Mirror Sinkhorn: Fast Online Optimization on Transport Polytopes

Optimal transport has arisen as an important tool in machine learning, allowing to capture geometric properties of the data. It is formulated as a linear program on transport polytopes. The problem of convex optimization on this set includes both OT and multiple related ones, such as point cloud registration. We present in this work an optimization algorithm that utilizes Sinkhorn matrix scaling and mirror descent to minimize convex objectives on this domain. This algorithm can be run online and is both adaptive and robust to noise. A mathematical analysis of the convergence rate of the algorithm for minimising convex functions is provided, as well as experiments that illustrate its performance on synthetic data and real-world data.

Stochastic Optimization for Regularized Wasserstein Estimators

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.