Markov Chain Analysis of Cumulative Step-size Adaptation on a Linear Constrained Problem

TL;DR

This paper models the behavior of a (1, λ)-Evolution Strategy on a linear constrained problem using Markov chains, analyzing stability and divergence with and without step-size adaptation.

Contribution

It introduces a Markov chain framework to analyze the stability and divergence of the algorithm under different step-size adaptation schemes.

Findings

Constant step-size leads to divergence of the algorithm.

Stability of the Markov chain is proven for the constant step-size case.

For cumulative step-size adaptation with parameter 1, stability is established, with discussions on extending to the full algorithm.

Abstract

This paper analyzes a (1, $\lambda$)-Evolution Strategy, a randomized comparison-based adaptive search algorithm, optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using cumulative step-size adaptation. We exhibit for each case a Markov chain describing the behaviour of the algorithm. Stability of the chain implies, by applying a law of large numbers, either convergence or divergence of the algorithm. Divergence is the desired behaviour. In the constant step-size case, we show stability of the Markov chain and prove the divergence of the algorithm. In the cumulative step-size adaptation case, we prove stability of the Markov chain in the simplified case where the cumulation parameter equals 1, and discuss steps to obtain similar results for the full (default) algorithm where the cumulation parameter is smaller than 1. The stability of the Markov chain allows us to deduce geometric divergence or convergence , depending on the dimension, constraint angle, population size and damping parameter, at a rate that we estimate. Our results complement previous studies where stability was assumed.