Lipschitz-Free Mirror Descent Methods for Relatively Strongly Convex Functions with/without Absolute and Relative Inexactness

TL;DR

This paper develops and analyzes Lipschitz-free mirror descent algorithms for relatively strongly convex non-smooth optimization problems, providing convergence guarantees without Lipschitz assumptions and validating results through numerical experiments.

Contribution

It introduces a novel analysis of mirror descent methods that do not depend on Lipschitz continuity, extending their applicability to broader classes of convex functions.

Findings

Convergence bounds for exact and inexact subgradient methods

Numerical validation of theoretical bounds

Demonstrated effectiveness of the proposed algorithms

Abstract

In this paper, we analyze the mirror descent algorithm for non-smooth optimization problems in which the objective function is relatively strongly convex, without relying on the standard Lipschitz continuity assumption commonly used in the literature. We provide convergence analyses for both exact and inexact subgradient information. Furthermore, through numerical experiments, we compare the derived bounds on the quality of the approximate solutions with existing estimates in the literature and demonstrate the effectiveness of the proposed results.