Linesearch-free adaptive Bregman proximal gradient for convex minimization without relative smoothness

TL;DR

This paper develops adaptive Bregman proximal gradient algorithms that operate without linesearch or global relative smoothness assumptions, enabling efficient convex minimization with only local curvature information.

Contribution

It introduces a novel linesearch-free adaptive method using Bregman distances and a new Bregman Young's inequality for convex minimization without global smoothness.

Findings

Algorithms perform competitively in numerical tests.

Eliminates need for backtracking linesearch.

Applicable to locally smooth convex problems.

Abstract

This paper introduces adaptive Bregman proximal gradient algorithms for solving convex composite minimization problems without relying on global relative smoothness or strong convexity assumptions. Building upon recent advances in adaptive stepsize selections, the proposed methods generate stepsizes based on local curvature estimates, entirely eliminating the need for backtracking linesearch. A key innovation is a Bregman generalization of Young's inequality, which allows controlling a critical inner product in terms of the same Bregman distances used in the updates. Our theory applies to problems where the differentiable term is merely locally smooth relative to a distance-generating function, without requiring the existence of global moduli or symmetry coefficients. Numerical experiments demonstrate their competitive performance compared to existing approaches across various problem classes.