Linear convergence of mirror descent via Bregman distance under more general conditions

Abstract

The mirror gradient algorithm is a first-order method based on the Bregman distance for convex optimization. Traditional analyses typically assume Lipschitz continuous gradients and strongly convex Bregman kernels. Recent works introduced relative smoothness to relax the Lipschitz condition. However, convergence guarantees under the absence of both assumptions remain limited. This paper establishes global convergence to a minimizer and linear convergence to the optimal solution set using only relative smoothness, without requiring strong convexity of the kernel or Lipschitz continuity of the objective gradient. These results broaden the applicability of mirror descent to a wider class of problems.