Linesearch-free adaptive Bregman proximal gradient for convex minimization under local relative smoothness

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 tool in our analysis is a Bregman generalization of Young's inequality, which allows the control of 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 \emph{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.