On Averaging the Best Samples in Evolutionary Computation

Abstract

Choosing the right selection rate is a long standing issue in evolutionary computation. In the continuous unconstrained case, we prove mathematically that a single parent μ = 1 leads to a suboptimal simple regret in the case of the sphere function. We provide a theoretically-based selection rate μ/λ that leads to better progress rates. With our choice of selection rate, we get a provable regret of order O(λ −1 ) which has to be compared with O(λ −2/d ) in the case where μ = 1. We complete our study with experiments to confirm our theoretical claims.

Asymptotic convergence rates for averaging strategies

“…In this paper, we are interested in the rates obtained by averaging the best samples for a wide class of functions. We extend the results of [23] which only hold for the sphere function.…”

“…In this paper we are able to deal with potentially non symmetric functions. One can remark, that if the sublevel sets are symmetric the bias term vanishes and we recover the rate of [23].…”

“…Then, in Section 4, we prove upper and lower convergence rates for random search for these functions. In Section 5, we extend [23] by showing that asymptotically in the number of samples , the handled functions satisfy a better convergence rate than random search. We then extend our results on wider classes of functions in Section 6.…”

“…1.1.1 Better than picking up the best. Given a finite number of samples 𝜆 equipped with their fitness values, we can simply pick up the best, or average the "best ones" [9,23], or apply a surrogate model [7,12,16,19,27,29]. Overall, the best is quite robust, but the surrogate or the averaging usually provides better convergence rates.…”

“…Although experiments show that these methods perform well, it is not still understood why taking the average of best points actually leads to a lower regret. In [23], it is proved in the case of quadratic functions that the regret is indeed lower for the averaging strategy than for pure random search. In this paper, we extend the result of this paper by proving convergence rates for a wide class of functions including three times continuously differentiable functions with unique optima.…”