Stochastic Optimization for Regularized Wasserstein Estimators

Abstract

Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss between distributions that has been used in many applications throughout machine learning and statistics. Recent algorithmic progress on this problem and its regularized versions have made these tools increasingly popular. However, existing techniques require solving an optimization problem to obtain a single gradient of the loss, thus slowing down first-order methods to minimize the sum of losses, that require many such gradient computations. In this work, we introduce an algorithm to solve a regularized version of this problem of Wasserstein estimators, with a time per step which is sublinear in the natural dimensions of the problem. We introduce a dual formulation, and optimize it with stochastic gradient steps that can be computed directly from samples, without solving additional optimization problems at each step. Doing so, the estimation and computation tasks are performed jointly. We show that this algorithm can be extended to other tasks, including estimation of Wasserstein barycenters. We provide theoretical guarantees and illustrate the performance of our algorithm with experiments on synthetic data.

References in corpus (10)

• Fast Computation of Wasserstein Barycenters
• Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes
• Sample Complexity of Sinkhorn divergences
• Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
• Wasserstein Discriminant Analysis
• Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance
• Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm
• optimalFlow: Optimal-transport approach to flow cytometry gating and population matching
• Utility/Privacy Trade-off through the lens of Optimal Transport
• Statistical Windows in Testing for the Initial Distribution of a Reversible Markov Chain

Cited by in corpus (3)

• Minibatch optimal transport distances; analysis and applications
• Stochastic Saddle-Point Optimization for Wasserstein Barycenters
• Stochastic Approximation versus Sample Average Approximation for population Wasserstein barycenters