A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization

TL;DR

This paper introduces a stochastic primal-dual algorithm using Bregman divergences for convex-concave saddle point problems, achieving various convergence guarantees under different assumptions, applicable to Wasserstein barycenter and inverse problems.

Contribution

It develops a general stochastic primal-dual method with Bregman divergences that converges under mild conditions, extending previous deterministic and strongly convex approaches.

Findings

Ergodic convergence rate of O(1/k) in expectation.

Almost sure weak convergence of iterates to saddle points.

Strong convergence under additional convexity assumptions.

Abstract

We study a stochastic first order primal-dual method for solving convex-concave saddle point problems over real reflexive Banach spaces using Bregman divergences and relative smoothness assumptions, in which we allow for stochastic error in the computation of gradient terms within the algorithm. We show ergodic convergence in expectation of the Lagrangian optimality gap with a rate of O(1/k) and that every almost sure weak cluster point of the ergodic sequence is a saddle point in expectation under mild assumptions. Under slightly stricter assumptions, we show almost sure weak convergence of the pointwise iterates to a saddle point. Under a relative strong convexity assumption on the objective functions and a total convexity assumption on the entropies of the Bregman divergences, we establish almost sure strong convergence of the pointwise iterates to a saddle point. Our framework is general and does not need strong convexity of the entropies inducing the Bregman divergences in the algorithm. Numerical applications are considered including entropically regularized Wasserstein barycenter problems and regularized inverse problems on the simplex.