Variable metric evolution strategies by mutation matrix adaptation

Abstract

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful evolutionary algorithms. CMA-ES incrementally learns the variable metric by evolving a full covariance matrix. Yet, it suffers from high computational overload. In this paper, we propose two efficient variants of CMA-ES, termed mutation matrix adaptation (MMA-ES) and exponential MMA-ES (xMMA-ES). These variants are derived by taking the first order approximation of the update of the covariance matrix in CMA-ES. Both variants avoid the computational costly matrix decomposition while keeping the simplicity of the update scheme of CMA-ES. We analyze the properties and connections of MMA-ES and xMMA-ES to other variants of evolution strategies. We have experimentally studied the proposed algorithms’ behaviors and performances. xMMA-ES and MMA-ES generally outperform or perform competitively to CMA-ES. We have investigated the performance of MMA-ES with the BiPop restart strategy on the BBOB benchmarks. The experimental results validate the performance of the proposed algorithms.