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Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions

Cheikh Touré 1 Anne Auger 1 Nikolaus Hansen 1 
1 RANDOPT - Randomized Optimisation
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Evolution Strategies (ES) are stochastic derivative-free optimization algorithms whose most prominent representative, the CMA-ES algorithm, is widely used to solve difficult numerical optimization problems. We provide the first rigorous investigation of the linear convergence of step-size adaptive ES involving a population and recombination, two ingredients crucially important in practice to be robust to local irregularities or multimodality. We investigate convergence of step-size adaptive ES with weighted recombination on composites of strictly increasing functions with continuously differentiable scaling-invariant functions with a global optimum. This function class includes functions with non-convex sublevel sets and discontinuous functions. We prove the existence of a constant r such that the logarithm of the distance to the optimum divided by the number of iterations converges to r. The constant is given as an expectation with respect to the stationary distribution of a Markov chain—its sign allows to infer linear convergence or divergence of the ES and is found numerically. Our main condition for convergence is the increase of the expected log step-size on linear functions. In contrast to previous results, our condition is equivalent to the almost sure geometric divergence of the step-size on linear functions.
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https://hal.inria.fr/hal-03286037
Contributor : Cheikh Toure Connect in order to contact the contributor
Submitted on : Friday, June 24, 2022 - 12:32:19 PM
Last modification on : Wednesday, June 29, 2022 - 3:29:23 AM

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  • HAL Id : hal-03286037, version 2
  • ARXIV : 2107.08847

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Cheikh Touré, Anne Auger, Nikolaus Hansen. Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions. 2022. ⟨hal-03286037v2⟩

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