Decentralized Algorithms for Wasserstein Barycenters

TL;DR

This thesis develops decentralized algorithms for Wasserstein barycenters, focusing on computational efficiency and statistical estimation, introducing new regularization techniques and dual methods that improve complexity bounds and enable practical distributed computation.

Contribution

It proposes novel primal and dual algorithms for Wasserstein barycenters, with a focus on decentralized execution and improved complexity through new regularization and dual formulations.

Findings

Dual approaches have lower oracle call complexity than primal methods.

New regularization improves approximation bounds.

Dual gradient computation is more efficient than primal in entropy-regularized cases.

Abstract

In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides. The statistical focus is estimating the sample size of measures necessary to calculate an approximation for Fr\'echet mean (barycenter) of a probability distribution with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied together with proposing a new regularization which contributes to the better complexity bounds in comparison with quadratic regularization. The computational focus is developing algorithms for calculating Wasserstein barycenters: both primal and dual algorithms which can be executed in a decentralized manner. The motivation for dual approaches is closed-forms for the dual formulation of entropy-regularized Wasserstein distances and their derivatives, whereas the primal formulation has closed-form expression only in some cases, e.g., for Gaussian measures. Moreover, the dual oracle returning the gradient of the dual representation for entropy-regularized Wasserstein distance can be computed for a cheaper price in comparison with the primal oracle returning the gradient of the entropy-regularized Wasserstein distance. The number of dual oracle calls, in this case, will also be less, i.e., the square root of the number of primal oracle calls. This explains the successful application of the first-order dual approaches for the Wasserstein barycenter problem.