CMA-ES: Evolution strategies and covariance matrix adaptation

Evolution Strategies (ESs) and many continuous domain Estimation of Distribution Algorithms (EDAs) are stochastic optimization procedures that sample a multivariate normal (Gaussian) distribution in the continuous search space, Rⁿ. Many of them can be formulated in a unified and comparatively simple framework. This introductory tutorial focuses on the most relevant algorithmic question: how should the parameters of the sample distribution be chosen and, in particular, updated in the generation sequence? First, two common approaches for step-size control are reviewed, one-fifth success rule and path length control. Then, Covariance Matrix Adaptation (CMA) is discussed in depth: rank-one update, the evolution path, rank-mu update. Invariance properties and the interpretation as natural gradient descent are touched upon. In the beginning, general difficulties in solving non-linear, non-convex optimization problems in continuous domain are revealed, for example non-separability, ill-conditioning and ruggedness. Algorithmic design aspects are related to these difficulties. In the end, the performance of the CMA-ES is related to other well-known evolutionary and non-evolutionary optimization algorithms, namely BFGS, DE, PSO,...