Convergence results for the (1, γ)-SA-ES using the theory of φ-irreducible Markov chains

This paper investigates theoretically the (1,λ)-SA-ES on the well known sphere function. We prove sufficient conditions on the parameters of the algorithm ensuring the convergence of 1/nln(∥X_n∥), where Xn is the parent at generation n. This in turn guarantees the asymptotic log-linear convergence or divergence of the algorithm. The technique used for this analysis calls upon the theory of Markov chains on a continuous state space and on the so-called Foster-Lyapunov drift conditions. Those conditions enable us to derive practical conditions that prove stability properties of Markov chains.