Optimal Mutation Rates for the $(1+\lambda)$ EA on OneMax

Abstract

The OneMax problem, alternatively known as the Hamming distance problem, is often referred to as the "drosophila of evolutionary computation (EC)", because of its high relevance in theoretical and empirical analyses of EC approaches. It is therefore surprising that even for the simplest of all mutation-based algorithms, Randomized Local Search and the (1+1) EA, the optimal mutation rates were determined only very recently, in a GECCO 2019 poster. In this work, we extend the analysis of optimal mutation rates to two variants of the $(1+\lambda)$ EA and to the $(1+\lambda)$ RLS. To do this, we use dynamic programming and, for the $(1+\lambda)$ EA, numeric optimization, both requiring $\Theta(n^3)$ time for problem dimension $n$. With this in hand, we compute for all population sizes $\lambda \in \{2^i \mid 0 \le i \le 18\}$ and for problem dimension $n \in \{1000, 2000, 5000\}$ which mutation rates minimize the expected running time and which ones maximize the expected progress. Our results do not only provide a lower bound against which we can measure common evolutionary approaches, but we also obtain insight into the structure of these optimal parameter choices. For example, we show that, for large population sizes, the best number of bits to flip is not monotone in the distance to the optimum. We also observe that the expected remaining running time are not necessarily unimodal for the $(1+\lambda)$ EA$_{0 \rightarrow 1}$ with shifted mutation.