Performance Analysis of Evolution Strategies with Multi- Recombination in High-Dimensional IRN-Search Spaces Disturbed by Noise

The presence of noise in real-world optimization problems poses difficulties to optimization strategies. It is frequently observed that evolutionary algorithms are quite capable of succeeding in noisy environments. Intuitively, the use of a population of candidate solutions alongside with some implicit or explicit form of averaging inherent in the algorithms is considered responsible. However, so as to arrive at a deeper understanding of the reasons for the capabilities of evolutionary algorithms, mathematical analyses of their performance in select environments are necessary. Such analyses can reveal how the performance of the algorithms scales with parameters of the problem — such as the dimensionality of the search space or the noise strength — or of the algorithms — such as population size or mutation strength. Recommendations regarding the optimal sizing of such parameters can then be derived. The present paper derives an asymptotically exact approximation to the progress rate of the (μ=μI;λ)-Evolution Strategy (ES) on a finite-dimensional noisy sphere. It is shown that, in contrast to results obtained in the limit of infinite search space dimensionality, there is a finite optimal population size above which the efficiency of the strategy declines, and that therefore it is not possible to attain the efficiency that can be achieved in the absence of noise by increasing the population size. It is also shown that nonetheless, the benefits of genetic repair and an increased mutation strength make it possible for the multi-parent (μ=μI;λ)-ES to far outperform simple one-parent strategies.