Applications of optimal transport have recently gained remarkable attention as a result of the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation to the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn approximation, proving that it enjoys the same smoothness of its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula is used to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.

It is important to learn various types of classifiers given training data with noisy labels. Noisy labels, in the most popular noise model hitherto, are corrupted from ground-truth labels by an unknown noise transition matrix. Thus, by estimating this matrix, classifiers can escape from overfitting those noisy labels. However, such estimation is practically difficult, due to either the indirect nature of two-step approaches, or not big enough data to afford end-to-end approaches. In this paper, we propose a human-assisted approach called ''Masking'' that conveys human cognition of invalid class transitions and naturally speculates the structure of the noise transition matrix. To this end, we derive a structure-aware probabilistic model incorporating a structure prior, and solve the challenges from structure extraction and structure alignment. Thanks to Masking, we only estimate unmasked noise transition probabilities and the burden of estimation is tremendously reduced. We conduct extensive experiments on CIFAR-10 and CIFAR-100 with three noise structures as well as the industrial-level Clothing1M with agnostic noise structure, and the results show that Masking can improve the robustness of classifiers significantly.

In this paper, we formalize the idea behind capsule nets of using a capsule vector rather than a neuron activation to predict the label of samples. To this end, we propose to learn a group of capsule subspaces onto which an input feature vector is projected. Then the lengths of resultant capsules are used to score the probability of belonging to different classes. We train such a Capsule Projection Network (CapProNet) by learning an orthogonal projection matrix for each capsule subspace, andshow that each capsule subspace is updated until it contains input feature vectors corresponding to the associated class. We will also show that the capsule projection can be viewed as normalizing the multiple columns of the weight matrix simultaneously to form an orthogonal basis, which makes it more effective in incorporating novel components of input features to update capsule representations. In other words, the capsule projection can be viewed as a multidimensional weight normalization in capsule subspaces, where the conventional weight normalization is simply a special case of the capsule projection onto 1D lines. Only a small negligible computing overhead is incurred to train the network in low-dimensional capsule subspaces or through an alternative hyper-power iteration to estimate the normalization matrix. Experiment results on image datasets show the presented model can greatly improve the performance of the state-of-the-art ResNet backbones by10 20% and that of the Densenet by 5 7% respectively at the same level of computing and memory expenses. The CapProNet establishes the competitive state-of-the-artperformance for the family of capsule nets by significantly reducing test errors on the benchmark datasets.

Both resources in the natural environment and concepts in a semantic space are distributed "patchily", with large gaps in between the patches. To describe people's internal and external foraging behavior, various random walk models have been proposed. In particular, internal foraging has been modeled as sampling: in order to gather relevant information for making a decision, people draw samples from a mental representation using random-walk algorithms such as Markov chain Monte Carlo (MCMC). However, two common empirical observations argue against people using simple sampling algorithms such as MCMC for internal foraging. First, the distance between samples is often best described by a Levy flight distribution: the probability of the distance between two successive locations follows a power-law on the distances. Second, humans and other animals produce long-range, slowly decaying autocorrelations characterized as 1/f-like fluctuations, instead of the 1/f^2 fluctuations produced by random walks. We propose that mental sampling is not done by simple MCMC, but is instead adapted to multimodal representations and is implemented by Metropolis-coupled Markov chain Monte Carlo (MC3), one of the first algorithms developed for sampling from multimodal distributions. MC3 involves running multiple Markov chains in parallel but with target distributions of different temperatures, and it swaps the states of the chains whenever a better location is found. Heated chains more readily traverse valleys in the probability landscape to propose moves to far-away peaks, while the colder chains make the local steps that explore the current peak or patch. We show that MC3 generates distances between successive samples that follow a Levy flight distribution and produce 1/f-like autocorrelations, providing a single mechanistic account of these two puzzling empirical phenomena of internal foraging.

The process of learning new behaviors over time is a problem of great interest in both neuroscience and artificial intelligence. However, most standard analyses of animal training data either treat behavior as fixed or track only coarse performance statistics (e.g., accuracy, bias), providing limited insight into the evolution of the policies governing behavior. To overcome these limitations, we propose a dynamic psychophysical model that efficiently tracks trial-to-trial changes in behavior over the course of training. Our model consists of a dynamic logistic regression model, parametrized by a set of time-varying weights that express dependence on sensory stimuli as well as task-irrelevant covariates, such as stimulus, choice, and answer history. Our implementation scales to large behavioral datasets, allowing us to infer 500K parameters (e.g. 10 weights over 50K trials) in minutes on a desktop computer. We optimize hyperparameters governing how rapidly each weight evolves over time using the decoupled Laplace approximation, an efficient method for maximizing marginal likelihood in non-conjugate models. To illustrate performance, we apply our method to psychophysical data from both rats and human subjects learning a delayed sensory discrimination task. The model successfully tracks the psychophysical weights of rats over the course of training, capturing day-to-day and trial-to-trial fluctuations that underlie changes in performance, choice bias, and dependencies on task history. Finally, we investigate why rats frequently make mistakes on easy trials, and suggest that apparent lapses can be explained by sub-optimal weighting of known task covariates.

Leverage score sampling provides an appealing way to perform approximate com- putations for large matrices. Indeed, it allows to derive faithful approximations with a complexity adapted to the problem at hand. Yet, performing leverage scores sampling is a challenge in its own right requiring further approximations. In this paper, we study the problem of leverage score sampling for positive definite ma- trices defined by a kernel. Our contribution is twofold. First we provide a novel algorithm for leverage score sampling and second, we exploit the proposed method in statistical learning by deriving a novel solver for kernel ridge regression. Our main technical contribution is showing that the proposed algorithms are currently the most efficient and accurate for these problems.

This paper examines the long-run behavior of learning with bandit feedback in non-cooperative concave games. The bandit framework accounts for extremely low-information environments where the agents may not even know they are playing a game; as such, the agents’ most sensible choice in this setting would be to employ a no-regret learning algorithm. In general, this does not mean that the players' behavior stabilizes in the long run: no-regret learning may lead to cycles, even with perfect gradient information. However, if a standard monotonicity condition is satisfied, our analysis shows that no-regret learning based on mirror descent with bandit feedback converges to Nash equilibrium with probability 1. We also derive an upper bound for the convergence rate of the process that nearly matches the best attainable rate for single-agent bandit stochastic optimization.

Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold-valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Promising experimental results on both simulated and real data complete our study.

We analyze stochastic gradient algorithms for optimizing nonconvex, nonsmooth finite-sum problems. In particular, the objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a possibly non-differentiable but convex component. We propose a proximal stochastic gradient algorithm based on variance reduction, called ProxSVRG+. Our main contribution lies in the analysis of ProxSVRG+. It recovers several existing convergence results and improves/generalizes them (in terms of the number of stochastic gradient oracle calls and proximal oracle calls). In particular, ProxSVRG+ generalizes the best results given by the SCSG algorithm, recently proposed by [Lei et al., NIPS'17] for the smooth nonconvex case. ProxSVRG+ is also more straightforward than SCSG and yields simpler analysis. Moreover, ProxSVRG+ outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, which partially solves an open problem proposed in [Reddi et al., NIPS'16]. Also, ProxSVRG+ uses much less proximal oracle calls than ProxSVRG [Reddi et al., NIPS'16]. Moreover, for nonconvex functions satisfied Polyak-\L{}ojasiewicz condition, we prove that ProxSVRG+ achieves a global linear convergence rate without restart unlike ProxSVRG. Thus, it can \emph{automatically} switch to the faster linear convergence in some regions as long as the objective function satisfies the PL condition locally in these regions. Finally, we conduct several experiments and the experimental results are consistent with the theoretical results.

We present a new algorithm for stochastic variational inference that targets at models with non-differentiable densities. One of the key challenges in stochastic variational inference is to come up with a low-variance estimator of the gradient of a variational objective. We tackle the challenge by generalizing the reparameterization trick, one of the most effective techniques for addressing the variance issue for differentiable models, so that the trick works for non-differentiable models as well. Our algorithm splits the space of latent variables into regions where the density of the variables is differentiable, and their boundaries where the density may fail to be differentiable. For each differentiable region, the algorithm applies the standard reparameterization trick and estimates the gradient restricted to the region. For each potentially non-differentiable boundary, it uses a form of manifold sampling and computes the direction for variational parameters that, if followed, would increase the boundary’s contribution to the variational objective. The sum of all the estimates becomes the gradient estimate of our algorithm. Our estimator enjoys the reduced variance of the reparameterization gradient while remaining unbiased even for non-differentiable models. The experiments with our preliminary implementation confirm the benefit of reduced variance and unbiasedness.