Lipschitz-Free Mirror Descent Methods for Non-Smooth Optimization Problems

TL;DR

This paper corrects previous convergence analysis of mirror descent with adaptive step sizes and introduces a Lipschitz-free mirror descent method that guarantees weak ergodic convergence without Lipschitz continuity assumptions.

Contribution

It provides a corrected convergence analysis for adaptive mirror descent and introduces a new Lipschitz-free variant with weak ergodic convergence guarantees.

Findings

Corrected convergence rate analysis for adaptive mirror descent.

Introduced Lipschitz-free mirror descent method with weak ergodic convergence.

Generalized convergence results without Lipschitz assumptions.

Abstract

The part of the analysis of the convergence rate of the mirror descent method that is connected with the adaptive time-varying step size rules due to Alkousa et al. (MOTOR 2024, pp. 3-18) is corrected. Moreover, a Lipschitz-free mirror descent method that achieves weak ergodic convergence is presented, generalizing the convergence results of the mirror descent method in the absence of the Lipschitz assumption.