Irreducibility of nonsmooth state-space models with an application to CMA-ES

TL;DR

This paper investigates the irreducibility of a Markov chain derived from the CMA-ES optimization algorithm, providing foundational results for analyzing its convergence properties in nonsmooth, scale-invariant settings.

Contribution

It establishes the irreducibility and aperiodicity of the Markov chain associated with CMA-ES, extending conditions for nonsmooth state-space models to facilitate convergence analysis.

Findings

Proves the Markov chain is irreducible and aperiodic.

Extends irreducibility conditions to nonsmooth state spaces.

Lays groundwork for analyzing CMA-ES convergence.

Abstract

We analyze a stochastic process resulting from the normalization of states in the zeroth-order optimization method CMA-ES. On a specific class of minimization problems where the objective function is scaling-invariant, this process defines a time-homogeneous Markov chain whose convergence at a geometric rate can imply the linear convergence of CMA-ES. However, the analysis of the intricate updates for this process constitute a great mathematical challenge. We establish that this Markov chain is an irreducible and aperiodic T-chain. These contributions represent a first major step for the convergence analysis towards a stationary distribution. We rely for this analysis on conditions for the irreducibility of nonsmooth state-space models on manifolds. To obtain our results, we extend these conditions to address the irreducibility in different hyperparameter settings that define different Markov chains, and to include nonsmooth state spaces.