Dimension-Independent Convergence Rate for Non-isotropic (1, λ) - ES

摘要

Based on the theory of non-negative super martingales, convergence results are proven for adaptive (1,λ) - ES (i.e. with Gaussian mutations), and geometrical convergence rates are derived. In the d-dimensional case (d > 1), the algorithm studied here uses a different step-size update in each direction. However, the critical value for the step-size, and the resulting convergence rate do not depend on the dimension. Those results are discussed with respect to previous works. Rigorous numerical investigations on some 1-dimensional functions validate the theoretical results. Trends for future research are indicated.