Maximizing drift is not optimal for solving OneMax

Abstract

It seems very intuitive that for the maximization of the OneMax problem f (x) := n i=1 x i the best that an elitist unary unbiased search algorithm can do is to store a best so far solution, and to modify it with the operator that yields the best possible expected progress in function value. This assumption has been implicitly used in several empirical works. In [Doerr, Doerr, Yang: GECCO 2016] it was formally proven that this approach is indeed almost optimal.In this work we prove that drift maximization is not optimal. More precisely, we show that for most fitness levels n/2 < /2 < 2n/3 the optimal mutation strengths are larger than the drift-maximizing ones. This implies that the optimal RLS is more risk-affine than the variant maximizing the step-wise expected progress. We show similar results for the mutation rates of the classic (1+1) Evolutionary Algorithm (EA) and its resampling variant, the (1+1) EA >0 .As a result of independent interest we show that the optimal mutation strengths, unlike the drift-maximizing ones, can be even.

Optimal Static Mutation Strength Distributions for the $(1+λ)$ Evolutionary Algorithm on OneMax

“…Our study continues the recent works [6] and [7], which provide optimal dynamic configurations for (1+1) and (1+𝜆)type algorithms, respectively. While their works are restricted to specific mutation operators (variants of SMB and the 𝑘-bit flips flip 𝑘 ), we study in this work in a generalization to arbitrary unary unbiased variation operators.…”

“…While their works are restricted to specific mutation operators (variants of SMB and the 𝑘-bit flips flip 𝑘 ), we study in this work in a generalization to arbitrary unary unbiased variation operators. In contrast to [6,7] we focus on static configurations, with the idea to build a rigorous baseline against which we can compare dynamic parameter control methods.…”

Blending Dynamic Programming with Monte Carlo Simulation for Bounding the Running Time of Evolutionary Algorithms

“…While doing it, we assume, similarly to [27], [28], that for all higher fitness values the best possible expected running times are already computed. However, since we aim at dealing with various fitness functions, we use the Monte Carlo approach to approximate transition probabilities instead.…”

“…In this section we outline several kinds of insights that can be derived from the results computed by the proposed algorithm. Some of them, namely lower bounds for parameter control methods, plots of optimal parameter values, and parameter efficiency heatmaps, have been previously proposed in [27], [28], and the regret plots are new to this paper.…”

“…Our contribution: We demonstrate in this work how the approach taken in [27], [28] can be extended to settings in which we do not necessarily have a closed-form expression from which we can derive exact parameter settings. Our overall approach is still based on dynamic programming, but we replace the exact formulae that model the probabilities to transition between different states by Monte Carlo simulations.…”