.


[8] shows how to find classifiers that preserve -differential privacy; however, their algorithm takes time exponential in d for inputs in R d . [3] provides a general method for publishing data-sets while preserving -differential privacy such that the answers to all queries of a certain type with low VC-dimension are approximately correct.

their algorithm can also be computationally inefficient.


these methods suffer from three main problems.


That is, only the features (rows of W ) that share a location (e.g., only the red nodes in figure 2) were orthonormalized using symmetric orthogonalization.

visualizing the filters learned by a TCNN with local orthonormalization, shows that many adjacent receptive fields end up learning the same (copied) filters due to the lack of an orthonormality constraint between them.


An exception is parallel WaveNet [18] which generates a sequence with a fully convolutional network for faster inference.

the parallel approach is trained to reproduce the output of a standard WaveNet, which means that faster inference comes at the cost of increased training time.


End-to-end approaches like these have been particularly successful in perceptual domains like computer vision where designing intermediate representations is a big challenge.

within program synthesis, there exists a great deal of structure and auxiliary information the model could learn to exploit in addition to the typically used input/output examples.


Tensor Comprehensions [41] combine both approaches, using black-box optimization to choose parameters of thread blocks and polyhedral optimization to generate internal loops.

black-box approaches can require many experiment trials to explore a huge S e . predefined cost models may not be sufficiently accurate to capture the complexity of modern hardware and must be manually redefined for each new hardware target.


The definition above is chosen for the sake of continuity with the next section, and it also makes the algorithm below more intuitive.

in order to test conditional independence statistically one does not have to use mutual information, but may resort to any other divergence measure instead.


We can use off-the-shelf softwares to solve this problem in polynomial time.

the whole optimization procedure may not be very efficient since we need to solve problem (3) and call QP solvers for multiple times.


A naive approach is to estimate the k-th surface integral for each k (in some way) and sum them up.

with L hyperplanes, the number K of regions can grow as fast as O (L n ), implying that the naive approach is slow.


Being universal function approximators [Gybenko, 1989], deep neural networks can easily uncover the complicated variations in data [Zhang et al., 2017], leading to powerful representations.

how to systematically incorporate a desired invariance into the learned representation in a controllable way remains an open problem.


It follows from Theorem 3 that no k-NN algorithm is Bayes-consistent on it.

the following theorem shows that KSU is weakly Bayes-consistent on this distribution.


inferred the network connectivity using a submodular optimization, called NETINF [4].

both CONNIE and NETINF assume that the transmission model for each pair of nodes is fixed with predefined transmission rate.


In this problem the goal requires regulation about the state where the pendulum is inverted and in an unstable equilibrium.

the nonlinearities of the problem limit the region of applicability of a linear policy, and non-trivial trajectories have to be created to cover a larger region.


The MV-LCB is specifically designed to minimize the probability of pulling the wrong arms, so whenever there are two equivalent arms (i.e., arms with the same mean-variance), the algorithm tends to pull them the same number of times, at the cost of potentially introducing an additional variance which might result in a constant regret.

expExp stops exploring the arms after τ rounds and then elicits one arm as the best and keeps pulling it for the remaining n − τ rounds.


To assess the quality of a learned model it is common to evaluate the likelihood of a set of held out documents.

this turns out to be a challenging inference problem on its own which has attracted significant attention [37,5,32,22].


To give the theoretical guarantees for the algorithms introduced in the next section, we assume not only the adaptive submodularity and the adaptive monotonicity, but also the policy-adaptive submodularity.

our theoretical analyses can still be applied to many applications.


In this sense, the model is non-parametric, and allows any target concept.

the complexity of learning depends upon the number of sets, which we are denoting by m, and would be trivial if m = |X |.


Because variational inference jointly learns the model parameters along with the inference network, we could theoretically use an inference algorithm such as SMC (Doucet et al., 2001) for more accurate estimates of the true posterior predictive distribution.

our cutoff prediction must be done in an amortized setting because we rely on it to be set for a gradient run prior to the updates returning from the workers.


Then, the prediction is calculated by weighted majority-voting of the outputs of all CNNs.

each CNN may not have sufficient training data, which is especially true with limited AU-coded data.


The consistency of this framework was shown by [13] in the context of robust optimization [14].

in practice it usually leads to a large or infinite number of constraints, and hence tractable formulations inevitably rely on approximating or restricting the invariance under consideration.


In the transformed graphical model, the training data x tr i , y tr i and the prior θ are conditionally independent.

since we have only a crude approximation to p(φ i | x tr i , y tr i , θ), this independence often doesn't actually hold.


Which image should be labeled (or relabeled) next must also be decided -it may be prudent, for example, to collect many labels for each image in order to increase one's confidence in that image's label.

if an image is easy and the labelers of that image are reliable, a few labels may be sufficient and valuable resources may be used to label other images.


Training typically involves a series of gradual adjustments of stimulus conditions and rewards in order to bring about learning.

training protocols are usually hand-designed, relying on a combination of intuition, guesswork, and trial-and-error, and often require weeks or months to achieve a desired level of task performance.


The determinantal point process, as described in section 2, is a powerful tool for modeling diverse subset selection.

video frames are more than items in a set.


Thus, deep (fully) convolutional architectures that are as translation-  invariant as possible are preferable as evidenced by the leading results on ImageNet classification [10,25,27].

the object detection task needs localization representations that are translation-variant to an extent.


In this simple case the posterior of ω can be calculated exactly.

in the applications we consider the z i are latent variables, and they are also simulated as part of a larger Gibbs sampler.


On one hand, we want to have a small k so that the sketched matrices can fit into memory.

the parameter k controls how much information is lost during sketching, and a larger k gives a more accurate estimation of the inner products.


Therefore, a component receives a high compensation weight γ k = 2λ( βk − µβ k ) if some neighboring spatial locations have already been selected (i.e., made "active") by the composite classifier.

the weight of a component is reduced (proportional to the magnitude of parameter µ) if it is already "active", i.e., β k > 0.


Thus, this type of a greedy update is a special case of a natural gradient update in which the step-size approaches infinity.

the requirement that θ k → ∞ will hold only during the first step using a constant α → ∞, assuming a bounded initial θ.


Consequently, the algorithm originally presented above does not perform well for small bandwidths.

few sources have a contribution greater than q i ǫ at low bandwidths, since the cut-off radius becomes very small.


Therefore we do not have monotonic convergence guarantees.

through better approximation and optimization of the discriminator we may expect the proposed dual GAN to work better than standard gradient based GAN training in some cases.


Surprisingly, we found the algorithm performed similarly irrespective of p and all outperformed DQN, as shown in Figure 5b.

this is strange and we discuss this phenomenon in Appendix D. in light of this empirical observation for Atari, we chose p=1 to save on minibatch passes.


Naively computing δ tar would involve explicitly enumerating all 2 p − 1 feature subsets and evaluating their respective minimum attainable p-values, something unfeasible even for moderate p. Existing work in significant discriminative pattern mining solves that limitation by exploiting specific properties of certain test statistics, such as Fisher's Exact Test or Pearson's χ 2 test, that allow to apply branch-andbound algorithms to evaluate δ tar .

the properties those algorithms rely on do not apply to conditional statistical association tests, such as the CMH test.


The expression in (1) provides a variational bound for matching p θ (x) to samples of q(x), thus one may naively think to simultaneously set a similar variational expression for q φ (z), with these two variational expressions optimized jointly.

to compute this additional variational expression we require an analytic expression for q φ (x, z) = q φ (z|x)q(x), which also means we need an analytic expression for q(x), which we do not have.


Once the tracker saturates, it jumps to the end of the recording.

when the location-aware network failed it just stopped aligning -no particular frames were selected for each phone.


For an individual with very few transactions (e.g., only one) we can "shrink" our predictive profile for that individual towards a general population profile.

for an individual with many transactions, their predictive model can be more individualized.


Like the coordinationfree approach, OCC exploits the infrequency of data-corrupting operations.

instead of allowing occasional data-corruption, OCC detects data-corrupting operations and applies correcting computation.


Often this is modeled as a system that optimally combines several sources of information about the same variable.

it has long been realized that this information combining depends on the interpretation of the data.


Adjusting the weights is computationally most straightforward: directly following belief update equation 1, the importance weight w(m) on model m is given by: where The advantage of simply reweighting the samples is that the belief update is extremely fast.

new experience may quickly render all of the current model samples unlikely.


This is due to the fact that the Gamma band LFP is a filtered version of the wide band LFP, making these signals highly correlated in the Gamma band.

in addition to this obvious linear dependency, we observe significant interactions in the lowest frequencies (0.5-2Hz) which can not be explained by linear interaction (and is thus not detected by the linear kernel).


Note the state-of-the-art (SOTA) models on these datasets are [49,21,25] respectively.

these SOTA results often rely on data augmentation [49], semi-supervised learning [25], additional training data (SNLI) [20], or specialized architectures [20].


In order to consider the performance of such an estimator, we need to compare the ROC curve of ŝ to the optimal ROC curve.

if the sequence { Ĉωi } i=1,...,K is not decreasing, the computation of the ROC curve as a function of the errors of the overlaying classifiers becomes complicated.


The aggregate Markov model (AMM) of Saul and Pereira [13] (also discussed by Hofmann and Puzicha [10] as a special case of the aspect model) factors the conditional probability table of a word given its context by a latent variable representing context "classes".

this latent variable approach is difficult to generalize to multiword contexts, as the size of the conditional probability table for class given context grows exponentially with the context length.


For a few points, the criterion picks a maximally smooth representation (the highest timescale considered here, h =12), possibly to increase the # criterion.

our preliminary experiments suggest that the adaptive time scales do not have a special classification advantage for this dataset.


Since SHAZOO works in the online transductive model, it can easily be applied to the more standard train/test (or "batch") transductive setting: one simply runs the algorithm on an arbitrary permutation of the training nodes, and obtains a predictive model for all test nodes.

the implementation might take advantage of knowing the set of training nodes beforehand.


It should also be noted that, because hypercubes are all k-regular graphs (in particular, the hypercube Q n is n-regular), the form of the normalized Laplacian becomes simple.

the usual formula for the normalized Laplacian is: since the graph is regular, we have D = kI, and so Also note that, because the formula for the combinatorial Laplacian is L = D − A, we also have L = 1 k L. The Laplacian also distributes over graph products, as shown in the following theorem.


In [19] the authors consider a model similar to ours, but instead of learning a single monotonic function, they learn multiple monotonic functions, one for each row of the matrix.

unlike in this paper, their focus is on a ranking problem and their proposed algorithms lack theoretical guarantees.


Electrophysiological studies performed in primary visual cortex agree in reporting high sparseness values for neural activity [7,8,9,10,11,12].

it is contested whether the high degree of sparseness is due to a neural representation which is optimally sparse, or is an epiphenomenon due to neural selectivity [10,12].


This is true also in the unconstrained optimal sampler.

e and I inputs are strongly anti-correlated in LS.


For both datasets, SGD-DET performs badly due to its biased gradient estimators.

sGD converges much faster and outperforms GD, where SGD for 10M converges much slower than that for 1M due to the larger dimension d = dr (see Theorem 5).


So, T iterations will take To achieve -accuracy, ignoring again the logarithmic factor, we need T = From the last expression, for a fixed number of processors n, there exists an optimal value for h that depends on the network size and communication graph G: If the network is a complete graph, using h opt yields τ ( ) = O(n); i.e., using more processors hurts performance when not communicating every iteration.

if the network is a k-regular expander then τ ( ) = c1 √ n + c 2 for constants c 1 , c 2 , and we obtain a diminishing speedup.


Bayesian model averaging is a coherent approach to dealing with the uncertainty in model selection, and it has been shown to improve the predictive performance of classifiers [2].

since the number of models in practically useful model spaces is enormous, exact model averaging over the entire model space is usually not feasible.


Following a standard procedure [11], we introduce a variational distribution q(Θ) and compute the Jensen's lower bound F v on the logarithm of ( 9), where θ denotes the model's parameters.

the above form of the lower bound is problematic because X (in the GP term p(F |X)) appears non-linearly inside the covariance matrix K N N making the integration over X difficult.


r t denotes the rewarding value of the outcome the animal receives at time t, which is a often set to a "fixed" value that can be either positive or negative, depending on whether the corresponding stimulus is appetitive or aversive, respectively.

animals motivation for outcomes is not fixed, but a function of their internal state: a food pellet is more rewarding to a hungry, than a sated rat.


Clearly, not all problems will benefit from this approach.

given the small computational overhead and there is no reason to not try.


Most commonly, STDP is viewed as a type of asymmetric Hebbian learning with a temporal dimension.

this perspective is hardly a fundamental computational rationale, and one would hope that such an intuitively sensible learning rule would emerge from a first-principle computational justification.


In both cases, a weaker analysis led to O(1/ 2 ) rates of convergence for nonsmooth loss functions.

our results establish a O(1/ ) rate for nonsmooth loss functions and O(log(1/ )) rates for smooth loss functions under mild technical assumptions.


As we have shown, our simulations confirm an average growth of spatial summation extent of excitation (and inhibition) at low contrast.

this growth is neither sufficient nor necessary to explain receptive field growth.


selection of expert, communication patterns or any random bits used.

the oblivious adversary may know the description of the algorithm.


End-to-end deep-learning networks recently demonstrated extremely good performance for stereo matching.

existing networks are difficult to use for practical applications since (1) they are memory-hungry and unable to process even modest-size images, (2) they have to be trained for a given disparity range.


The disadvantage of this approach is that we need this extra data to encode a given input.

this coding scheme works even if the variance of the approximate posterior is dependent on the input.


for modern data-intensive applications, minimization of empirical loss with an off-line optimization solver could suffer from very poor scalability.


Compared to the baseline model, the linear-combination cache model (ResNet32-Cache3; green) significantly reduces the first singular value but slightly increases the lower-order singular values.

although the cache-only model does not reduce the first singular value as much as the linear-combination cache model does, it produces a more consistent reduction in the singular values (ResNet32-CacheOnly; black).


We see that the best test accuracies are given by depths just below where these numerical issues occur.

if we were to predict that the optimal init is the one minimizing (0) / (L) 1, then we would be wrong -in fact it is exactly the opposite.


In practice, current approaches make approximations for the computation of the posterior entropy [12,9].

a second difficulty arises; if n new data points are available for selection, with |{−1, 1}| = 2 possible values for y.


However, [39] consider only finite MDPs, which poses a severe limitation on the algorithm's applicability.

our approach can handle high-dimensional tasks.


Therefore, in low dimensions, the causal order can be found by exhaustive search over all orders.

as mentioned earlier for large dimensions, this is infeasible.


Additionally, under certain assumptions, "unrolling" the model in time effectively removes the cycles, which is used in methods such as vector auto-regressive models, which are popular in econometrics, or more generally, Dynamic Bayesian Networks [5] and ordinary differential equation models.

all these methods need time series data where the temporal resolution of the measurements is high relative to the characteristic time scale of the feedback loops in order to rule out instantaneous cyclic relationships.


calculating q(x i |x S t , y) may vary according to different Qdistributions.


Since L Y is a (k − 1) × (k −1) matrix, the per-iteration cost is O(k 3 ).

this complexity can be reduced by applying Equation 3 on every iteration to update the inverse.


It is therefore not surprising that the best performing models considered (in terms of area under the ROC curve, given in Table 1) are the distance models, with the eigenmodels close behind.

the latent class models perform poorly, and the results suggest that increasing K for this model would not improve its performance.


To simplify notations, we consider TGL with a single tree, in the proof.

all major results are directly applicable to TGL with multiple trees, as they are independent from each other.


In [6], exact clustering of TSC is guaranteed if max a↵ = O(1/ log(nL)).

these algorithms perform empirically better as the number of data points increases.


The general proof strategy is closest to the one of UCRL-γ [6] and the obtained bounds are similar if we replace the time horizon H with the equivalent in the discounted case 1/(1 − γ).

there are important differences that we highlight now briefly.


This way we keep the same invariant (η is never larger than the right-hand side of (9), with equality when the budget is depleted), and the frequency of doubling remains logarithmic in L * T with a constant determined by φ, so both approaches are equally valid.

controlling the sequence of values of η allows for easier comparison to AdaHedge.


The advantage of this "greedy" homotopy path is that it provides a set of action-values that are a greedy fixed point for all settings of the L 1 regularization parameter.

this additional flexibility comes with increased computational complexity.


Besides, our relaxation algorithm has O(n) parameters in the SDP cone and O(n) linear equality constraints, which involves a worst-case computational complexity of O(n 4.5 ).

in the previous relaxation algorithms [1,14], there are approximately O(n 2 ) parameters in the SDP cone, which involve a worst-case computational complexity in the scale of O(n 6.5 ).


Interestingly, most methods performed well on REC-STREET and YT-CLEAN.

the visual encoder and separation block are necessary for more complex videos as in YT-MUSIC and YT-ALL.


A deterministic apprentice policy π A that closely imitates the expert will either set π A (s) = a s 1 or π A (s) = a s 2 , but in either case the classification error will not be less than 1 2 .

since π E is optimal, both actions a s 1 and a s 2 must be optimal actions for state s, and so the apprentice policy π A will be optimal as well.


We say the stochastic gradient has second moment at most Observe that any stochastic gradient with second moment bound B is automatically also a stochastic gradient with variance bound Second, in convex optimization, one often assumes a second moment bound Data: Local copy of the parameter vector x 1 for each iteration t do 2 Let g i t be an independent stochastic gradient ; )) //encode gradients ;  when dealing with non-smooth convex optimization, and a variance bound when dealing with smooth convex optimization.

for us it will be convenient to consistently assume a second moment bound.


These are common failure modes of monocular depth algorithms [26].

these qualitative results show that epistemic uncertainty captures difficulties due to lack of data.


We explored whether any combinations of target contrast and number of distractors would produce significantly different predictions of the ideal observer vs the Max model approximation and found none in the case where the visual system can estimate decision thresholds in advance.

our simulations predict different error rates when interleaving images containing diverse contrast levels and distractor numbers.


To do inference we would like to calculate max h P (h v), but this is difficult because of the partition function.

we can consider the mean field approximation, which can be solved by regarding the equation as a dynamics and iterating it.


Second, the error in the PI estimate of location grows unbounded with time, as expected due to the accumulating effects of noise in the motion estimates (Figure 2, gray).

the errors in the network and PF -which make use of the same motion estimates -remain bounded.


The first result is about properties of TRW.

the TRW algorithm provides provable upper bound on log-partition function for arbitrary graph [3]to the best of authors' knowledge the error is not quantified.


Incorporation of such priors will require careful attention to the interchange between taking the limit of and integrating over the inverse temperature parameter.

in many practical situations 1 and 2 -penalty parameters are pre-determined by cross-validation.


[3], Liu and Wright [19] found that the upper bound depends on the problem dimension as well as the diagonal dominance of the Hessian matrix of the objective.

it still lacks a comprehensive and generic analysis to comprehend all pieces and show how these factors jointly affect the speedup property.


[3] showed that temporal information in videos and spatial context information in images can be utilized as a convenient supervisory signal for learning feature representation with CNNs.

the computational cost of the training algorithm is enormous since the approach in [3] needs to tackle all possible pair-wise image relationships requiring training set that scales quadratically with the number of samples.


In other words, the solver without screening needs about eight and a half hours to solve the 100 SGL problems for each value of α.

combined with TLFre, the solver needs only six to eight minutes.


Any new subjects or new data will be referenced to the original SRM.

if we independently learn two SRMs, the group shared responses S 1 , S 2 , may not be registered (use the same Q).


This strategy was pioneered in [5], who obtained the first halfway statistically satisfactory learning procedures.

here we set out from the second mechanism by which C influences model variance, namely, choose C such that the variance of C V itself is minimized.


In particular, note that the regret does not depend on n. This seemingly contradicts the result of Lai and Robbins (1985) who showed that the expected regret of any algorithm is at least 1)) log n where p i⇤ and p i are the reward distributions of the optimal arm and arm i respectively and D is the Kullback-Leibler divergence.

our algorithm receives as an input, and thus its expected regret depends on .


When we sample from the global policy π θ (u t |x t ), our method resembles policy gradient methods with KL-divergence constraints [14,13,17].

policy gradient methods update the policy π θ (u t |x t ) at each iteration by linearizing with respect to the policy parameters, which often requires small steps for complex, nonlinear policies, such as neural networks.


The sound alone doesn't provide us with sufficient information to infer the semantic context.

if the sound is followed by applause, we guess the context to be baseball, screams or sirens suggest an accident, while monotonic repetitions of the metallic sound suggest someone using a hammer.


In drawing BNs we will not include the plates to avoid cluttering the figures.

it is always possible to infer the number of times variables in the BN are replicated by looking at its indices.


Other approaches solve it in the primal form with stochastic functional gradient descent.

these algorithms need to store all the training points seen so far.


Determining how much of the sensory information carried by a neural code contributes to behavioral performance is key to understand sensory function and neural information flow.

there are as yet no analytical tools to compute this information that lies at the intersection between sensory coding and behavioral readout.


In contrast, geometric arguments can be made to show that uncorrelated sources with large n offer the best opportunity for local minima avoidance.

when strong correlations are present as well as high noise levels, the method of [14] (which explicitly attempts to model correlations) could offer a worthwhile alternative, albeit at a high computational cost.


Here, we considered one dimensional inputs and stationary processes to most clearly elucidate the key properties of Lévy kernel processes.

one could generalize this process to multidimensional non-stationary kernel learning by jointly inferring properties of transformations over inputs alongside the kernel hyperparameters.


4 with respect to all possible values of λ ′ [10], and cross-validation is then employed to find an appropriate solution.

it is not easy to obtain extra data for crossvalidation in BCI since real acoustic environments are often time-varying.


. . , y T ) by iterating the following equation: The RNN can easily map sequences to sequences whenever the alignment between the inputs the outputs is known ahead of time.

it is not clear how to apply an RNN to problems whose input and the output sequences have different lengths with complicated and non-monotonic relationships.


However, the binary descriptors are trained using pair-wise learning methods, which substantially limits their applicability to new tasks.

binary descriptors can be learned with hashing algorithms that aim at preserving original distances between images in binary spaces, such as in [2,20,30,8].


For a fixed empirical distribution p(x F , x T ), we define d(F ), a set function of the FVS F as the minimum value of (1), i.e., When the FVS is unknown, the ML estimate can be computed exactly by enumerating all possible n k FVSs of size k to find the F that minimizes d(F ).

hence, the exact solution can be obtained with complexity O(n k+2 k), which is polynomial in n for fixed k. as our empirical results suggest, choosing k = O(log(n)) works well, leading to quasi-polynomial complexity even for this exact algorithm.


Similarly, we can derive the update equation for the sparseness parameter π.

as the approximation only considers states � s with a maximum of γ non-zero entries, the update has to correct for an underestimation of π (compare Suppl.


At test time, a target-domain observation is first mapped into this shared feature space, then classified.

few of the existing works can model the relationship among different domains, which we note is important for several reasons.


This is the so-called "graph connectivity" problem studied in [19].

failure to achieve SEP does not necessarily imply clustering error either, as the spectral clustering step may give a (sometimes perfect) solution even when there are connections between blocks.


Combining these gradients with a discriminative postprocessor [17], the resulting DRIP Fisher kernel substantially improved upon all state-of-the-art methods for downstream analysis on a large number of datasets.

while DRIP significantly improves several facets of MS/MS analysis due to its parameter learning capabilities, these improvements come at high runtime cost.


When θ was small, the improvement afforded by the LC was negligible (Figure 3B).

when the threshold was significantly greater than noise, the improvement was substantial (Figure 3C).


It should be noted that the second-layer representation does not discard precise time and frequency information (this information is carried in the times and frequencies of the second-layer spikes).

the identities of the features that are active remain invariant to pitch and frequency modulations.


Intuitively, it describes how many outliers can be in the data without the estimator becoming unreliable.

the literature is full of slightly inconsistent and informal definitions of this concept.


The most dominant step is estimating W; when doing Gibbs sampling, if done naïvely, it would O(DK 3 ) time if sampling W row-wise, and O(KD 3 ) time if sampling column-wise.

if using the EM algorithm, estimating W can be done much more efficiently, e.g., using Conjugate Gradient updates because, it is not even required to solved for W exactly in each iteration of the EM algorithm [20].


For tanh and ↵-ReLU with ↵ < 1, they are still subexponential and polynomial (Thm B.21).

while (0) = exp(⇥(l)) (l) for a ReLU resnet, its weight gradients have size independent of layer, within O(1) (Thm B.21)!


With these assumptions, the probability of an assignment ι C,k,m that maps k named instances to m distinct objects, when C contains n objects, is given by When n > m, the world contains objects unreferenced by any of the terms.

these filler objects are obviously irrelevant (if they affected the attributes of some named term, they would have been named as functions of that term.)


[8] proposed a somewhat related method for genome-wide analysis of protein interaction data, based on protein domains.

their method is focused on predicting protein-protein interactions and not on revealing the site of interaction, and they do not allow for the possibility that some domains are inactive.


For a fixed noise and privacy level, a smaller λ leads to small impact of noise on prices, meaning very good accuracy.

decreasing λ does not come for free: the worst-case financial loss of to the market designer scales as 1/λ.


Munro and Hernandez numerically showed that a network can retrieve spatio-temporal patterns even in a noisy environment owing to LTD [9].

they did not discuss the reason why TAH was effective in terms of the storage and retrieval of the spatio-temporal patterns.


[13] showed the pyramidal width for P k only with k = 1.

[18] suffers from multiple inherent restrictions.


To capture this information, we can adapt the attention mechanism [2,17] -first compute a weight of the importance of each token in the sentence, and then output a weighted sum of the embeddings of these tokens.

our intuition suggests that the weight for each token depends not only on the token itself, but also the overall sentence structure.


The goal of the classification task is to find a good hypothesis f with small excess risk R 0−1 And to find the good hypothesis based on the samples, one minimizes the empirical risk.

using 0-1 loss, it is hard to find the global minimizer of the empirical risk because the loss function is discontinuous and non-convex.


This GLM mean response is a known function of η Zd in all standard cases.

e q [µ(η Zd ) Zd ] has   an exact solution only in some cases (e.g.


[8] for monotone submodular objective functions are quite efficient, requiring only O(k) memory (k is the size of the largest feasible set) and using only O(km) value and independence oracle queries for processing a single element of the stream (m is the number of matroids used to define the p-matchoid constraint).

the algorithms developed for non-monotone submodular objectives are much less efficient (see Table 1 for their exact parameters).


Similar to the fast bipartiteness/K-colorability test, there is a loss in the power of the fast max clique test due to both the estimation and the sketching procedures.

unlike the fast bipartiteness/K-colorability test, the sampling procedure also contributes to partial of the type-I error in the fast max clique test which implies a more stringent requirement on the sampling percentage.


Figure 2 (bottom) shows that both VASE and CCI-VAE learn progressively more informative latent representations when exposed to each dataset s, as evidenced by the increasing classification accuracy and decreasing mean squared error (MSE) measures within each stage of training.

with CCI-VAE, the accuracy and MSE measures degrade sharply once a domain shift occurs.


The main challenge in the unsupervised setting is that since the alignment information is not available, the Hankel matrices (as in [4]) can no longer be directly estimated from observable statistics.

a key observation is that we can nevertheless compute observable statistics that can constraint the coefficients of the Hankel matrix.


In (4), τ (k) > 0 is a "global" scaling parameter for all the elements of {ω (k) ij |i = 1, ..., n, j < i} and generally fixed to be one [18], and γ With the Laplacian prior (4), many off-diagonal elements of Ω (k) are encouraged to be close to zero.

in the inference procedure, the above Laplacian shrinkage prior ( 4) is inconvenient due to the lack of analytic updating expressions.


:::l 100000) number of states corresponding to individual atoms of speech.

in the mel-cepstral domain, the noisy reverberant speech has a strong non-linear relationship to the clean speech.


Recently, online crowdsourcing platforms have dramatically decreased the labeling cost by dividing the workload into small parts, then distributing micro-tasks to a crowd of ordinary web workers [17,20].

the labeling accuracy of web workers could be lower than expected due to their various backgrounds or lack of knowledge.


Too small values of τ mean that the contribution from the very first nodes is vanishingly small, effectively decreasing the reservoir dimensionality, which has a strong impact on the performance both of the ideal and the experimental reservoir.

larger values of τ impact the performance of the experimental readout, as the residual term in equation 7 gets larger.


Proposition 1 For A = ab ∈ A k,q with strength γ = γ(a, b), there exist universal constants Our proof, presented in the appendix, follows the scheme proposed in [7] and used for the trace norm and 1 norm.

ω k,q is not decomposable and requires some work to obtain precise upper bounds on various quantities.


In general, learning to accurately classify object images from a single training example is unfeasible due to overfitting effects of high dimensional data.

if a certain distance function over the instances guarantees that all within-class distances are smaller than any betweenclass distance, then nearest neighbor classification could achieve perfect performance with a single training example.


Local-SVRG relies on local smoothness to certify that some ∆ t i = ▽φ i x ⊤ i w t − ▽φ i x ⊤ i w are small.

empirical Affinity SVRG (Algorithm 4) takes ∆ t i > t to be evidence that a loss is active; when ∆ t i = 0 several times, that is evidence of local affinity of the loss, hence it can be sampled less often.


As a result, it could easily be extended to other domains like speech, or text, where the backpropagation through the decoder can be really challenging because of the non-differentiable text generation process for instance.

methods commonly used in vision to assess the visual quality of the generated images, like PatchGAN, could totally be applied on top of our model.


multinomial logistic regression) that has as many units as the number of classes (see, for instance [11]).

as the number of classes grows, the distinction between classes blurs, and it becomes increasingly difficult to obtain sufficient numbers of training images for rare concepts.


The standard objective here is a myopic, backwards looking one -in retrospect, we desire that our performance is not much worse than had we chosen any single expert on the sequence of examples provided by Nature.

a reinforcement learning setting typically makes the much stronger assumption of a fixed environment, typically a Markov decision process (MDP), and the forward looking objective is to maximize some measure of the future reward with respect to this fixed environment.


When deriving asymptotic lower bounds, this is circumvented by only considering consistent algorithms whose regret is sub-polynomial for any problem [12].

this asymptotic notion of consistency is not applicable to finite-horizon problems.


The difficulty in defining multiresolution analysis on discrete spaces is that there is no natural analog of dilation, as required by Mallat's fourth axiom.

in the specific case of the symmetric group, we do at least have a natural multiscale structure on our domain.


This result suggests that completely informative features are extremely powerful for matrix completion, and the algorithm has been successfully applied in many applications [29,37].

this model is still quite restrictive since if features are not perfect, it fails to guarantee recoverability and could even suffer poor performance in practice.


First, it is known that for some hypothesis classes, active learning cannot improve over passive learning for certain distributions (Dasgupta, 2005), and the same is true for auditing.

exponential speedups are possible for active learning on certain classes of distributions (Balcan et al., 2006;Dasgupta et al., 2008), in particular ones with a small disagreement coefficient (Hanneke, 2007a).


The basic idea is to identify "important" co-ordinates by constructing suitable biased estimators of each signal coefficient, followed by a specific eigendecomposition.

the initialization in CoPRAM is far simpler than the approaches in [22,23]; requiring no pre-processing of the measurements and or tuning parameters other than the sparsity level s. A drawback of the theoretical results of [23] is that they impose a requirement on signal coefficients: this assumption disobeys the power-law decay observed in real world signals.


In contrast to a single-layer, two-frame model, the reconstruction error is no longer directly applicable (although a naive way to train the model would be to minimize reconstruction error for each pair of adjacent nodes in each layer).

a natural way of training the model on sequential data is to replace the reconstruction task with the objective of predicting future frames as we discuss next.


The majority of these methods do not scale to large graphs or are designed for whole-graph classification (or both) [4,9,8,24].

our approach is closely related to the graph convolutional network (GCN), introduced by Kipf et al.


We do not yet have such a closed-form lower bound on the cumulative price distribution.

as we shall see in Section 5, the price distributions seen in large simulation results do indeed show power-law behavior.


Several methods, such as junction trees [14], belief propagation [15], and sampling-based methods such as Markov Chain Monte Carlo (MCMC) [16,17], have been proposed to find approximate solutions.

we show that this can be calculated exactly when the aforementioned definitions of the factor functions are utilized.


( 2) shows that it is equal to 2/N times the number of £ckle agents at time t, which is Poisson distributed with a mean of N/ √ 2πR = ∆/2, where ∆ ≡ N 2/πR is the average step size.

since the attractor is formed by steps which reverse the sign of A µ , the average step size in the attractor is larger than that in the transient state, because a long jump in the vicinity of the attractor is more likely to get trapped.


The vector parameter β * (x) ∈ R D is the estimated local gradient using local data, and this vector often overfits in a high-dimensional space resulting in a poor solution of α.

lLR asymptotically produces the bias of Eq.


As indicated by Figure 1 (b), due to the sample bias, the approach preferring informative instances tends to choose the data points close to the horizontal line, leading to incorrect decision boundaries.

as indicated by Figure 1 (c), the approach preferring representative instances is able to identify the approximately correct decision boundary but with a slow convergence.


Such guidelines are based largely on the presence of symptoms associated with an existing C. diff infection, and thus are not useful for predicting whether a patient will become infected.

risk stratification models aim to identify patients at high risk of becoming infected.


As the " -NN results suggest, other state-of-the-art techniques like support vector machines will probably lead to better numerical categorization results than the HSOM.

since the main purpose of the HSOM is the visualization of relationships between texts and text categories, we believe that the observed categorization performance of the HSOM compares sufficiently well with the more specialized (non-visualization) techniques to warrant its efficient use for creating insightful maps of large bodies of document data.


The right figure demonstrates that when α increases, more prediction is required during the training time, and the speedup increases from a factor of 1 (no change) to a factor of as much as 9.

as the history length increases, the speedup is more modest due to low cache hits.


Traditionally, meteorological radars, such as the National Weather Service NEXRAD system, are tasked to always scan 360 degrees.

the Collaborative Adaptive Sensing of the Atmosphere (CASA) Engineering Research Center [5] is developing a new generation of small, low-power but agile radars that can perform sector scanning, targeting sensing when and where the user needs are greatest.


In general, this does not mean that the players' behavior stabilizes in the long run: no-regret learning may lead to cycles, even with perfect gradient information.

if a standard monotonicity condition is satisfied, our analysis shows that no-regret learning based on mirror descent with bandit feedback converges to Nash equilibrium with probability 1.


The Deep Voice 2 vocal model is based on a WaveNet architecture (Oord et al., 2016) with a two-layer bidirectional QRNN (Bradbury et al., 2017) conditioning network, similar to Deep Voice 1.

we remove the 1 ⇥ 1 convolution between the gated tanh nonlinearity and the residual connection.


Patterns of use and other factors can result in a drift effect, where the scale varies over time [Siegel and Castellan, 1988].

human beings do excel at comparing options and expressing a preference for one over others [Kingsley, 2006].


The SM kernel has been used for GP regression of a scalar output (i.e., single function, or observation "task"), achieving impressive results in extrapolating atmospheric CO 2 concentrations [24]; image inpainting [25]; and feature extraction from electrophysiological signals [21].

the SM kernel is not defined for multiple outputs (multiple correlated functions).


In our current problem setting, the goal is to train a prediction model f c : x → y for each cluster c of similar treatment sequences, where x denotes the input features and y denotes the label.

the straightforward approach to achieve this is to train a prediction model by using only the samples in cluster c. since the available treatment history for some samples might be incomplete, totally excluding the samples from all other clusters ( = c) ignores relevant information about the model f c . Furthermore, the cluster-specific tasks are related and the samples from the other clusters -especially those close to the cluster boundaries of cluster c -also carry valuable information for the model f c . Therefore, we use a multi-task learning approach where a separate model is trained for each cluster by not only using the training samples from the target cluster, but also the available training samples from the remaining clusters with appropriate sample-specific weights.


The gradients of convolutional layers are 4 dimensional tensors with shape of [x, y, k, k] where x, y are two spatial dimensions and k is the size of the convolutional kernel.

matrices are required to compute the SVD for spectral-ATOMO, and we choose to reshape each layer into a matrix of size [xy/2, 2k 2 ].


Extracting structure in one random variable that is relevant for another variable has been principally addressed recently via the information bottleneck method [15].

such auxiliary variables often contain more information than is actually required due to structures that are irrelevant for the task.


A robot running the optimal 4D planner would still be near the beginning of its path 25 seconds after receiving a goal location (Figure 5d).

in the same amount of time the robot running ARA* has advanced much further (Figure 5f), and its plan by now has converged to optimal ( has decreased to 1).


On the other hand, we could also use the non-convex regularizers which are "steeper" than the 1 -norm for greater sparsity [5].

we will not trade convexity with denoising performance here, because local minima often put practitioners into trouble.


We again use Gurobi 8.0 MILP solver to solve the best response problem for the team.

in the case of HCG, no time limit can be set on Gurobi without invalidating the theoretical convergence guarantee of the algorithm.


Normally the best action to take at time t depends on the entire history of actions and observations that the agent has taken so far.

the probability distribution over current states, known as the belief, is a sufficient statistic for a history of actions and observations.


In general, it is not easy to quantify the difference to the lower bound of Lai and Robbins lim inf Even for Gaussian distributions, there is no general closed-form formula for T * (µ) and w * (µ) except when K = 2.

we conjecture that the worst case is when µ 1 and µ 2 are much larger than the other means: then, the regret is almost the same as in the 2-arm case, and ETC strategies are suboptimal by a factor 2.


If the separation between the trajectory classes is large compared to the temporal correlations (i.e., the first term in (11) dominates for the resulting w) the slowest feature will be similar to the weight vector found by FLD on the corresponding classification problem.

as the temporal correlations of the trajectories increase, i.e., the trajectories themselves become smoother, the slowest feature will tend to favor exploiting this temporal structure of the trajectories over the separation of different classes (in this case, (11) is dominated by the second term for the resulting w).


We include this term for preciseness of the result-even though we will not estimate it-because for smaller T , A T is unlikely to be invertible.

we expect this big-O term to get small quickly, and be dominated by the other terms.


In [17] and [12], however, additional assumptions on the interplay between H and L 2 (P X ) are required, and [17] actually considers a different exponent in the regularization term of (2).

[12] shows that the rate n /( +p) is often asymptotically optimal in a minmax sense.


Here, we see that the quality of the TAP approximation becomes less good.

our results provide still a reasonable estimate for the bootstrapped generalization error at sample size S = N . While for practical applications of estimating the "true" generalization error using Efron's 0.632 bootstrap estimator the case S = N is of main importance, it is also interesting to discuss the limit of extreme oversampling S → ∞.


The GEM principle developed by Hero [4] for determining MV sets circumvents the above difficulties by using the asymptotic theory of random Euclidean graphs instead of function approximation.

the GEM based K-kNNG anomaly detection scheme proposed in [4] is computationally difficult.


1 The likelihood values increase with the number of components, as expected from nested models.

the marginal likelihood stagnates after K = 16.


A banal, but nonetheless valid, behaviorist observation is that hungry animals work harder to get food [1].

associated with this observation are two stranger experimental facts and a large theoretical failing.


Then, the following stopping conditions can be applied, which will help to determine the number of channels to be selected.

given a predefined parameter κ l in problem (7), Algorithm 2 will be stopped if ||W|| 2,0 >κ l . in practice, the parameter κ l is hard to be determined.


If an observation x i is contaminated with large noise so that p(x i |θ , σ i , x i ) has a flat shape, then we can pick a x i that is very different from x i which predicts y i well.

if an observation x i is contaminated with very small noise, then (1) and ( 2) penalize a parameter θ such that p(x i , y i |θ) is small.


An early example of ECA is adaptive direction sampling (ADS) [4], in which each sample is taken along a random direction that is chosen based on the samples from a set of chains.

the validity of ADS can be established only when the size of ensemble is greater than the number of dimensions, otherwise the samples are trapped in a subspace.


From a decoding perspective, a fully labeled concept shows those stimuli that have activated at least the set of neurons in the intent.

the stimuli associated with a concept in reduced labeling will activate the set of neurons in the intent, but no others.


This ensures that the Cholesky update will remain positive definite and doubles the marginal variance instead.

in practice we never encountered any warnings in our experiments if the hyperparameters were initialized sensibly so that the prior was tight compared to the likelihood.


The key step of such gradient-type algorithms is to compute the proximal map (of the nonsmooth regularizer), which is available in closed-form for some specific regularizers.

the proximal map becomes highly nontrivial when we start to combine regularizers.


Different from the setting of samples being corrupted by sparse noise, [25,24] and [7] solve robust PCA in the case that a few samples are completely corrupted.

all of these RPCA methods are implemented in batch manner and cannot be directly adapted to the online setup.


The p ij need to be estimated: [9] use Swendsen-Wang sampling, while [13] use randomized trees sampling.

these trees are not used to define a proximity graph, unlike in our work.


Furthermore, as expected, the projection matrix inferred by the classical CCA had no exact zero entries and even after thresholding significantly small absolute values to zero, the uncovered sparsity was only about 25%.

the projection matrix inferred by the infinite CCA model had 57% exact zero entries and 62% zero entries after thresholding very small values, thereby demonstrating its effectiveness in also capturing the sparsity patterns.


Benefited from the end-to-end training, such attention schemes improve the tracking accuracy by strengthening the discriminative power of features.

the feature weights learned in single frames are unlikely to enable classifiers to concentrate on robust features over a long temporal span.


For example, this situation fits into the product-of-experts framework [6], with one semantic expert and one syntactic expert that must agree on a single structure.

since we are presently only interested in finding most-likely parses, no global renormalization constants need to be calculated.


While t depends on p we will assume it does so only via X.

for example t could depend on p through the marginal p X on X or possible group action on X; it is a manifestation in the data X, possibly over time, of underlying structure in the true joint distribution p 2 P. The representation t captures structure in X induced by p. the regression function itself depends only on the conditional p(y|t(x)).


The few labeled examples are used to further train F to satisfy F (x) = y.

as explained above, the problem with this method is that the network can learn to generate the categorical distribution by ignoring some part of the input distribution.


Ever since, a series of applications such as tracking and action recognition [10], body-joint position estimation [12] and multi-class object detection [22] have been presented.

hough Forests typically produce non-distinctive object hypotheses in the Hough space and hence there is the need to perform non-maximum suppression (NMS) for obtaining the final results.


Context-free grammars (CFGs) underlie most high-performance parsers in one way or another [13,12,14].

a CFG which simply takes the empirical productions and probabilities off of a treebank does not perform well.


The Dawid-Skene estimator has been widely used for inferring the true labels from the noisy labels provided by non-expert crowdsourcing workers.

since the estimator maximizes a non-convex log-likelihood function, it is hard to theoretically justify its performance.


For instance, the CDF F (z| ) of a univariate distribution provides such a function, mapping samples from it to samples from the uniform distribution over [0, 1].

inverting the CDF is often complicated and expensive, and computing its derivative is even harder.


In addition, we note that Fisher Information does not provide an accurate account of the performance of box-like tuning functions: it fails as soon as the nonlinearity in the tuning functions becomes effective and overestimates the true minimal discrimination error E. Similarly, the approximate neurometric functions E d (∆θ) obtained from equation 6 do not capture the shape of neurometric functions E(∆θ) but underestimate the minimal discrimination error.

the deviation between both curves stays rather small for cosine tuning functions.


Based on our experimental results on the benchmark problems, DNG-MCTS typically needs about 2 to 4 times (depending on problems and the iterating stage of the algorithms) of computational time more than UCT algorithm for a single iteration.

if the simulations are expensive (e.g., computational physics in 3D environment where the cost of executing the simulation steps greatly exceeds the time needed by action-selection steps in MCTS), DNG-MCTS can obtain much better performance than UCT in terms of computational complexity because DNG-MCTS is expected to have lower sample complexity.


While there is certainly some optimal trade off between depth and cell size, with enough data, simply upscaling both has yielded remarkable results in neural machine translation (NMT) [6].

1 massive upscaling is impractical in all but the least hardware constrained settings and fails to remedy fundamental architecture issues, such as poor gradient flow inherent in recurrent cells [8].


For the case when c s = 0.1, we notice that the accuracy and search time are relatively unchanged for all the policies.

c-DAC has a notable advantage in terms of number of switches, while the number of switches remain unchanged for Infomax.


is generic-it can be used with any type of latent variables and any model.

the gradient estimates typically suffer from high variance, which can lead to slow convergence.


We consider here the case of binary classification for simplicity.

we note that one can easily extend the construction (and obtain analogous guarantees) for multi-class classification.


Empirical results have shown the benefits of clustering in a low dimensional discriminative space rather than in the principal component space (generative).

the integration between subspace selection and clustering in DisCluster is not well understood, due to the intertwined and iterative nature of the algorithm.


In this case the offset is minimal, thus recovering virtually all revenue.

even if we set the optimal reserve price for every cluster, the inherent variance of each cluster makes us leave some revenue on the table.


It has applied to PageRank [7], sparse PCA [19,9], private PCA [4] and spectral clustering [18].

its convergence rate depends on | 2 |/| 1 |, the ratio of magnitude of the top two dominant eigenvalues [14].


This approach to neural circuit mapping can utilize the light microscope, provides a high-throughput and the potential to monitor the circuits over time, and complements the dense, small scale connectomic studies using electron microscopy [3] with its large field-of-view.

its use has been limited due to its reliance on manual segmentation.


The entire expression can be multiplied out to give a sum of 2 J product forms, and exact "QuickS core" inference can be performed by combining the results of exact inference in each of the 2 J product forms (Heckerman 1989).

this exponential time complexity makes large problems, such as QMR-DT, intractable.


Most likely, as 3(b) suggests, their performances are not significantly different.

one reason ACS might perform better is that learning a sparsity basis in compressed space tunes the sparsity basis with respect to the measurement matrix whereas performing dictionary learning for CS estimates the sparsity basis independently of the measurement matrix.


It is often hoped that the use of automatic decision support systems trained on observational data will remove human bias and improve accuracy.

factors such as data quality and model choice may encode unintentional discrimination, resulting in systematic disparate impact.


no outliers), all methods perform well; their RMSEs are close to optimal (1/2, the standard deviation of i ).

when outliers start to appear, the result of least squares is significantly skewed, while the results of classic robust statistics methods, Huber, L1 and LTS, indeed turn out to be more robust than the least squares, but nevertheless are still affected significantly.


If c x,S h 1 were 1, c x,S h would be the uniform distribution u S h and we would have δ ≥ .


One can note that our non-linear model relies on the max function, which is not differentiable everywhere.

[8] has shown that gradient ascent can also be applied to generalized differentiable functions, which is the case of our criterion.


An MDP is defined as the 5-tuple (S, A, P, R, γ) [18], where S is a finite state space, A is a finite action space, The use of the 1-step policy improvement in Reinforcement Learning (RL) was theoretically investigated under several frameworks, e.g., Policy Iteration (PI) [18], approximate PI [2,9,13], and Actor-Critic [10]; its practical uses are abundant [22,12,25].

single-step based improvement is not necessarily the optimal choice.


In contrast, the UCB algorithm [5] achieves non-trivial regret after T = O(k) rounds.

our lower bound in Section 3.2 shows that any fair algorithm must suffer constant per-round regret for T k 3 rounds on some instances.


Quasiorthogonalization makes use of evaluations of Hessians of the fourth cumulant function to construct a matrix of the form C = ADA T where D has all positive diagonal entries-a task which is complicated by the possibility that the latent signals S i may have fourth order cumulants of differing signs-and requires taking the matrix square root of a positive definite matrix of this form.

the algorithm used for constructing C under sampling error is not always positive definite in practice, which can make the preprocessing step fail.


By combining all the base models, the clustering ensemble approaches (MCLA and HBGF) can improve the performance over each single model.

these two methods are not designed for classification, and the reported accuracy is the upper bound of their "true" accuracy.


[2009b] studied its subgradient variants.

this type of algorithms can only converge to a ball of the optimal solution, whose diameter depends on the steplength.


However, these studies do not attempt to infer functional connections between non-simultaneously observed neurons.

a few studies have presented statistical methods for dealing with sub-sampled observations of neural activity or connectivity, but these approaches are not applicable to our problem: A recent study [12] presented a method for predicting noise correlations between non-simultaneously recorded neurons, but this method requires the strong assumption that noise correlations are monotonically related to stimulus correlations.


Click-through from web search engines could provide valuable supervision.


Therefore, the same maximization problem as in standard SSVMs needs to be solved during prediction, which is given by This maximization problem is over the space of bounding box coordinates.

this problem involves a very large search space and therefore cannot be solved exhaustively.


This problem has also been addressed by recent algorithms of [31,3], who provide algorithms that are range-independent and satisfy approximate differential privacy, a relaxed version of differential privacy.

none of these algorithms is general; they explicitly fail unless additional special conditions on f hold.


Spike words are overwhelmingly sparse, with w 0 by far the most common word.

we compare the prior empirical synchrony distribution sampled using 10 6 samples from the NSB prior (π ∼ Dir(α, .


during a computation step, node k locally solves the following minimization problem min where (2) Crucially, this subproblem only depends on the local data A [k] , and local vectors v l from the neighborhood of the current node k. in COCOA [Smith et al., 2018] the subproblem is defined in terms of a global aggregated shared vector v c := Ax 2 R d , which is not available in the decentralized setting.


Unfortunately computing the marginal-likelihood is computationally intensive, and this precludes tiling the sparseness/over-completeness space.

an alternative is to learn the optimal overcompleteness at a given sparseness using automatic relevance determination (ARD) [7,8].


For example, rotation, translation and additive noise are typical methods of augmentation used in computer vision, especially for classification and detection tasks.

models trained naïvely on the augmented dataset become robust to limited forms of nuisance by learning to associate every seen variation of such factors to the target variables.


This was done in both [19] (in which messages are approximated using a using a mixture of delta functions, which in practice results in retaining the K largest message components) and [20] (which uses an additional uniform distribution in the approximating distribution to ensure non-zero weights for all states in the discretization).

these approaches appear to suffer when multiplying messages with disjoint peaks whose tails have been truncated to enforce sparsity: such peaks are unable to fuse their evidence correctly.


At this stage, the generative models do not perform well, as the number of available examples is low compared to the number of parameters in the model.

for very large sample sizes, with 100 times more unla- beled examples than labeled examples, the generative approach eventually becomes more accurate than the diagnosis approach.


2 would seem to indicate the appropriateness of the Gabor regression scheme for audio signal enhancement.

listening tests reveal that the algorithm, while improving upon the shortcomings of standard approaches discussed in 5 1.2, still suffers from the same 'musical' residual noise.


Even this simple approach demonstrates interesting performance and capabilities.

in a more elaborate setup the input density associated with each expert might itself be an infinite mixture of simpler distributions (for instance, an infinite mixture of Gaussians [5]) to allow for the most flexible partitioning of input space amongst the experts.


Inspection of the right-hand side of (13) suggests that the usual Ω(log N ) lower bounds [14,27,24] can be obtained if φ(u) behaves like u log u for large u.

if φ(u) behaves like − log u for small u, then the lower bounds will be of the form Ω log 1 δ .


There may still be cases where the average similarity of an external object is the same as that of an internal object, mining the cluster's external incoherency.

since x is an ESS, from (7) we see that whenever we try to extend a cluster with small shares of external objects, the cluster's overall average similarity drops.


3 as: Although the individual components in this objective function are solvable optimization problems, the combination of them makes it difficult to solve.

the objective function resembles that of a standard graph energy minimization problem and hence can be solved using graph-cut algorithms [18].


Most past work in neural programs induction assumes that a very large amount of training data is available to train a particular task, and ignores data sparsity issues entirely.

in a practical scenario such as the FlashFill domain described in Section 1 or the real-world Karel analogue , the approach we take is to randomly sample a number of k-sized sets and performing ensembling of the softmax log probabilities for each output token.


If the marginal payoffs were completely known, then the greedy policy would provide a competitive solution to the optimum.

one cannot run the greedy policy without knowing the marginal gains.


This, however, requires maximization of the possibly multimodal likelihood function of latent tree models.


To take a more attainable view, we also computed a standard ensemble Ensemble(Ours, SAN) and compared to an Ensemble(SAN, SAN) model, outperforming it by 1.24% for VQA v2 but falling short by 0.15% for VQA v1.

ensemble(GVQA, SAN) improves VQA v2 performance by only 0.54%.


This assumption greatly simplifies the problem, since we do not need to judge whether a data point is an outlier or not, which is very challenging.

this assumption also results in bad performance of these methods when there exists a large number of outliers, as frequently met in many real-world applications.


dataset, all algorithms performed similarly, with LASSO regression performing a little better than the rest.

oLS, stepwise regression, LASSO regression and PLS performed far worse on the Sergio & Kalaska dataset, with OLS regression attaining the worst error.


In addition, if each synapse has a n stable states (i.e.

its weight has to traverse n states to go from the lower bound to the upped bound), then the number of patterns p can grow quadratically n. this can happen only in unrealistic scenarios, where fine tuning of the network's parameters is allowed.


In the past, a significant amount of work has been done in semantic labeling of 2D images.

a lot of valuable information about the shape and geometric layout of objects is lost when a 2D image is formed from the corresponding 3D world.


For any prior distribution over MDPs, the optimal exploration strategy is available through dynamic programming in the Bayesian belief state space.

the exact solution is intractable even for very simple systems [8].


Equation ( 6), in turn, leads to an optimal rule for finding the features h b to merge with the current optimal solution h a : the set which minimizes The equation also leads to a straightforward procedure for updating the optimal cost once this set is determined.

when the cost function is BE, the equivalent expression is Note that the non-linearity introduced by the !


The trained neural network has these maps (warm colors indicate recognition) that show that K, Y, and so on are correctly identified.

the maps for 7 and 9 show several false positives.


The energy takes the form The RBM can be trained by following the gradient of the negative log-likelihood where F (x) is the free energy (unnormalized log-probability associated with P (x)).

this gradient is intractable because the second expectation is combinatorial.


To address this, [9] proposed to store and maintain U = DW , which reduced the cost to O(p) flops per update.

this is not a strategy we can use when dealing with very large data sets: storing the p by p dense matrices U and W in memory would be prohibitive.


The other families use the full grammar, given in Table 1.

the limited set of rules means that if any solution is found, it can at most be O n 3 , rather than exponential in n, as the naive evaluations would be.


This technique is easy to implement using existing automatic differentiation software packages.

if estimating the score function independently is computationally feasible, and a practitioner has evidence that the variance induced by Monte Carlo integration will reduce the overall variance away from the optimum point, we recommend establishing an annealling schedule for the optimal scaling constant that converges to 1.


Notice that in the statement of Theorem 3, the approximation guarantee of ESC-Streaming is given with respect to a memory only large enough to fit the offline greedy algorithm's solution.

if we allow our memory M to be as large as k ⇤ /✏, then Theorem 1 follows immediately for ✏ = 1/ ln(1/✏).


Initially, tree-based methods Gray, 2012, Koenigstein et al., 2012] were proposed for MIPS, which use the idea of branch and bound similar to k-d tree [Friedman and Tukey, 1974].

these methods suffer from the curse of dimensionality and their performance can be even worse than linear scan when feature dimension is as low as 20 [Weber et al., 1998].


In principle we can optimize the data likelihood to estimate B and π.

p(A; B, π) involves a sum over a complicated large finite set (the cardinality of this set is K n ), and hence is not easy to deal with.