In the theory of lossy compression, the rate-distortion (R-D) function R(D) describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining R(D) for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate R(D) from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport.

In this paper, we analyze the trajectories of stochastic gradient descent (SGD) with the aim of understanding their convergence properties in non-convex problems. We first show that the sequence of iterates generated by SGD remains bounded and converges with probability 1 under a very broad range of step-size schedules. Subsequently, we prove that the algorithm's rate of convergence to local minimizers with a positive-definite Hessian is O(1/n p) if the method is run with a Θ(1/n p) step-size. This provides an important guideline for tuning the algorithm's step-size as it suggests that a cool-down phase with a vanishing step-size could lead to significant performance gains; we demonstrate this heuristic using ResNet architectures on CIFAR. Finally, going beyond existing positive probability guarantees, we show that SGD avoids strict saddle points/manifolds with probability 1 for the entire spectrum of step-size policies considered.

A challenge that machine learning practitioners in the industry face is the task of selecting the best model to deploy in production. As a model is often an intermediate component of a production system, online controlled experiments such as A/B tests yield the most reliable estimation of the effectiveness of the whole system, but can only compare two or a few models due to budget constraints. We propose an automated online experimentation mechanism that can efficiently perform model selection from a large pool of models with a small number of online experiments. We derive the probability distribution of the metric of interest that contains the model uncertainty from our Bayesian surrogate model trained using historical logs. Our method efficiently identifies the best model by sequentially selecting and deploying a list of models from the candidate set that balance exploration-exploitation.

In this paper, we consider the problem of computing the barycenter of a set of probability distributions under the Sinkhorn divergence. This problem has recently found applications across various domains, including graphics, learning, and vision, as it provides a meaningful mechanism to aggregate knowledge. Unlike previous approaches which directly operate in the space of probability measures, we recast the Sinkhorn barycenter problem as an instance of unconstrained functional optimization and develop a novel functional gradient descent method named \texttt{Sinkhorn Descent} (\texttt{SD}). We prove that \texttt{SD} converges to a stationary point at a sublinear rate, and under reasonable assumptions, we further show that it asymptotically finds a global minimizer of the Sinkhorn barycenter problem. Moreover, by providing a mean-field analysis, we show that \texttt{SD} preserves the {weak convergence} of empirical measures.

This is a canonical problem that admits two difficulties in optimization: 1) non-convexity and 2) non-smoothness (due to unbalancedness of \mathbf{U} and \mathbf{V}). This is also a prototype for more complex problems such as asymmetric matrix sensing and matrix completion. Despite being non-convex and non-smooth, it has been observed empirically that the randomly initialized gradient descent algorithm can solve this problem in polynomial time. Existing theories to explain this phenomenon all require artificial modifications of the algorithm, such as adding noise in each iteration and adding a balancing regularizer to balance the \mathbf{U} and \mathbf{V} .This paper presents the first proof that shows randomly initialized gradient descent converges to a global minimum of the asymmetric low-rank factorization problem with a polynomial rate. For the proof, we develop 1) a new symmetrization technique to capture the magnitudes of the symmetry and asymmetry, and 2) a quantitative perturbation analysis to approximate matrix derivatives.

We consider off-policy evaluation and optimization with continuous action spaces. We focus on observational data where the data collection policy is unknown and needs to be estimated from data. We take a semi-parametric approach where the value function takes a known parametric form in the treatment, but we are agnostic on how it depends on the observed contexts. We propose a doubly robust off-policy estimate for this setting and show that off-policy optimization based on this doubly robust estimate is robust to estimation errors of the policy function or the regression model. We also show that the variance of our off-policy estimate achieves the semi-parametric efficiency bound.

Recent developments in differentially private (DP) machine learning and DP Bayesian learning have enabled learning under strong privacy guarantees for the training data subjects. In this paper, we further extend the applicability of DP Bayesian learning by presenting the first general DP Markov chain Monte Carlo (MCMC) algorithm whose privacy-guarantees are not subject to unrealistic assumptions on Markov chain convergence and that is applicable to posterior inference in arbitrary models. Our algorithm is based on a decomposition of the Barker acceptance test that allows evaluating the Rényi DP privacy cost of the accept-reject choice. We further show how to improve the DP guarantee through data subsampling and approximate acceptance tests.

In this paper we study the training dynamics for gradient flow on over-parametrized tensor decomposition problems. Empirically, such training process often first fits larger components and then discovers smaller components, which is similar to a tensor deflation process that is commonly used in tensor decomposition algorithms. We prove that for orthogonally decomposable tensor, a slightly modified version of gradient flow would follow a tensor deflation process and recover all the tensor components. Our proof suggests that for orthogonal tensors, gradient flow dynamics works similarly as greedy low-rank learning in the matrix setting, which is a first step towards understanding the implicit regularization effect of over-parametrized models for low-rank tensors.

Over the last decade, a single algorithm has changed many facets of our lives - Stochastic Gradient Descent (SGD). In the era of ever decreasing loss functions, SGD and its various offspring have become the go-to optimization tool in machine learning and are a key component of the success of deep neural networks (DNNs). While SGD is guaranteed to converge to a local optimum (under loose assumptions), in some cases it may matter which local optimum is found, and this is often context-dependent. Examples frequently arise in machine learning, from shape-versus-texture-features to ensemble methods and zero-shot coordination. In these settings, there are desired solutions which SGD on standard' loss functions will not find, since it instead converges to theeasy' solutions.

We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function h and the corresponding penalized estimator \hbeta, we construct a quantity \eta, the first order expansion of \hbeta, such that the distance between \hbeta and \eta is an order of magnitude smaller than the estimation error \ \hat{\beta} - \beta *\ . In this sense, the first order expansion \eta can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of \hat{\beta} is asymptotically the same as the risk of \eta which leads to a precise characterization of the MSE of \hbeta; this characterization takes a particularly simple form for isotropic design.