Randomized methods for computing optimal transport without regularization and their convergence analysis

TL;DR

This paper introduces randomized block coordinate descent methods for solving large-scale optimal transport problems directly as linear programs, providing convergence guarantees and demonstrating competitive efficiency and memory savings compared to existing algorithms.

Contribution

The paper develops and analyzes the RBCD framework for OT, including convergence proofs, acceleration techniques, and inexact variants, advancing direct LP solutions for OT without regularization.

Findings

RBCD methods converge almost surely with linear rate.

Accelerated RBCD outperforms Sinkhorn's algorithm at high accuracy.

Proposed methods save memory and improve efficiency in large-scale OT.

Abstract

The optimal transport (OT) problem can be reduced to a linear programming (LP) problem through discretization. In this paper, we introduced the random block coordinate descent (RBCD) methods to directly solve this LP problem. Our approach involves restricting the potentially large-scale optimization problem to small LP subproblems constructed via randomly chosen working sets. By using a random Gauss-Southwell-$q$ rule to select these working sets, we equip the vanilla version of (${\bf{\rm RBCD}_0}$) with almost sure convergence and a linear convergence rate to solve general standard LP problems. To further improve the efficiency of the (${\bf{\rm RBCD}_0}$) method, we explore the special structure of constraints in the OT problems and leverage the theory of linear systems to propose several approaches for refining the random working set selection and accelerating the vanilla method. Inexact versions of the RBCD methods are also discussed. Our preliminary numerical experiments demonstrate that the accelerated random block coordinate descent (${\bf {\rm ARBCD}}$) method compares well with other solvers including Sinkhorn's algorithm when seeking solutions with relatively high accuracy, and offers the advantage of saving memory.