Stochastic Block Bregman Projection with Polyak-like Stepsize for Possibly Inconsistent Convex Feasibility Problems

TL;DR

This paper introduces a stochastic block Bregman projection method with Polyak-like stepsizes for convex feasibility problems, achieving convergence guarantees and robustness, especially in inconsistent cases.

Contribution

It proposes a unified bilevel reformulation and a new stochastic projection algorithm with Polyak-like stepsizes, extending existing methods and ensuring convergence in challenging scenarios.

Findings

Establishes ergodic sublinear convergence rates for the method.

Achieves linear convergence under a Bregman distance growth condition.

Demonstrates robustness and effectiveness through numerical experiments.

Abstract

Stochastic projection algorithms for solving convex feasibility problems (CFPs) have attracted considerable attention due to their broad applicability. In this paper, we propose a unified stochastic bilevel reformulation for possibly inconsistent CFPs that combines proximity function minimization and structural regularization, leading to a feasible bilevel model with a unique and stable regularized solution. From the algorithmic perspective, we develop the stochastic block Bregman projection method with Polyak-like and projective stepsizes, which not only subsumes several recent stochastic projection algorithms but also induces new schemes tailored to specific problems. Moreover, we establish ergodic sublinear convergence rates for the expected inner function, as well as linear convergence in expectation to the inner minimizer set under a Bregman distance growth condition. In particular, the proposed Polyak-like stepsize ensures exact convergence in expectation for possibly inconsistent CFPs. Finally, numerical experiments demonstrate the effectiveness of the proposed method and its robustness to noise.