Sample Reuse via Importance Sampling in Information Geometric Optimization

In this paper we propose a technique to reduce the number of function evaluations, which is often the bottleneck of the black-box optimization, in the information geometric optimization (IGO) that is a generic framework of the probability model-based black-box optimization algorithms and generalizes several well-known evolutionary algorithms, such as the population-based incremental learning (PBIL) and the pure rank-$\mu$ update covariance matrix adaptation evolution strategy (CMA-ES). In each iteration, the IGO algorithms update the parameters of the probability distribution to the natural gradient direction estimated by Monte-Carlo with the samples drawn from the current distribution. Our strategy is to reuse previously generated and evaluated samples based on the importance sampling. It is a technique to reduce the estimation variance without introducing a bias in Monte-Carlo estimation. We apply the sample reuse technique to the PBIL and the pure rank-$\mu$ update CMA-ES and empirically investigate its effect. The experimental results show that the sample reuse helps to reduce the number of function evaluations on many benchmark functions for both the PBIL and the pure rank-$\mu$ update CMA-ES. Moreover, we demonstrate how to combine the importance sampling technique with a variant of the CMA-ES involving an algorithmic component that is not derived in the IGO framework.