Existing RDA methods [21] and many other stochastic gradient methods (e.g., [14,10]) can only show the convergence rate for the averaged iterates: xN = N t=1 t x t / N t=1 t , where the { t } are nonnegative weights.

in general, the average iterates xN cannot keep the structure that the regularizer tends to enforce (e.g., sparsity, low-rank, etc).


It still remains an open problem to us what is the convergence bound of this practical variant.

next we establish a connection between DisDCA and ADMM that sheds light on the motivation behind the practical variant and the differences between the two algorithms.


Realistically, it may be expected that the number of latent components in the process underlying any dataset is unrestricted.

the number of components that can be discovered by LDA or PLSA is restricted by the cardinality of the data, e.g.


The persistent and graded activity often observed in cortical circuits is sometimes seen as a signature of autoassociative retrieval of memories stored earlier in synaptic efficacies.

despite decades of theoretical work on the subject, the mechanisms that support the storage and retrieval of memories remain unclear.


This has previously been explored in a range of domains: from well-known algorithmic problems like knapsack packing [19], through bioinformatics [26] to music [10].

we are not aware of any previous work related to discovering mathematical formulas using grammars, and learning in such framework.


The standard benchmark for the algorithm in this scenario is the external regret, that is the difference between its cumulative loss and that of the best (static) expert in hindsight.

while this benchmark is useful in a variety of contexts and has led to the design of numerous effective online learning algorithms, it may not constitute a useful criterion in common cases where no single fixed expert performs well over the full course of the algorithm's interaction with the environment.


To summarize, for every training/test partition and for every pair (% patterns, # of scalings) we look at the pair (C, σ) that outputs the smaller value of Q D . In this case, the use of the Prior PAC-Bayes Bound to perform the model selection increases the computational burden of using the PAC-Bayes one in the training of one classifier (the one used to learn the prior), in comparison to the extra N classifiers needed by N -fold cross validation.

one could have included the tuning of the pair (% patterns, # of scalings) in the model selection.


3 2 nd best LPs for general graphs -Spanning tree inequalities When the graph G is not a tree, the marginal polytope M(G) generally requires an exponential number of inequalities.

as mentioned above, it does have an exact description in terms of marginals over cliques and separators of a junction tree.


In the context of memory, hard attention has been used in RLNTM -a version of the Neural Turing Machine modified to use stochastic hard addressing [22].

rLNTM has been trained using REINFORCE rather than variational inference.


This hinders research and consequently there have been extensive efforts devoted to speeding up training procedure.

there are relatively few efforts aimed at improving the test-time performance of the models.


For example, either simply adding L 1 regularization to the gradient of an online weight update or simply rounding small weights to zero are problematic.

these two ideas are closely related to the algorithm we propose and more detailed discussions are found in section 3.


In the general case, the MPLP fixed point may not correspond to a primal optimum because of the local optima problem with coordinate descent.

when the variables are binary, fixed points do correspond to primal solutions, as the following proposition states.


Moreover, the theoretical analysis of this model relies on the assumption that the learner maintains access to training data for all previously observed tasks, which allows to formulate a joint optimization problem.

it is unlikely that an autonomous robot is able to keep all this data.


One possible method to fit the c b is to maximize the likelihood in (3) over all samples drawn from all meta-training functions [7].

solving for the optimal c b requires an iterative algorithm whose inner loop requires Cholesky decomposition of a matrix of dimension the number of meta-training samples.


We then predict using ŷi = sgn(ȳ i ), for i = 1, . . . , n. The common justification behind the above learning paradigm [14,2] is that minimizing the cut (1) encourages neighbouring vertices to be similarly labelled.

we now demonstrate that in the online setting such a regime will perform poorly on certain graph constructions -there exists a trial sequence on which the method will make at least θ( √ n) mistakes.


Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM).

usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities.


The learned function f can be used to determine the pair of motions and the interpolation parameter α that produce the best approximation to ē.

such an approach is destined to fail as Y can be any arbitrarily long and complex sequence, possibly consisting of several movements performed one after the other.


In the experiments, we observe that CCALin and ALS CCA outperform MEG and capped-MSG when recovering the top CCA component, in terms of progress per-iteration.

capped-MSG is the best in terms of the overall runtime.


Hence, bisimilar states have the same optimal value (Givan et al., 2003).

bisimulation is not robust to small changes in the rewards or the transition probabilities.


We expect them to have the same state-action pair embedding with their respective optimal actions.

the optimal actions are "right" for s 1 and "up" for s 2 .


will not be an exact posterior sample, since ⌘real was not sampled from p(⌘ |D).

the true hyperparameters ⌘real which generated D ought to be in a region of high posterior mass unless the prior p(⌘) concentrates most of its mass away from ⌘real .


These results are basically consistent with those obtained by other authors (Härdle et al., 2004).

using C p as the selection criterion, the variables indux, age, dist, and tax are estimated to be irrelevant, a result not seen in other studies.


Vanishing gradients in residual networks Generally, data flows along all paths in residual networks.

not all paths carry the same amount of gradient.


Without this assumption, we are not able to identify the rank of W * , and therefore the estimated low-rank tensor W could be very far from the true tensor W * .

in many practical applications such as face recognition and image/video denoising, a portion of the observation tensor Y might be contaminated by gross error due to illumination, occlusion or pepper/salt noise.


The decoding approaches in these demonstrations usually employed classical classification/regression algorithms that derive a linear mapping between brain signals and outputs.

they typically only incorporate little prior information about the target kinematic parameter.


State-of-the-art SR results have been achieved for the landmark LR images, for which similar HR images can be retrieved from a large image set.

it has limitations for general LR images (i.e., it reduces to be the conventional sparsity-based SR method), for which correlated HR images cannot be found in the image database.


Note that this modification to the MCGD method is supported by Theorem 2 and the sublinear convergence rate holds.

closed form update of α is not supported by prior works such as [24,9,19,2].


The behaviour data are clustered based on the inferred reward functions, where the reward functions are defined per cluster.

the number of clusters (hence the number of reward functions) has to be specified as a parameter in order to use the approach.


According to all theories, this mapping requires reference to previous sensation-response pairings.

the theories differ with respect to the reference set.


In this case, by applying vanilla Gradient Descent (GD) or Accelerated Gradient Descent (AGD, [16]), one obtains iteration complexity of respectively, where κ := L/µ denotes the condition number of the problem and Õ hides logarithmic factors in the problem parameters.

whereas such bounds enjoy logarithmic dependence on the accuracy level, the multiplicative dependence on n renders this approach unsuitable for modern applications where n is very large.


MCML assumes that each class may be collapsed to a single point, at least approximately, and thus is only suitable for unimodal class distributions (or for simply connected sets if kernelization is used).

if points belonging to a single class appear in several disconnected clusters in input (or feature) space, it is unlikely that MCML could collapse the class into a single point.


Social and communication networks have been the subject of intense study over the last few years.

while these networks comprise a rich source of information for science, they also contain highly sensitive private information.


This would imply that X u ⊥ X v | X S is unfaithful.

testing for paths this way does not necessarily rule out all possible paths in G W\S .


These 20 experiments also clearly show that CCCP often converges in a few iterations.

pGD, EG and SMA are on par with each other since they are all first-order methods.


Performance improvements in these more recent models are mainly due to using better image features such as those obtained by region-based convolutional neural networks (R-CNN), or using more complex attention mechanisms [1] to provide a better context vector for caption generation, or using an ensemble of multiple LSTMs, among others.

the LSTM is still playing a core role in these works and we believe improvement over the core LSTM by turbo learning is still very valuable and orthogonal to most existing works; that is why we compare the turbo LSTM with a native LSTM.


We only know that they are potentially corrupted.

to generate the untrusted labels from the true labels in D, we first obtain a corruption matrix C. Then, for an example with true label i, we sample the corrupted label from the categorical distribution parameterized by the ith row of C. Note that this does not satisfy the conditional independence assumption required for our estimate of C. we find that the GLC still works well in practice, perhaps because this assumption is also satisfied when the data are separable, in the sense that each x has a single true y, which is approximately true in many of our experiments.


While the Fisher vector is frequently used with a Gaussian mixture model (GMM) [13,17], any generative model p(x; θ) can be used.

the information matrix is not always easy to compute.


In contrast, the MacKay update rules are considerably faster to converge [15].

the idea here is to form the gradient of (2), equate to zero, and then form the fixed-point update neither the EM nor MacKay updates are guaranteed to converge to a local minimum or even a saddle point of L(γ); both have fixed points whenever a γ i = 0, whether at a minimizing solution or not.


More comparisons are hard to make because [13] only provided an existential proof of pyramidal width, along with its value for simplex and hypercube only.

2 H s is clearly not intrinsic of the polytope.


Matrix completion aims to recover the underlying matrix only, since the factors are not uniquely defined due to invariance under rotations.

for orthogonal CP models, we can hope to recover the individual singular vectors u i 's exactly.


For a set Ω k,d that contains all supports of size k with atleast k − d active variables from S * , define γ d as where Popular sparse regression algorithms, such as the Lasso and the OMP, can perform accurate support recovery when We will show in Section 3.2 that SWAP can perform accurate support recovery when γ d < 1.

although the form of γ d is similar to ζ, there are several key differences, which we highlight as follows: vector, where d ≤ k. ζ is the 1 norm of a k × 1 vector.


The CNN analysis in Section 3 was restricted to one-hot representations (called binarized CNNs herein).

the filter weights and inputs of the normal CNNs are real numbers.


The supervised LDA (sLDA) model [2], by linking the empirical topics to the label of each pattern, is able to learn classifiers using Generalized Linear Models.

both LDA and sLDA are in essence dimensionality reduction techniques, and cannot be employed directly for the M 3 C tasks.


Because of the log-sum encountered, this problem is intractable in general.

there exist practical approximate optimization procedures, one of them is Expectation Maximization (EM) [3,4,12].


Techniques [6,21] like differentiated softmax and recurrent projection are employed to reduce the size of the output-embedding matrix.

they only slightly compress the model, and the number of parameters is still in the same order of the vocabulary size.


Algorithms based on cubic regularized Newton's method [28] and its variants [1,7,12,23,33,31] have been proposed to find such approximate local minima.

all of them need to solve the cubic problems exactly [28] or approximately [1,7] in each iteration, which poses a rather heavy computational overhead.


strong action invariance predicts better fit to ventral model) for monkey S (r=-0.35).

for monkey G, the correlation is moderately positive (r=0.43), contrary to expectation.


Most of the psychological literature on word completion involves the effects of priming certain responses with recent experience (Shacter and Tulving [18]).

priming is only able to account for about five percent of the variance in a typical fragment completion task (Olofsson and Nyberg [13], Hintzman and Hartry [6]).


The typical proof of the regret bound (1) suggests that the following can be guaranteed as well.

for each choice of probability distribution q on experts, there is an algorithm that guarantees for each expert k. (2) it is not immediately obvious how this can be achieved.


The edit distance between these sequences for two alignments then measures how much they differ.

even when this distance is large, the difference between two alignments may not be significant if both choices' scores are nearly the same.


Other works attempt to integrate tasks such as object detection and multi-class image segmentation into a single CRF model.

these models either use a different representation for object and non-object regions [23] or rely on a pixel-level representation [16].


Note that the GP for expert one appears to fit the data of the first function comparably well to that of expert two.

the gating network does not support this: the means of the GMM for expert one does not cover the region of the first function.


In general, polynomial models (linear and quadratic) are too restrictive and appear to underfit the data under both the null and the alternative, producing inflated Type I error and diminished power.

lower-order Matérn kernels (Matérn 1/2 and Matérn 3/2, dark blue lines) appear to be too flexible.


These tasks give the best indication that subjects maintain something more than a point estimate of the weights, and instead strongly suggest that they maintain a full joint distribution over them.

as we will show here, this effect can actually emerge due to covariance information being implicitly represented in the ensemble of beliefs over subjects, even if all the individuals are one-particle samplers.


Since we can sum two functions in O(1) time, we have that the worst case time complexity of the linear hinge loss algorithm is O(n log n).

for the case where (x) = x 2 , the worst case could involve going through the n breakpoints.


When each µ j is a discrete measure, the exact barycenter can be found via linear programming [2], and many of the regularization techniques apply for approximating it [16,17].

the problem size grows quickly with the size of the support.


This might be a difficult situation for the latent SMM and SVD+SMM because they cannot observe enough training data to estimate their own latent word vectors.

it would be an advantageous situation for SVMs using BoW features because they can use low frequency words that are useful for classification to compute their kernel values.


where m ro is the average overlap of the readout layer, and can be calculated using the recursion equations ( 8)- (11).

since generally f 6 = f ro , the activity factor need to be replaced in the proper positions in the equations.


It is interesting, and unfortunate, to note that the F does not asymptotically converge to the Hessian, so conjugate gradient methods might be more sensible asymptotically.

far from the converge point, the Hessian is not necessarily informative, and the natural gradient could be more efficient (as demonstrated in Tetris).


Intuitively, S-R habits correspond well to the policies learned by TD methods such as actor/critic [14,15], and rule-based cognitive planning strategies seem to mirror model-based algorithms.

the implication that this distinction fundamentally concerns the use or non-use of a world model in representation and algorithm seems somewhat at odds with the conception in psychology.


In theory, deterministic mappings guarantee cycle-consistency in ALI's framework.

to achieve this, the model has to fit a delta distribution (deterministic mapping) to another distribution in the sense of KL divergence (see Lemma 3).


Further, if A ∈ R 2×2 and det(A) = 1 then the gradient of the corresponding Möbius transformation is the convex function So the gradient of the index is the difference of the sums of a convex function of the linear-system orbits.

such sums are Schur-convex functions and it follows that the index is increasing because one orbit weakly supermajorises the other, as we now show for the case 0w1 (noting that the proof is easier for words 0, 1).


Although HMM-based algorithms can be considered arithmetic efficient, they demand many irregular memory accesses and require a large memory foot-print usually exceeding a few hundred MBs.

rNN based ASR has the advantage of low memory footprint; however, it demands many arithmetic operations for real-time inference.


For instance, due to the pose of the girl in Figure 1, most would predict that her arms are stationary but her leg is moving.

the exact motion is often unpredictable due to an intrinsic ambiguity.


However, if we keep the gradient of the noise component, or, equivalently, let α 1 = α 1 and leading to highly correlated branches.

if we consider the noise component as a constant, then only large E [ẑ 1,k ẑ2,k ] will be encouraged, which results in much weaker correlation, and hence the better generalization performance observed in practice.


As we are not aware of any predictive results on this task, we suppose this low accuracy is inherent to this task.

the experiments show that the algorithm can be run on very large graph datasets.


Consider the set Then the projection π 12 C onto the first two components is an 2 ball.

the projection π 13 C onto the first and third components is an 1 ball.


in our model MCMC is performed over the internal, neurally embodied latent variables that were learned from data.


Existing algorithm can formulate it as a quadratic programming problem [19], which can be solved by interior point method or active set method.

the computational cost is expensive, which limits its use in practice.


From a qualitative perspective, this question has been addressed in a large body of work (see e.g., [5]).

from a quantitative perspective little is known.


Then the bargaining process could focus on choosing the values of the weights w (i) .


Similarly to our work, this variance is computed via an ensemble.

they do not adapt to the uncertainty of individual estimates, only the overall ratio of real to imagined data.


Ignoring computational constraints, the algorithm achieves a regret bound of O( χ(G) log(k)T ).

with computational constraints, its regret bound is O( c log(k)T ), where c is the size of the minimal clique partition one can efficiently find for G. note that for general graphs, it is NP-hard to find a clique partition for which c = O(k 1− ) for any > 0.


For purely sequential policies (i.e., b = 1), every point is chosen based on a model informed by all previous observations.

for batch policies (b > 1), points are typically chosen with less information available.


For a start, in the former setting there is only one object to care about because the distribution from which examples are drawn is the target machine.

the latter involves two distinct objects: a distribution according to which strings are drawn, and a target weighted automaton assigning labels to these strings.


Maps corresponding to hard-thresholded CanICA [22] components have also been included, and have been called tCanICA.

the maps from the SODL [16] and our proposed Smooth-SODL (2) have not been thresholded.


In most works, the variables are either discrete or the distribution is assumed to be Gaussian .

many continuous real-world phenomenon are far from Gaussian, and can have a complex multi-modal structure.


Their method includes an extension for computing low-dimensional non-linear features using the kernel trick.

this approach computes dimensionality reductions through a two-step solution which involves first solving for a possibly full-rank metric and then estimating the low-rank approximation via spectral decomposition.


At the end of the procedure, additional gains can be achieved with an ensemble of multiple students generations [7].

multiple self-distillation processes multiply the total training time proportionally; an ensemble of multiple student generations increases the inference time accordingly as well.


It has been proved in [5] and [10] that the regression and classification based losses used in the pointwise approach are upper bounds of (1−NDCG).

for the pairwise and listwise approaches, which are regarded as the state-of-the-art of learning to rank [3,11], limited results have been obtained.


We see in Table 1 that for both G the proximal problems have an explicit solution.

note that smooth convex terms can also be directly exploited [25].


Batching is not compatible with the random-access nature of most training algorithms.

vector Quantized HOG features into 256 clusters would need 1 Byte per cell.


We include several failure cases in Figure 7, which share similar errors with the results listed in Figure 1.


Thus (13) indeed represents the tight complexity of the delay graph with a stochastic gradient oracle, and the wait-and-collect approach is optimal.

this answer is somewhat disappointing and leaves an intriguing open question: can a more natural, and seemingly more efficient (no wasted oracle accesses) delayed update SGD algorithm also match the lower bound?


Although these factors affect our results quantitatively, they do not change the statistical significance nor the qualitative trends.

a more rigorous analysis needs to be carried out.


The loss Γ : and R m x k( x ) dx < ∞, Note, that existence of a density is not necessary for consistency.

in order to keep the proofs simple, we restrict ourselves to this setting.


The well-known Langevin sampling (LS) recipe, so far the only sampling algorithm for continuous variables of which a neural implementation has been suggested, naturally fits into this dynamical framework.

we first show analytically and through simulations that the symmetry of the synaptic weight matrix implied by LS yields critically slow mixing when the posterior is high-dimensional.


If k < 15, we know λ i ∈ (5, 5 + ∆) for i = 1, 2, . . . , k + 1, and λ i ∈ (θ, θ + ∆) for i = k + 2, . . . , n. Therefore, the rate can be estimated as r small := Õ (k + 1 + (n − k − 1)(θ + ∆)/5) .

if k ≥ 15, we know that λ i ∈ (θ, θ + ∆) for i = k + 1, . . . , n, and hence the rate can be estimated as r large := Õ (k + 1 + (n − k − 1)(θ + ∆)/θ) .


RTRL allows updating the model parameters online after observing each input/output pair; we explain it in detail in Section 3.

its large running time of order O(n 4 ) and memory requirements of order O(n 3 ), where n is the number of units of a fully connected RNN, make it unpractical for large networks.


P (S t |S t−1 ) can be simply obtained by counting.

here we need to enforce the constraints described in section 2 (5).


2D intersymbol interference (ISI) channels).

in the MIMO channel model we assume that the channel matrix elements are iid and Gaussian; hence we cannot assume that the channel matrix H is sparse.


To motivate the approach, we again note that error bounds for the on-policy TD algorithm follow from the Markov chain property that P x D ≤ x D for all x when D is the stationary distribution.

finding a D that satisfies this condition is no easier than computing the stationary distribution directly and thus is not a feasible approach.


In contrast, algorithms in the convex setting are typically analyzed with the suboptimality gap, f (x) − f (x * ), where x * is an optimal point, as the convergence criterion.

it is not possible to provide meaningful guarantees for such criteria for general nonconvex problems due to the hardness of the problem.


We conclude with a few remarks: • For EH, projections are simply a renormalization of the weight vector.

iterative Bregman projections are often needed for projecting back into the polytope used by CH [25,19].


Note that this doesn't limit the size of parts, since all pairs of faces are potentially reachable via a sequence of adjacent links.

it does guarantee that only spatially contiguous parts have non-zero probability under the prior.


We are aware of one recent approach to skill discovery [8,9] which shares our assumption that each exercise depends on a single skill.

it differs from our approach in that it does not try to exploit expert labels and presumes a fixed number of skills.


Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely local or purely global in character.

by building a hierarchy of nested subgraphs, the Multiscale Laplacian Graph kernels (MLG kernels) that we define in this paper can account for structure at a range of different scales.


Note, for example, how LADDER clusters the codewords associated with speed limit signs, which were initially scattered around the unit circle.

all traffic sign codewords are pushed away from that of the background image class.


Using all KM nodes to obtain stochastic gradient estimates at the same point, we can perform T iterations of A-MB-SGD with a mini-batch size of KM , yielding an upper bound of This is a reasonable and common approach, and it is optimal (up to constant factors) when KM = O( L 2 H 2 T 3 ) so that the statistical term is limiting.

comparing (18) to the lower bound ( 16) we see a gap by a factor of K 2 in the optimization term, indicating the possibility for significant gains when K is large (i.e.


One might ask if monotonic feature classification is equivalent to cotraining with the monotonic features serving as one view, and the other features forming the other.

co-training requires labeled data to train classifiers for each view, unlike monotonic feature classification which can operate without any labeled data.


A wide spectrum of discriminative methods is increasingly used in diverse applications for classification or regression tasks.

many existing discriminative methods assume that the input data is nearly noise-free, which limits their applications to solve real-world problems.


Second, our pooling units are superficially similar to the pooling units used in convolutional networks.

ours differ both in functional form and purpose: we use pooling as a way of sharing information between cells in the matrices that are processed through our network by taking maximums across entire rows or columns, while in computer vision, max-pooling units are used to produce invariance to small translations of the input image by taking maximums in a small local neighborhood.


For the classification problem, the most natural loss function is the 0 − 1 loss.

0 − 1 loss function is non-convex, non-smooth, and we cannot get a non-trivial function value bound in terms of β − β * as we did for the logistic loss function.


Efforts to increase statistical power tend to focus on developing new features for use with the existing machine learning classifiers -these high-level features are non-linear functions of the low-level measurements, derived using knowledge of the underlying physical processes.

the abundance of labeled simulation training data and the complex underlying structure make this an ideal application for large, deep neural networks.


An iterative Gaussianization scheme transforms any source model to a spherical Gaussian by alternating between linear ICA transformations and nonlinear histogram matching to map marginal densities to Gaussians [28].

in general, the overall transformation of iterative Gaussianization is an alternating concatenation of many linear/nonlinear transformations, and results in a substantial distortion of the original source space.


Specifically, we say that the loss function L satisfies RSC(k) with parameter κ l if it satisfies RSC with parameter κ l for the set {∆ ∈ R p : ∆ 0 ≤ k}.

we say that the loss function satisfies restricted strong smoothness (RSS) with parameter κ u with respect to a set S if for all ∆ ∈ S. We can define RSS(k) similarly.


In practice, convex recovery is usually much more expensive than nonconvex recovery, because it requires optimizing over an n × n matrix variable instead of an n × r vector-like variable.

it is statistically consistent [3], and guaranteed to succeed with m ≥ 1 2 n(n + 1) noiseless, linearly independent measurements.


Recency effects have been demonstrated in domains as varied as legal reasoning and jury evidence interpretation (Furnham, 1986;Hogarth & Einhorn, 1992) and clinical assessments (Mumma & Wilson, 2006).

the most carefully controlled laboratory studies of sequential dependencies, dating back to the the 1950's (discussed by Miller, 1956), involve the rating of unidimensional stimuli, such as the loudness of a tone or the length of a line.


For estimating such connectivity, magnetoencephalography (MEG) and electroencephalography (EEG) are well-suited tools because of their millisecond temporal resolution.

localizing source activity in the brain requires solving an under-determined linear problem.


Time delays along each segment are to, t" and t2.

the total time delay down the first branch is T,= to +f" and the second - we can apply this result to axonal branching and arrive at a testable prediction about the relationship among branch diameters without knowing the relative cost.


An interesting aspect here is that the medium risk invoice could have been perturbed towards low risk or high risk.

our method found that it is closer (minimum perturbation) to being high risk and thus suggested a pertinent negative that takes it into that class.


For each color, training data was pooled data across all locations (rather than learn a separate model for each coordinate).

features were provided that gave the position of the model (there was a binary feature for each column and each row), so the tree could choose to attend to position information if necessary.


when the random variable ξ is absolutely continuous with respect to the Lebesgue measure and Σ is non-singular, the density of X exists and has the form where g(•) is a scale function uniquely determined by the distribution of ξ.


We are unable to compare directly to NTPT because no public implementation is available.

our graph task is a more difficult version of a task from [14], who report on their shortest-path task "2% of random restarts successfully converge to a program that generalizes" (see their supplementary material).


Both SCAL and TUCRL quickly achieve sub-linear regret as predicted by theory.

sCAL and TUCRL seem to achieve different growth rates in regret: while SCAL appears to reach a logarithmic growth, the regret of TUCRL seems to grow as √ T with periodic "jumps" that are increasingly distant (in time) from each other.


In particular, the context matrix C SVD is orthonormal while the word matrix W SVD is not.

the factorization achieved by SGNS's training procedure is much more "symmetric", in the sense that neither W W2V nor C W2V is orthonormal, and no particular bias is given to either of the matrices in the training objective.


Regularization technique has become a principled tool for statistics and machine learning research and practice.

in most situations, these regularization terms are not well interpreted, especially on how they are related to the loss function and data.


This reduces the multi-agent problem into a single-agent one.

the central node is often unscalable, susceptible to malicious attacks, and even infeasible in large-scale applications.


In the test scenario, BayesOpt chooses the hyperparameters corresponding to the left-most bar in Figure 1b because those hyperparameters minimized error on the validation set.

figure 1b shows that other hyperparameter settings outperform this selection when testing.


One strategy that makes use of Monte Carlo estimates of both Z n and Z 0 is given by [3].

the computation time required to find accurate estimates can be extremely high [20] (see Section 6).


We are given a training set of image and label pairs {(I t , l t )} N t=1 and the task is to predict the value of l t (e.g.

a class label l t ∈ {1, . . . , C}) given the associated image I t . The standard machine learning approach would consist in extracting features from the whole image I t and from those directly learn to predict l t . since we wish to incorporate the notion of fixation into our problem framework, we need to introduce some constraints on how information from I t is acquired.


The models can sometimes count up to five or six.

as shown in the second picture of Figure 3, the ability is fairly weak as they do not count correctly when different object types are present.


Bayesian nonparametric methods based on the Dirichlet Process (DP), gamma process and beta process, have proven effective in capturing aspects of various datasets arising in machine learning.

it is now recognized that such processes have their limitations in terms of the ability to capture power law behavior.


In the "strong data" limit, where the latent variables f are fixed by the likelihood L, Algorithm 1 would be ideal.

the likelihood terms in the example are not so strong that the prior can be ignored.


This fact is often referred to as the propagation error problem in hierarchical classification.

flat classifiers rely on a single decision and are not prone to this problem (even though the decision to be made is harder).


All likelihood ratio values bigger than t will be clipped to t. The estimator was derived from Fenchel duality for f -divergence [18].

the optimization for the estimator is not convex if one uses log-linear models.


Figures 1(a) and 1(b) show that in general the MLE rule does achieve greater accuracy than approval voting.

the increase is at most 4.5%, which may not be significant in some contexts.


Given the nature of the connectivity patterns in our chip, the correlation among neighboring neurons is increased throughout the array, independent of the input sources, hence the mean correlation coefficient is higher throughout the whole network.

the difference in correlation between the base level and the group with highest correlation is significantly higher when cooperation and competition are enabled, with respect to the feed-forward case.


[21] apply gradient-based optimization to find an image I which minimizes where P is a simple natural image prior, such as the total variation (TV) regularizer.

this method produces images which are roughly natural and have features similar to the input features, corresponding to Φ −1 R . due to the simplistic prior, reconstructions from fully connected layers of AlexNet do not look much like natural images (Fig.


We obtain a crisp analysis of the computational complexity of perceptron learning under the p,-margin success requirement: On one hand, for every p, > 0 we present an efficient p,-margin successful learning algorithm (that is, an algorithm that runs in time polynomial in both the input dimension and the sample size).

unless P=NP, no algorithm whose running time is polynomial in the sample size and dimension and in 1/ p, can be p,-margin successful for all p, > O.


Our algorithm can be thought as an extension of fully corrective greedy in [17] to nonlinear features from multiple kernels where we optimize the risk over the class (2).

in MFGA, we work with the empirical risk (rather than the true risk in [17]), which happens in practice as we do not know P X Y .


For both tasks, SGD made fast initial progress, but slowed down dramatically very soon.

as predicted by our theory, natural gradient descent caused the test error to decrease exponentially and reach the minimum achievable loss (limited by Σ) after only a few passes through the training set (Fig.


This approach expands the focus from one neuron (in feature visualization) to a set of neurons.

while these existing techniques are related to attribute witness extraction in our technique, they are not sufficient for our purpose.


At the same time the sum of the squared norms of the gradients will typically be of the order of O( √ T ), giving us a O( √ T ) regret bound (see for example the proofs in [4]).

following this approach in DFEG we would have that the sum of norms of the squared gradients grows much faster than O( √ T ).


In the two-class case, the transformation matrix W is just a vector, which is in the same direction as the discriminant in the corresponding optimal Bayes classifier.

it has been shown that LDA is suboptimal for multi-class problems [2].


A similar attempt to compare both approaches on discrete models (for stochastic context free grammars) was presented in [23].


In the setting where log p is of a lower order than n 1 2α+1 , the n − 2α 2α+1 term dominates, and the rate of convergence is determined by the smoothness parameter α.

when log p is much larger than n 1 2α+1 , p has a much greater influence on the minimax rate of convergence.


Ideally, an oracle should lead to a subset of features having the maximum reward.

we have too large a state space to exhaustedly search for the optimal subset of features.


These non-parametric resampling techniques and their numerous extensions and improvements (see [27] for review) are capable of producing high quality natural textures very efficiently.

they do not define an actual model for natural textures but rather give a mechanistic procedure for how one can randomise a source texture without changing its perceptual properties.


The visualizations for the neurons that are trained on regular images often show coherent global structures, realistic-looking colors and sharpness.

the visualizations for neurons that are trained on modified images indeed show cut-up objects (Fig.


This can then be propagated to H − 2 and so on.

these bounds can be very loose and they do not exploit the properties of creg functions.


[2016] adopt the dual-learning framework for machine translation.

they all require two domain entities, such as image domains (sketch and photo) and language domains (English and French).


The Dirichlet is conjugate to the multinomial and hence mathematically convenient for Bayesian inference, and the number of parameters is conveniently linear in the size of the sample space.

the Dirichlet is a distribution over the entire probability simplex, and for many problems this is simply the wrong domain if there is application-specific prior knowledge that the pmfs come from a restricted subset of the simplex.


Although it is discussed in [5] and mentioned to have the same sample complexity as their algorithm, it turns out that it is more efficient.

its sample complexity is slightly better (or the same, depending on the parameter regime) compared to the one of the algorithm for the personalized setting presented in [5], which is in the setting of the lower bound in [5] where k = Θ(d), there is a gap of log(k) multiplicatively between the sample complexity of Algorithm R1 and the lower bound.


In general, bias b can be a probability distribution indicating a mixture of several guesses, but in this paper we only use two simpler scenarios: a binary vector to indicate whether there is a particular object with its associated weights U G , or a group bias b G with equal values for all objects, which indicates the presence of some object in general.

a simple strategy would be to feed the segmented input y into a classifier C. this suffers from the loss of details during M 's reconstruction and is prone to hallucinations, i.e.


As illustrated in Figure 1, the learning objective of G is to produce a synthetic sentence that receives higher ranking score than those drawn from real data.

the goal of R is to rank the synthetic sentence lower than human-written sentences.


For these reasons, we use this objection function in the remainder of this work.

the Adept framework is general and can be used with many different objective functions, including all presented in this section.


Intuitively, the single probe condition taps the marginal probability of the memory for the target item, p(ŝ t ), where t represents the index of the target item.

the minimal color change condition taps the conditional probability of the memory for the target given the memories for the distracters, p(ŝ t |ŝ −t = s −t ) where −t represents the indices of the distracter items, because the actual dis- tracters s −t are shown during test.


From the neuroscientific point of view, drug addiction is a complex disorder characterized by compromised inhibitory control and reward processing.

individuals with compromised mechanisms of control and reward are difficult to identify unless they are directly subjected to challenging conditions.


Thus one way to compile a BN into an AC is via a junction tree.

when the network contains context-specific independences, a much more compact circuit can be obtained.


One can prove that w(γ) = y(γ)/1 T k y(γ) converges to w * as γ → 0.

the convergence behavior of w(γ) is quite uncertain for small γ > 0.


Our first set of results is deterministic, depending on specific instantiations of the observed variables {(y i , z i )} n i=1 .

we are also interested in proving results that hold with high probability when the x i 's and z i 's are drawn at random from some distribution.


Kernel methods provide an elegant and effective framework to develop nonparametric statistical approaches to learning [1].

memory requirements make these methods unfeasible when dealing with large datasets.


In the spirit of context-specific independence [9], we can view tree learning as a method for learning compact partition-based dependencies.

tree learning induces recursive subpartitions, which limits their ability to partition the joint state space.


Seeking a distribution-specific tight analysis, one simple option to try to tighten these bounds is to consider their minimum, min(d, E[ X 2 ]/γ 2 )/ǫ 2 , which, trivially, is also an upper bound on the sample complexity.

this simple combination is also not tight: Consider a distribution in which there are a few directions with very high variance, but the combined variance in all other directions is small.


Natural neural networks [6] also aim at normalizing the variance of activations by reparametrization of the weights.

training with normalization techniques is perturbed by stochastic gradient descent (SGD), stochastic regularization (like dropout), and the estimation of the normalization parameters.


After scaling by the number of predictors p, a similar rate of convergence is observed for the stationary   phase approximation to the partition function for both the diabetes and leukemia data (Figure 1b,c).

convergence of the posterior expectation values xτ to the maximum-likelihood coefficients x, as measured by the ∞ -norm difference xτ − x ∞ = max j |x τ,j − xj | is noticeably slower, particularly in the p n setting of the leukemia data (Figure 1d-f).


We could instead calculate the number of length t walks from v to each vertex more quickly, but this count would probably be dominated by walks that go to a high degree vertex and then leave and return to it repeatedly, which would throw the calculations off.

most reasonably short nonbacktracking walks are likely to be paths, so counting each vertex a number of times equal to the number of nonbacktracking walks of length t from v to it seems like a reasonable modification.


[22] described a Dependency Tree Recursive Neural Network that puts entire sentences into correspondence with visual data.

these methods reason about the image only on the global level using a single, fixed-sized representation from the top layer of a Convolutional Neural Network as a description for the entire image.


Most machine learning (ML) algorithms assume that given instances are represented in numerical vectors.

much real-world data is not represented as numerical vectors, but as more complicated structures, such as sequences, trees, or graphs.


A straightforward convex combination of ToMs would result in yet another linear mapping, incapable of capturing nonlinear inter-taxonomic interactions.

our kernel approach retains ToM's granularity-specific features but also enables nontrivial (nonlinear) combinations, especially when coupled with a novel hierarchical regularizer, which we will define next.


An intuition for why this helps is that each step of the procedure is supposed to generate increasingly realistic samples.

if we passed gradients through the iterative procedure, then this gradient could encourage the earlier steps to store features which have downstream value instead of immediate realistic x-values.


Note that using spatially varying weights w ab provides better results.

the main aim of this experiment is to demonstrate the accuracy and speed of our approach and not to design the best possible Gibbs energy.


In the definition of the TOP kernel (4.5), we did not include any normalization of feature vectors.

in practical situations, it is effective to normalize feature s for improving classification performance.


the term not proportional to τ ) cannot be improved, since it is already optimal for objective functions defined on only one machine (see for example Theorem 3.13 p. 280 in [18]).

quite surprisingly, the error due to communication time may benefit from fast convergence rates in O(RL g /ε).


Some methods are not sensitive to initializations but tend to return less accurate clustering.

some other methods can find more accurate results but require comprehensive initialization.


The advantage of this model is that we are able to estimate the temporal causal structure in variances.

this model provides no information about the contemporaneous causal relations among the sources if there indeed exist such causal relations.


Our theoretical analysis and quantitative evaluations confirm DN as an effective efficient coding transform for natural sensory signals.

we also observe a previously unreported phenomenon that DN may increase statistical dependencies when the size of pooling is small.


Kernel methods with polynomial kernels k(z, z ) = z T z q = α z α z α provide an implicit mechanism to compute inner products in the feature space associated with P q .

they require O(n 3 ) computation for solving associated kernelized (ridge) regression problems and O(n 2 ) storage of dense n × n Gram matrices K (given by K ij = k(z i , z j )), and therefore do not scale well.


It is easy to see that, X and Y are dependent in the post interventional graph too because of the direct causal relationship.

if X is not a parent of Y , then in the post interventional graph D `[{X }] even with or without the latent L between X and Y , X is independent of Y since X is intervened on.


By introducing the Dirichlet process mixture (DPM) prior, infinite mixture of GPs models are able to infer the number of experts, both hyperparameters and parameters via Gibbs sampling [1,2].

these models are trained by MCMC methods, which demand expensive training and testing time (as collected samples are usually combined to give predictive distributions).


Note that in [31] more sophisticated diversity measures were used e.g., the Hamming Ball.

the DivMBest method (7) with this measure requires to run a very time-consuming HOP-MAP [37] inference technique.


To model L, we will choose a distribution for positive semi-definite matrices analogous to the Gaussian distribution: a scaled Wishart matrix with expectation L. Note that, although it captures the trait that L is positive semi-definite, this distribution does not accurately model every feature of L. For example, a scaled Wishart matrix does not necessarily have ones along its diagonal.

the mode of the density is at L, a Laplacian; and for large values of the scale parameter, most of the mass will be on matrices close to L. Appendix B of [8] provides a more detailed heuristic justification for the use of the Wishart distribution.


Not all model parameter settings lead to shrink; as in any gradient-based algorithm, step sizes that are too large do not lead to converge.

such parameter settings make little sense in the context of the learning objective.


Throughout the paper we have taken particular care to avoid functions or systems which are linear, or approximately linear, as in these cases our model can be reduced to standard GP regression.

for the problems for which NIGP has been designed, such as the various non-linear problems we have presented in this paper, our model outperforms current methods.


Many RL problems can be reduced to compute gradients of expectations (e.g., in policy gradient methods) and there has been series of exploration in this area for natural gradients.

we would suggest that it might be interesting to consider where stochastic backpropagation fits in our framework and how 2 nd order computations can help.


Consequently, G may collapse to these modes.

to this end, we generate pseudo labels through C for some unlabeled data and use these pairs as positive samples of D. The cost is on introducing some bias to the target distribution of D, which is a mixture of p c and p instead of the pure p. this is acceptable as C converges quickly and p c and p are close (See results in Sec.5).


Probabilistic search in these settings is significantly more computationally demanding than A* search, both in theory and practice, primarily due to key differences between the min and softmin functions.

despite this, we found significant performance improvements compared to other IOC inference methods by employing heuristic-guided search ideas.


Although OO frameworks are easier to implement, their runtime performance falls short of that of frameworks using SCT for workloads that do not spend most of their time in hand-optimized compute primitives.

existing frameworks that use SCT require the user to metaprogram computation graphs, significantly complicating the definition of ML models.


For very small settings of m (i.e., m = 2, 6), OUTLIERS-MR and GREEDY-MR perform slightly worse than sequential OUTLIERS and GREEDY.

for practical settings of m (i.e., m ≥ 10), OUTLIERS-MR matches OUTLIERS and GREEDY-MR matches GREEDY.


Using the random assignment variation, both approximations have negligible costs, since they simply choose any combination of points within a bin.

in high dimensions, the time required by the uniform bin pyramid with the optimal per-bin matching approaches the time required by the optimal matching itself.


In this case, the ReLU and input constraints of the SDP are identical to the constraints of the QCQP.

this rank-one constraint on P would make the feasible set non-convex.


When workers cannot form a fully connected communication topology or the communication latency is high (e.g., in sensor networks or mobile networks), decentralizing the communication comes to the rescue.

when the amount of data sent through the network is an optimization objective (maybe to lower the cost or energy consumption), or the network bandwidth is low, compressing the traffic, either via sparsification [Wangni et al., 2017, Konečnỳ andRichtárik, 2016] or quantization , Suresh et al., 2017 is a popular strategy.


Finally, these results hold only for a fixed w * and are not universal in general.

[6] studied RLSR with less stringent assumptions, allowing arbitrary corruptions in response variables as well as in the data matrix X, and proposed a trimmed inner product based algorithm for the problem.


However, it is based on a tricky decomposition of the risk, where surrogate losses for risk minimization and risk analysis are different and labels of U data are needed for risk evaluation, so that no further bound is implied.

ours utilizes the same surrogate loss for risk minimization and analysis and requires no label of U data for risk evaluation, so that an estimation error bound is possible.


Under the functional point of view, each 2D image is a function defined on the regular 2D lattice, so all images are functions over a common underlying parameterizing domain.

with discretized 3D shapes, the probe functions are generally defined on heterogeneous shape graphs/meshes, whose nodes are points on each individual shape and edges link adjacent points.


The output of G1 is blurry because the L1 loss encourages the result to be an average of all possible cases [10].

g1 does capture the global structural information specified by the target pose, as shown in Figure 2, as well as other low-frequency information such as the color of clothes.


we find the cost of attribute-parallel algorithm is relative to the size of training data N , and the cost of data-parallel algorithm is relative to the number of attributes d. the cost of PV-Tree is constant.


Some preliminary investigations have been proposed to combine numerical iterations and learnable architectures for deep propagations design.

due to these naive combination strategies (e.g., directly replace iterations by architectures), it is still challenging to provide strict convergence analysis on their resulted deep models.


• We show that, for any fixed k, the k-nearest neighbor KL entropy estimator nearly achieves the minimax rates without knowing the smoothness parameter s. In the functional estimation literature, designing estimators that can be theoretically proved to adapt to unknown levels of smoothness is usually achieved using the Lepski method [39,22,45,44,27], which is not known to be performing well in general in practice.

a simple plug-in approach can achieves the rate of n −s/(s+d) , but only when s is known [26].


The determinantal point process (DPP), a probabilistic model of subset selection that prefers diverse sets, is a natural fit for these kinds of applications.

while DPPs offer efficient algorithms for probabilistic operations like marginalization, conditioning, and sampling, in practice we often need to select a single "best" set, and this can be more challenging.


The closest work to ours is the Weisfeiler-Lehman Neural Machine (WLNM) [12], which also learns from local subgraphs.


Scissors, whose only equilibrium has full support ( 1 3 , 1 3 , 1 3 ).

there is evidence that support sizes shrink for many games as a function of episode length, how much hidden information is revealed and/or affects it has on the payoff [65,86,10].


The construction and analysis from [23] crucially relies on the fact that in the setting of that paper, first the random misclassification noise rate η is chosen to take some particular value in (0, 1/2), and then the margin parameter γ is selected in a way that depends on η.

in this paper the situation is reversed: in our setting first the margin parameter γ is selected, and then given this value we study how high a malicious noise rate η can be tolerated.


Different from the non-regularized case, the introduction of the regularization term will add bias in θT −i (t), which gives a biased evaluation of indexes.

we prove the convergence of θT −i (t) (so again the indexes will converge properly) in the following lemma, which enables an easy adaption of our previous results for nonregularized case to ridge regression: The basic idea for extending the results to non-linear regression is inspired by the consistency results on M-estimator [14], when the error of training data satisfies zero mean.


One way to write the global cost function is as a weighted combination of the two.

such a formulation does not lend itself to efficient optimization routines (c.f.


Similarly to the 'One bounding box' scenario, we augment O s in order to force tracklets to belong to either the corresponding action or the background depending on the spatial overlap criterion.

here an action instance has potentially several annotations, therefore the spatial overlap with the track is defined as the minimum IoU between the corresponding bounding boxes.


A lot of recent work has focused on transferring policies trained in simulation to a real robot [4,14,19,20].

there has been very limited work on transferring knowledge and skills between different robots [6,21].


Given the features f and training data D that consists of fully observed assignments to X and E, the optimal feature weights w * maximize the conditional log-likelihood (CLLH) of the data: The problem (2) does not have a closed form solution, but has a unique global optimum that can be found using any gradient-based optimization technique because of the following fact [1]: Fact 1 Conditional log-likelihood (2), abbreviated CLLH, is concave in w. Moreover, where E P denotes expectation with respect to a distribution P. Convexity of the negative CLLH objective and the closed-form expression for the gradient lets us use convex optimization techniques such as L-BFGS [9] to find the unique optimum w * .

the gradient (3) contains the conditional distribution over X i X j , so computing (3) requires inference in the model for every datapoint.


They have found numerous applications in machine learning, including viral marketing [1], dictionary learning [2] network monitoring [3,4], sensor placement [5], product recommendation [6,7], document and corpus summarization [8] data summarization [9], crowd teaching [10,11], and probabilistic models [12,13].

submodularity is in general a property that goes beyond set functions and can be defined for continuous functions.


In [6] and [7], this EM approach has been applied to more complicated probabilistic models for image labeling tasks.

the performance of these approaches are only empirically evaluated, and there is no analysis that proves performance guarantees.


Similar phenomena were observed in [13], that decreasing the non-zeros by ignoring non-frequent dimensions can be around 150 times faster.


If we perform search first, up to some memory limit, and then sample, which we refer to as two-stage sampling, our probabilistic bounds will proceed from an improved baseline, giving better bounds at moderate to long computation times.

doing so sacrifices the quick improvement early on given by basic importance sampling.


On the other hand, when ∆ = Ω(T ( )), we also prove an information theoretic lower bound that any algorithm that outputs such a x must use super polynomial number of evaluations of f . Optimization of convex functions over a convex domain is a well studied problem in machine learning, where a variety of algorithms exist to solve the problem efficiently.

in recent years, practitioners face ever more often non-convex objectives -e.g.


Hence, a low-rank approximation may actually improve the performance of, for example, a lifted collective classification algorithm.

the approximation made in Example 6 may not be desirable if we are querying attributes of the constant c, and we may prefer to approximate other areas of the evidence matrix instead.


The work in [16] considers an algorithm for online distributed submodular maximization with an application to sensor selection.

their approach requires k stages of communication, which is unrealistic for large k in a MapReduce style model.


From the perspective of eigenflows, a graph representing a set of weakly coupled clusters produces eigenflows between the various clusters which decay with long halflives.

the eigenflows within each cluster decay much more rapidly.


For many classic structured prediction problems, probability distributions over the dependent variables can be efficiently computed using widely-known algorithms and data structures (such as forward-backward, and its corresponding trellis for exact probability distributions in Markov models).

we know of no previous work studying efficient representations of exact distributions over clusterings.


This locality explains why sparse coding works well on MNIST data.

local coordinate coding is able to remove the unusual coefficients and further improve the locality.


The above discussion shows that for appropriate choices of q, i.e., q ∈ (2, ∞), the RKBS embeddings in Examples 1-3 are useful in practice as γ K (P m , Q n ) is consistent and has a closed form expression.

the drawback of the RKBS framework is that the computation of γ K (P m , Q n ) is more involved than its RKHS counterpart.


Moreover, efficient algorithms for its optimization have been proposed which produce much better cuts than the standard spectral relaxation.

the multi-cut problem has still to be solved via the greedy recursive splitting technique.


That presenting Henry should decrease the reporting of Adam is obvious, and is commented on in the paper.

that presenting anti-Henry should decrease the reporting of Adam is less obvious, since, by removing Henry as a competitor, one might have expected Adam to have received an additional boost.


We can train a separate set of inference networks to perform this operation, potentially also using gradient descent within the inference network.

these networks do not receive any gradient information during Algorithm 1 Meta-training, differences from MAML in red Require: p(T ): distribution over tasks 1: initialize Θ := {µ θ , σ 2 θ , vq, γp, γq} 2: while not done do 3: Sample batch of tasks Ti ∼ p(T ) 4: for all Ti do 5: D tr , D test = Ti 6: Evaluate Evaluate ∇ θ L(θ, D tr ) 9: Compute adapted parameters with gradient descent: Compute ∇Θ T i L(φi, D test ) +DKL(q(θ|D test ) || p(θ|D tr )) 12: Update Θ using Adam meta-training, and may not work well in practice.


As long as we pick η t as a constant η < 1/L, we have linear convergence of O((1 − γ/L) t ) Nesterov [2004].

for SGD, due to the variance of random sampling, we generally need to choose η t = O(1/t) and obtain a slower sub-linear convergence rate of O(1/t).


when h = 0.01, ξ becomes much larger than A.


Even the simplest 2 balls 2 bins game satisfies these properties (two pure Nash and unique best responses) which shows the strength of this impossibility result.

our convergence result holds for any adversarial scheduling with the minimal fairness assumption that given any mixed state at least one agent who is not best responding eventually will be given the possibility to update their behavior, answering open questions in [26,25].


Compared with these naive unrolling based methods (e.g., [7,33,30,17,2,20]), the main advantage of PODM is that we can generate iterations, which strictly converge to the critical point of the given optimization model, even in the complex nonconvex and nonsmooth scenarios.

by slightly relaxing the exact optimality constraints during propagations, we can also obtain an interpretable framework to integrate mathematical principles (i.e., formulated by model based building-block) and experience of the tasks (i.e., network structures designed in heuristic manners) for collaborative end-to-end learning.


Asynchronous updates provide an alternative solution to disguise the communication overhead to a certain amount [19].

those methods usually rely on a sparsity assumption on the updates [25,31], which is not realistic e.g.


the peak of the inverted U should shift to the right).

across all volatility conditions, lose shift rate increases as learning rate increases (Fig 2b), except for learning rate=0.


2 (e) indicates that Noisy-DQN wastes significant amount of time visiting explored states.

div-DQN is the only method that is capable of exploring the gridworld uniformly and systematically, as illustrated in Fig.


Moreover, in this setting (i.e., when  = O(1)), the number of samples required to exactly recover w ⇤ (with = 0) is given by: n = ⌦(s + k ⇤ log M ) [16], where s is the maximum support size of a union of k ⇤ groups and M is the number of groups.

if one were to directly use sparse regression techniques (by ignoring group sparsity altogether) then the number of samples is given by n = ⌦(s log p).


The notion of regret has been extensively studied, and there exist plenty of algorithms and theories for minimizing regret [Shalev-Shwartz et al., 2007, Srebro et al., 2010, Duchi et al., 2011, Shalev-Shwartz, 2011, Zhang et al., 2013.

when the environment is changing, the traditional regret is no longer a suitable measure, since it compares the learner against a static point.


These architectures together with recent large databases (e.g., ImageNet [3]) have obtained astonishing performance on object classification tasks [12,5,20].

the baseline performance reached by these networks on scene classification tasks is within the range of performance based on hand-designed features and sophisticated classifiers [24,21,4].


(2017), where the authors prove that their algorithm, ASAGA, can obtain a linear speedup even without sparsity in the wellconditioned regime.

pROXASAGA always requires some sparsity.


This lemma is of independent interest, and relates the SVD of a truncated matrix, here M (B) , to that of the initial matrix M . More precisely: Lemma 2 If δ ≤ m −8/9 , there exists a × k matrix V (B) such that its column vectors are orthonormal, and with probability 1 − exp(−m 1/7 ), for all i ≤ k satisfying that Note that as suggested by the above lemma, it might be impossible to recover V (B) i when the corresponding singular value s i (M ) is small (more precisely, when s 2 i (M ) ≤ n δ m log( )).

the singular vectors corresponding to such small singular values generate very little error for lowrank approximation.


One can simply replace deconvolution with convolution, which results in an architecture that is very similar to recently proposed very deep fully convolutional neural networks [19,7].

there exist differences between a fully convolution model and our model.


Thus, our measures are inherently equality preferring.

the key advantage of our measures of social welfare over those focusing on inequality manifests when, as we saw in the above example, comparing two benefit distributions of different means.


(d) However, in the semi-sparse regime, there are too few common neighbors with any node, and hence the heuristic does not work.

we show that with a simple additional "cleaning" step, we regain the theoretical properties shown for the semi-dense case.


In a recent breakthrough, Recht et.al [20] extend the techniques of compressed sensing to rank minimization with affine constraints.

the results of Recht et.al do not apply to the case of matrix completion as the constraints in matrix completion do not satisfy the restricted isoperimetry property they assume.


In particular, note that the error term ||c − c || does not apply to the weights of the reveal-network that receives the container image and extracts the secret image.

all of the networks receive the error signal β||s − s || for reconstructing the hidden image.


This is to find optimized Θ 0 from which the task-train posterior can be obtained quickly and closely to the true task-posterior.

this is intractable because we neither have access to the true posterior p ∞ τ nor its samples Θ ∞ τ .


One way to solve the above problem is to use regression, using That is, we estimate the data embedding z i as z i = W x i , and minimize their distances to the correct label embeddings u yi ∈ R m where the dimension for y i is set to 1 and every other dimension is set to 0: The above ridge regression will project each instance close to its correct embedding.

it does not guarantee that the resulting embeddings are well separated.


Hence, we see that the VBF is effective at reconstruction, but is unable to function stably in prediction.

we see that the Deep Koopman model, aided by its ability to construct state mappings that approximately span an invariant subspace, is able to generate stable predictions for much longer time horizons.


Consistent with this prediction, in this condition the three two-way interaction terms between the effect features (f24, f34, f23) are greater than those interactions in the control condition.

the common-effect schema does not imply that the three cause features will be correlated, and in fact in that condition the interactions between the cause attributes (f12, f13, f23) did not differ from those in the control condition (Figure 3).


), the logarithm of the likelihood can be calculated as where N is the size of the entire dataset.

as already mentioned, it is computationally infeasible to deal with the entire large-scale dataset at each timestep as would typically be required in MCMC and HMC methods.


Mohassel and Zhang [25] proposed a ReLU-based alternative function to softmax for privacy-preserving machine learning since softmax is expensive to compute inside a secure computation.

it leads to a division by zero since all outputs of ReLUs frequently become zeros and the denominator for normalization becomes zero.


From our argument above, we see that in the causal direction the Kolmogorov complexity of p(Y |X = x) is independent of the particular value x of the cause X, i.e.

this will not hold in the anticausal direction as the input and mechanism are not algorithmically independent in that direction, i.e.


Typically, this is not problematic; for example, for most realistic probability densities, p and p 2π have similar orders of smoothness, and p 2π = q 2π if and only if p = q.

there are (meagre) sets of exceptions; for example, if q is a translation of p by exactly 2π, then p 2π = q 2π , and one can craft a highly discontinuous function p such that p 2π is uniform on X .


[5] used this idea with SVMs as base classifiers and subsequently in unsupervised learning by [6].

co-EM also suffers from local maxima problems, and while each iteration's optimization step is clear, the co-EM is not really an expectation maximization algorithm (i.e., it lacks a clearly defined overall log-likelihood that monotonically improves across iterations).


For chordal graphs, graph coloring can be solved efficiently so the minimum size intervention design problem can also be solved efficiently.

the minimum cost intervention design problem is NP-hard, even on chordal graphs.


The remaining times, it did escape in accordance with theory; however, it converged to another local minimum.

when we initialize other LSM algorithms at an SBL local minima, we always remain trapped as expected.


When F (x) is essentially locally strongly convex [9] in terms of • such that 5 then we can see that the local error bound holds with θ = 1/2, which implies the iteration complexity of HOPS is Õ( 1 √ ), which is up to a logarithmic factor the same as the result in [6] for a strongly convex function.

here only local strong convexity is sufficient and there is no need to develop a different algorithm and different analysis from the non-strongly convex case as done in [6].


For discrete (or mixed) variables the GP model could still be used with appropriate choice of covariance function.

the resulting sum (analogous to equation 1) may be difficult to evaluate.


For non-parametric classes, and non-smooth Lipschitz loss, such as the hinge-loss, the excess risk might scale as 1/n and not 1/n, even in the separable case.

for H-smooth non-negative loss functions, where the second derivative of φ(t, y) w.r.t.


Because of the convex nature of the SMS problem, both algorithms converge to the same optimal value regardless of the initial values.

the multiplicative updates for SMS usually achieve two order speed up in terms of the number of iterations and typically about 10 times faster in running time when compared to the gradient based algorithm.


It is also commonly believed that all systematic scans mix at the same asymptotic rate, which is equivalent to choosing These conjectures imply that using systematic scan instead of random scan will not result in significant consequences, at least asymptotically, and that the particular permutation used for systematic scan is not important.

we show that neither conjecture is true by constructing models (listed in Table 1) in which the scan order has substantial asymptotic effects on mixing time.


[7] proposed sublinear-time algorithms for special cases of quadratic function minimization.

it is "sublinear" with respect to the number of pairwise interactions of the variables, which is O(n 2 ), and their algorithms require O(n log c n) time for some c ≥ 1.


SCWRL is considered one of the leading algorithms for modeling side-chain conformations.

in the last couple of years several groups reported better results due to more accurate energy function [7], better searching algorithm [8] , or extended rotamer library [14].


In practice, we found that optimizing the objective in Damianou and Lawrence (2013) results in layers being 'turned off' (the signal to noise ratio tends to zero).

our posterior retains the full conditional structure of the true model.


Note that if we simply set the maximum number of clusters to K limit = N , we would spend O(N log N ) time to estimate parameters.

in practice, the optimal value of K is low relative to N , and it is possible to detect when we cannot lower cost further by increasing K or lowering p max , thus allowing the search to terminate early.


The assumption is that this stacking will not only provide a strong model for local patterns, but also create a large receptive field to capture long range dependencies, e.g., contextual relations between an object and other elements of the scene.

this approach for modeling context is highly inefficient.


Thus, the advantage of using group convolutions is compromised.

our group channel-wise convolutions can overcome the problem without more layers, as shown by experiments in Section 4.5.


On the other hand, for mortgage underwriting, POLARIS needs to invoke Algorithm 2 for multiple times per input which searches under a combination of different features.

a single invocation to Algorithm 2 is still faster for mortgage underwriting.


This enables the use of kernels composed of XNOR and bit-count operations to perform highly efficient convolutions.

additional high-precision multipliers are still needed to perform the scaling after each convolution which limits its performance.


When the signal of interest is known without uncertainty, the frequency representation can be obtained by means of the Fourier transform [6].

real-world applications usually only provide us with a limited number of observations corrupted by noise.


The top-left box illustrates how the influence of isolated outliers is mitigated by the standard mixture.

a repeated observation (box two on the left) causes the EP solution to collapse onto the spike at the data (the log scale is deceptive: the second peak contributes only about 8% of the posterior mass).


The first justification follows directly from the standard approximation error bounds: using a lower discount factor may decrease the approximation error bounds.

we also show that these bounds are loose, thus their decrease does not entirely justify the improved solution quality.


For example, one can think of the desired object clusters as "topics of an image stack" and apply one of the standard topic discovery methods like probabilistic latent semantic analysis (pLSA) [8] or Latent Dirichlet Allocation (LDA) [2].

we found that a simple K-means algorithm applied to the indicator vectors produced good results.


In particular, AdaGrad allows for a more aggressive learning rate by using But, at a high level, AdaGrad and dropout appear to both be aiming for the same goal: scaling the features by the Fisher information to make the level-curves of the objective more circular.

l 2 -regularization makes no attempt to sphere the level curves, and AROW [18]-another popular adaptive method for online learning-only attempts to normalize the effective feature matrix but does not consider the sensitivity of the loss to changes in the model weights.


In all experiments we used identical input parameters; further tuning of these parameters may improve results.

in a clinical setting parameter tuning is impractical, and thus more work to investigate automated parameter tuning is needed.


In this case, it has been observed that social relationships reveal similarities in tastes and interests [15].

social links can also be asymmetric (e.g., followers of celebrities).


SA has been successfully applied to a number of combinatorial search problems.

many combinatorial problems, especially those with intricate constraint structure, are beyond the reach of SA and related MCMC methods.


Our approach shows a marginal gain over a GAN.

we can improve the underlying estimation technique by leveraging the multi-scale structure of the LAPGAN model.


Using this approach the network was able to learn representations of the handwritten and spoken digits.

these representation and the associations between them were not as distinctive as for synaptic sampling and the classification performance was significantly worse (only first learning phase shown).


It is easy to see that the query selection procedure of structural QBC is a rejection sampler where each query q is chosen with probability proportional to ⌫(q)u(q; ⇡ t ) (in the case of the 0-1 loss) or ⌫(q)var(q; ⇡ t ) (for general losses).

it is possible for the rejection rate to be quite high, even when there are many queries that have much higher uncertainty or variance than the rest.


Recent theoretical studies focus on the convex relaxation of the MMV problem based on the (2, 1)-norm minimization, which is an extension of the well-known 1-norm minimization employed in SMV.

the resulting convex optimization problem in MMV is significantly much more difficult to solve than the one in SMV.


( 4), but with the additional restriction that V t is rank one, i.e, V t arg min Note that in this case it follows that V t is a unit trace rank-one matrix which corresponds to the leading eigenvector of the matrix rf (X t ) + ⌘ t X t . However, when V t is rank-one, the regularization kV t X t k 2 F makes little sense in general, since unless X ⇤ is rank-one, we do not expect X t to be such (note however, that if X ⇤ is rank one, this modification will already result in a linear convergence rate).

we can think of solving a set of decoupled component-wise regularized problems: where the equivalence in the first line follows since kvv > k F = 1, and thus the minimizer of the LHS is w.l.o.g.


Note that although the variance contribution to the noise in the field, ro vanishes, the contribution from the correlations, Q 1 , remains finite and is responsible for the fact that m ro < 1 and ✏ > 0 [1].

in a deep network, if the initial overlap is within the basin of attraction the m = 1 solution, the overlap quickly approach m = 1 [figure (2).C].


The minimal sizes of T and H that are core w.r.t.

f are The proof is deferred to Appendix C. The theorem states that, as expected, the higher rank(f ; M ), the smaller T . it also implies that the more irrelevant information f contains, the larger H needs to be, which might seem counter-intuitive.


Initially, the network starts with some random connectivity, and obeys the dynamical equations, We then apply the following modified learning rules for D and Ω, (20) We note that in both cases, the learning remains local.

similar to the rate dynamics, the dynamics of learning now requires a slower synaptic process that computes the averaged signal inputs and presynaptic firing rates.


As mentioned earlier, the performance Figure 2: Results from the experiments on MNIST dataset of the regularized algorithm should be insensitive to the feature induction method, assuming inference is exact.

in practice, because inference is approximate, an induction algorithm that introduces spurious features will affect the quality of inference, and therefore the performance of the algorithm.


In contrast, if samples are allocated uniformly to the alternatives under the same conditions, then the most natural procedures require K max i =i ∆ −2 i log(K/δ) samples before returning i with probability at least 1 − δ.

standard best-arm bandit algorithms do not incorporate asymmetry as induced by nullhypotheses as in definition (1) by default.


The optimal learning rate scaling for GD and DA decrease by two orders of magnitude as the delays increase.

even with this tuning they do not obtain the performance of AdaptiveRevision.


Recently, [26] studies a nested batch knowledge gradient policy.

they optimize over a finite discrete feasible set, where the gradient of KG does not exist.


Specifically, the distribution of Optimizing the probability of X (r) under this conditional distribution with respect to A (r) corresponds to computing the supervised NMF using KL divergence [7].

to perform the correct update in our model, we must also incorporate the distribution of A (r) , and so, instead of using the standard multiplicative NMF updates, we use exponentiated gradient ascent [11] on the log objective.


While such priors allow for tractable EM algorithms, they have the effect of promoting smoothing rather than sparsity.

sparsity-inducing priors are invariably convex in their contribution.


The greedy algorithm does pay a price for this advantage, since it has to find the optimal block to update at each iteration.

for the problem we study here this can be done much more efficiently using a priority queue.


It is important to emphasize that for both data sets there are task-descriptor features available.

as we have described throughout this paper, our approach learns task similarity directly without the need for task-dependent features.


This method was possibly dependent on the observation window size (W O ).

as shown in Figure 6a, the optimal W N was found to be independent of W O . Experimentally, the W O values were selected in the range of 50-60 samples, which corresponded to largest DOM between nonnormalized active and idle data.


3 On one hand, our work shares similar spirit to the exact penalty methods as both approaches attempt to transfer constraints as penalties into the objective function.

our approach differs from the exact penalty methods in the following three folds: 1) as stated in [18], most of the results on exact penalty methods require the feasible set compact, which is not satisfied in our problem (2); 2) the exact penalty methods replace all the constraints with certain penalty functions, while our reformulated problem (3) still has the nonnegative constriant; 3) in general the exact penalty methods use either nonsmooth penalty functions or differentiable exact penalty functions which are based on replacing the conventional multipliers with continuously differentiable multiplier functions, while the penalty in ( 3) is smooth and the parameter λ is fixed and is independent of U and V . We comment that our result builds on the specific structure of the objective function in (3) while the exact penalty methods focus on general nonlinear programming problems.


This severely limited the complexity of the allowable operations: for instance the engine only allowed for collision between balls and the ground plane.

because our method analytically differentiates the LCP, it can be substituted within the traditional computations of most existing physics engines, only requiring the added differentiability within the LCP portion itself; indeed, in our system we use existing efficient methods for portions of the simulator such as a collision detection or Euler stabilization.


This gene was not identified as a frequently mutated gene in GBM [26].

there is recent evidence that CXCL13 plays a critical role in driving cancerous pathways in breast, prostate, and ovarian tissue [27,28].


We fit our model parameters to the psychometric function in the neutral case, with τ = 8.20ms and t nd = 312.50ms, and predict both the psychometric function and the reaction times in the biased case.

our results do not match the monkey data as well as the human data when Pr(d R ) = 0.8.


The initial PHAM program was constructed on the assumption that the agent should patrol among A, B, C, D unless there is mail to be delivered.

the specific rewards are such that the optimal behavior is to loiter in the mail room until mail arrives, thereby avoiding costly delays in mail delivery.


HEM uses one common bandwidth for all dimensions, which can be easily selected by cross validation.

it will be desirable to learn different σ d for different dimensions; this allows a form of feature selection.


Since the proposed approach models the shape of curved object contours and reduces the dimensionality of the representation, we expect it to be of particular value for objects that are characterized by their shape and where their contours can be extracted using state-of-the-art methods.

a significant number of images form PASCAL VOC are corrupted due to noise or compression artifacts (see Fig.


These differences may be due to the performance of the optimization with respect to the regularization parameter λ.

the results clearly demonstrate that the mixture weight method achieves comparable accuracies at a much decreased cost in network bandwidth -upwards of 1000x.


Theoretically understanding the empirical performance of PSOS(4) as stated remains open.

similar cycle constraints have proved useful in analyzing LP relaxations [WRS16].


[1] for this setting up to logarithmic factors.

the only high-probability regret bound for this setting due to Alon et al.


A relevant paper along this research line is [15], which studies the efficient reinforcement learning with value function generalization in the KWIK framework (see [16]), and reduces the efficient reinforcement learning problem to the efficient KWIK online regression problem.

the authors do not show how to solve the general KWIK online regression problem efficiently, and it is not even clear whether this is possible.


4(b), weights with smaller magnitudes are pruned while influences of larger weights have been discarded.

our proposed algorithm not only handles the smaller weights but also considers impacts of those larger weights.


[23] modified the original formulation using graph laplacian regularization to reduce the size of the SDP.

recent work has shown that even this reduced formulation of MVU can be solved more efficiently by reframing it as a semidefinite quadratic linear programming (SQLP) [25].


The previous two subsections provide exact algorithms for loss-augmented inference that reduce the time require for training an AP-SVM.

despite these improvements, AP-SVM might be slower to learn compared to simpler frameworks such as the binary SVM, which optimizes the surrogate 0-1 loss.


While this is a reasonable strategy, it is both memory-and computationally-expensive and challenging when unrolling through many iterations because the time-and space-complexity of the backward pass grows linearly with the forward pass.

we address this issue by showing how to analytically differentiate through the fixed point of a nonlinear MPC solver.


Therefore, columns 2, . . . , n c in AH T cl will be close to zero vectors and have small norms.

since the first row in H nc has all its entries equal to 1/ √ n c , the first column of , the scaled average of all vectors in this cluster.


Intuitively, the reparameterization trick provides more informative gradients by exposing the dependence of sampled latent variables z on variational parameters φ.

the REINFORCE gradient estimate only depends on the relationship between the density function log q φ (z|x, φ) and its parameters.


By definition, the transition matrices T a,z have eigenvalues with magnitude ≤ 1.

lossy compressed transition matrices T a,z are not guaranteed to have this property.


Our work extends the theory on asynchronous BCD algorithms such as [18,14,13].

their analyses rely on the independence assumption and assume bounded delays.


One can viably assume that the date is available noise-free and the CO 2 sensors are affected by signal-independent sensor noise.

in many datasets, either or both of these assumptions are not valid and lead to poor modelling performance.


We have that the rate of convergence to the optimal -risk is Õ(T − 3β 2β+1 ), if E (f ρ ) = 0, and Õ(T − 2β 2β+1 ) otherwise.

for any finite T the rate of convergence is Õ(T − 2β β+1 ) for any In other words, we can expect a first regime at faster convergence, that saturates when the number of samples becomes big enough, see Fig.


To make the comparison fair, once the regularization parameters are selected, we estimate the oracle and StARS graphs only based on a subsampled dataset with size b(n) = ⌊10 √ n⌋.

the K-CV, BIC, and AIC graphs are estimated using the full dataset.


• M-Step: Fix the mask M (t+1) and find optimized permutations α (t+1) i such that the masked values are minimized, i.e. . We can use unit permutations as the initial configuration for all dimensions and run above EM algorithm iteratively until it converges.

the M-Step is still an optimization problem.


Over the past few decades numerous variants have been considered, with many algorithms designed to achieve sub-linear regret in the worst case.

this level of robustness comes at a cost.


A possible explanation for this result is that multiclass-based models have more flexibility in creating decision boundaries, hence they perform better if the training data size is sufficient.

since multiclass-based models have many more parameters than threshold-based models (m|Y| parameters rather than m + |Y| − 1 parameters), multiclass methods may need more data, and hence, may not perform well on relatively small datasets.


Clustering and outlier detection are often studied as separate problems [1].

it is natural to consider them simultaneously.


Compared to approximate linear programming, in addition to being far more scalable, our algorithm performed comparably, giving better policies on just over half of the models (53/96) that ALP could be run on.

when we restrict to low-treewidth "chain" models, we find that the ALP algorithm appears to perform better on larger models; it outperforms our FVI algorithm on only 4/32 models of size 6, but this increases to 14/32 at size 10, and 25/32 at size 20.


Although the resulting distribution is not Gaussian, its first and second moments can be calculated analytically.

these calculations carry a high computational load and previous experiments showed this exact treatment provided no significant improvement over the much quicker approximate method we now describe.


In our experiments with shorter sequences where ELBO can be computed within a reasonable amount of time, we did not observe a significant difference in the coverged ELBOs among different choices of non-linearity.

we observed that the converged ELBO is sensitive to the number of hidden units and the number of hidden units needs to be adapted to the data set and computing budget.


To this end, observe that if a worker skips all the questions then the additive payment mechanism will incur an expenditure of µT .

the proposed payment mechanism of Algorithm 1 pays an exponentially smaller amount of µT G (recall that T < 1).


As λ decreases, R emp increases, indicating loss of discrimination on the training set.

generalization error decreases, implying more robustness.


( 1) where the score functions are indexed by input x and parameterized by w. We consider a classification task where given a training set {x n , y n } N n=1 , where y n ∈ {1, . . . , K}, we wish to fit the parameters w by maximizing the log likelihood,  When the number of training instances is very large, the above maximization can be carried out by applying stochastic gradient descent (by minimizing −L) where we cycle over minibatches.

this stochastic optimization procedure cannot deal with large values of K because the normalizing constant in the softmax couples all scores functions so that the log likelihood cannot be expressed as a sum across class labels.


Alternatively, one can be greedy and select ε to minimize the value of the loss function (3) after the change α j ← α j + ε: where Setting the derivative of ( 6) to 0 yields: Using , which can be used as a step size.

for the following slightly more conservative step size we can prove algorithm convergence: Theorem 1.


On the other hand, for α = 2 Kleinberg establishes that in his model not only that there is small (though order log(n) 2 rather than constant) diameter, but that short paths can be navigated by a naive greedy routing algorithm.

simulation results discussed in Section 5 suggest that the equilibrium networks of our model do support fast routing as well.


Our goal is to accurately estimate the count.

we evade the hard task of learning to detect and localize individual object instances.


Using synthetic data also allows us to generate perfect labels for the training set with the exact location of a good grasp for each object.

collecting and manually labeling a comparably sized set of real images would have been extremely time-consuming.


The techniques of [4,5], [13], as will be explained later, crucially rely on these properties of Erdős-Rényi graphs.

for most large real-world graphs such as the World Wide Web ( [1]), the degree distribution deviates significantly from the Poisson distribution and has high variance.


The idea of Nemirovski and Yudin is to build a sequence of pyramids, each capturing the variation of function in certain directions, in such a way that with O(d log d) pyramids we can explore all the directions.

as mentioned earlier, their approach fails to give a low regret.


This control case uses perturbations with identical statistics to adv up to a flip of the vertical axis.

this control breaks the pixel-to-pixel correspondence between the adversarial perturbation and the image.


Thus, no one has ever been experimented on massive data sets with, say, one million unlabeled examples.

the Core Vector Machine (CVM) is recently proposed for scaling up kernel methods in both supervised (including classification [12] and regression [13]) and unsupervised learning (e.g., novelty detection).


Second, SPGP-DR does not place a prior on the linear projection matrix W; it is instead fitted using Maximum Likelihood, just as the pseudo-inputs.

vDMGP does place a prior on W and variationally integrates it out.


While SGD is computationally inexpensive, the best current guarantee for finding an ✏-approximate local minimum (see Definition 1) requires O(✏ 4 poly(d)) iterations [Ge et al., 2015], which is inefficient in the high-dimensional regime.

second-order methods which have access to the Hessian of f can exploit curvature to more effectively escape saddles and arrive at local minima.


Theorem 6 Let P n denote a profile of n votes generated i.

when the data are generated from M 2 ϕ , we have a different story.


In all experiments, the performance of the standard methods heavily depends on the stepsize parameter.

in case of the proposed method, the default setting showed remarkable consistency.


Then, for all x < λb , we have ALG = OPT = x.

for x ≥ λb , we have Thus, Algorithm 2 gives an option to trade-off consistency and robustness.


In the first 4 steps, the estimation error can be obtained immediately by calculating the estimation loss function L e (x t , x t ) for t = 1, . . . , 4.

when t = 5, 6, 7, we cannot get the error immediately since the true values are missing.


Density estimation is one of several types of generative modelling, with the focus on obtaining accurate densities.

we know that accurate densities do not necessarily imply good performance in other tasks, such as in data generation [31].


This charging argument is similar to the usual primal-dual analysis for allocation problems.

since the objective functions are not linear, it may not be possible to interpret the proof using a primal-dual technique.


In this paper, we hypothesize that this discrepancy may simply be due to differences in the implicit reward (cost) structure of the two tasks: the NoGo response incurs a higher imposed waiting cost than the Go response, since the NoGo response must wait until the response deadline has passed to register, while a Go response immediately terminates the trial.

in the 2AFC task, the cost function is symmetric for the two alternatives, whether in terms of error or delay.


Finding each binary classifier is computationally efficient, reducing to a convex problem with O(n) variables.

nearly all previous approaches require solving a non-convex problem and resort to alternating optimization [25,19] or greedy approximation [11].


Works such as [15,14] propose to use networks with two branches to predict these two parts respectively.

to compute final predictions y, they have to solve dense matrix inversion or a system of linear equations, which is time-consuming and unstable (the matrix to be inverted may be singular.).


A variant with better consistency properties.

we show that the addition of an extra post-processing step (henceforth, the "cleaning" step) still enables common neighbors to identify nodes from its own cluster, while reducing the number of off-cluster nodes to zero with probability tending to one as n → ∞.


From the common point of view, a variety of reinforcement learning algorithms including DPG need the assumption that the samples are independently and identically distributed [24].

the sequential data from practical tasks usually violate the assumption, which is also known as high temporal correlations.


The mutation-type operator will easier to generalize, because it only requires a bias towards a random distribution.

the appropriate setting of the parameters may be difficult to ascertain.