The method of Bregman projections in deterministic and stochastic convex feasibility problems

TL;DR

This paper investigates Bregman projection methods for deterministic and stochastic convex feasibility problems, providing convergence analysis and rates, with applications to linear systems and optimal transport algorithms.

Contribution

It introduces a comprehensive analysis of Bregman projection methods with various control sequences, extending convergence results to new applications like entropic-regularized optimal transport.

Findings

Iterates converge Q-linearly with explicit rates.

Generalizes randomized methods for linear systems.

Applies results to Sinkhorn and Greenhorn algorithms.

Abstract

In this work we study the method of Bregman projections for deterministic and stochastic convex feasibility problems with three types of control sequences for the selection of sets during the algorithmic procedure: greedy, random, and adaptive random. We analyze in depth the case of affine feasibility problems showing that the iterates generated by the proposed methods converge Q-linearly and providing also explicit global and local rates of convergence. This work generalizes from one hand recent developments in randomized methods for the solution of linear systems based on orthogonal projection methods. On the other hand, our results yield global and local Q-linear rates of convergence for the Sinkhorn and Greenhorn algorithms in discrete entropic-regularized optimal transport, for the first time, even in the multimarginal setting.