What Does the Evolution Path Learn in CMA-ES?

The Covariance matrix adaptation evolution strategy (CMA-ES) evolves a multivariate Gaussian distribution for continuous optimization. The evolution path, which accumulates historical search direction in successive generations, plays a crucial role in the adaptation of covariance matrix. In this paper, we investigate what the evolution path approximates in the optimization procedure. We show that the evolution path accumulates natural gradient with respect to the distribution mean, and works as a momentum under stationary condition. The experimental results suggest that it approximates a scale direction expanded by singular values along corresponding eigenvectors of the inverse Hessian. Further, we show that the outer product of evolution path serves as a rank-1 momentum term for the covariance matrix. This paper will be presented at The 14th International Conference on Parallel Problem Solving from Nature (PPSN XIV), which will be organized by Edinburgh Napier University, Edinburgh, Scotland, UK on September 17-21st, 2016. Supervisor: Prof Qingfu Zhang Research Interests: Evolutionary Computation and Machine Learning.