On the Parallel Speed-Up of Estimation of Multivariate Normal Algorithm and Evolution Strategies

Abstract

To cite this version:Abstract. Motivated by parallel optimization, we experiment EDA-like adaptation-rules in the case of λ large. The rule we use, essentially based on estimation of multivariate normal algorithm, is (i) compliant with all families of distributions for which a density estimation algorithm exists (ii) simple (iii) parameter-free (iv) better than current rules in this framework of λ large. The speed-up as a function of λ is consistent with theoretical bounds.

A New Selection Ratio for Large Population Sizes

“…Their main advantages are robustness and simplicity, but, because they are population based they are also well suitable for parallelization. However, it has been shown in [2,6] that ES are not very efficient for large population sizes whereas we can notice that the number of parallel machines, supercomputers and grids, has increased, suggesting big population sizes. We define λ as the population size, µ as the number of offspring used for the recombination of the parent, and N as the dimensionality.…”

“…It is widely believed that (µ/µ, λ)-ES are more parallel (efficient for large population sizes) than (1, λ)-ES, which is shown by rough calculus as in e.g. [1] and by experiments with intermediate values of λ. Experimentally however, (µ/µ, λ)-ES are less efficient than (1, λ)-ES for a (sufficiently) large population size λ, for the usual parametrization of the algorithms, as shown in [6]; in particular, the theoretical speedups of ES for parallel optimization (the best possible speed-ups, for optimal algorithms) are far better than the results of current implementations. In this paper, we show that changing the parametrization of self-adaptive (µ/µ, λ)-ES leads to a huge improvement for large λ, and that, with this improvement we can reach the theoretical bounds shown in [7].…”

Log-Linear Convergence of the Scale-Invariant (μ/μ w ,λ)-ES and Optimal μ for Intermediate Recombination for Large Population Sizes

“…CMA-ES which has been designed to work well on small population sizes uses μ = λ 2 as a default parameter. However, when using a large population size λ, the convergence rate of some real-world algorithms tested in [15,8] using the rules μ = λ 4 or μ = λ 2 as recommended in [14,6] is worse than the theoretical prediction of [16]. This is due to the fact that the rules used in these tests for choosing μ, are recommended by the studies performed under the approximation (d → +∞) [14,6] and thus under the assumption λ d. For some values of λ and d such that λ d, Beyer [17] computed, using some approximations permitted by the assumption (d → +∞), optimal choices for μ when minimizing spherical functions.…”