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

Mohammad S. Alkousa, Fedor S. Stonyakin

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.

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

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.
Paper Structure (6 sections, 3 theorems, 54 equations, 6 figures, 1 algorithm)

This paper contains 6 sections, 3 theorems, 54 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fundamentals
  3. Lipschitz-free mirror descent method for strongly convex functions
  4. Lipschitz-free mirror descent method for strongly convex functions with inexact information
  5. Numerical experiments
  6. Conclusions

Key Result

Lemma 2.1

(Three points identity) Chen1993 Suppose that $\psi: \mathbf{E} \longrightarrow (- \infty, \infty]$ is a proper closed, convex, and differentiable function over $\operatorname{dom}(\partial \psi)$. Let $a, b \in \operatorname{dom}(\partial(\psi))$ and $c \in \operatorname{dom} (\psi)$. Then

Figures (6)

  • Figure 1: Results of comparison of theoretical estimates with respect to the functional values \ref{['estim_subgrad_func']} and \ref{['extime2_sub_strogly']}, for Example \ref{['ex_1']}, with $n=1000, R = 10$ (left). And with $n = 1000, R = 100$ (right).
  • Figure 2: Results of comparison of theoretical estimates with respect to the variable \ref{['estim_subgrad_var']} and \ref{['estimate_variable_new']}, for Example \ref{['ex_1']}, with $n=1000, R = 10$ (left). And with $n = 1000, R = 100$ (right).
  • Figure 3: Results of comparison of theoretical estimates with respect to the functional values \ref{['estim_subgrad_func']} and \ref{['extime2_sub_strogly']}, for Example \ref{['ex_2']}, with $m = 50, n=1000, R = 10$ (left). And with $m = 50, n = 1000, R = 100$ (right).
  • Figure 4: Results of comparison of theoretical estimates with respect to the variable \ref{['estim_subgrad_var']} and \ref{['estimate_variable_new']}, for Example \ref{['ex_2']}, with $n=1000, R = 10$ (left). And with $n = 1000, R = 100$ (right).
  • Figure 5: Theoretical estimates with respect to the functional values \ref{['extime_relative_mirror_free']} and variable \ref{['extime_var_relative']}, for Example \ref{['ex_1']}, with $n=1000,$ and $R = 10$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Example 5.1
  • Example 5.2