Precise Runtime Analysis for Plateau Functions

TL;DR

This paper provides a precise theoretical analysis of how the (1+1) EA optimizes plateau functions, revealing that runtime depends mainly on the probability of flipping between 1 and k bits, and suggesting an optimal mutation rate.

Contribution

It introduces the Plateau_k benchmark and derives a surprising geometric distribution for runtime, highlighting the key role of specific bit-flip probabilities in evolutionary algorithms.

Findings

Expected runtime is inversely proportional to the probability of flipping between 1 and k bits.

Optimal mutation rate is approximately k/(en) for standard bit mutation.

The analysis approach combining Markov chains on search and Hamming levels is broadly useful.

Abstract

To gain a better theoretical understanding of how evolutionary algorithms (EAs) cope with plateaus of constant fitness, we propose the $n$-dimensional Plateau$_k$ function as natural benchmark and analyze how different variants of the $(1 + 1)$ EA optimize it. The Plateau$_k$ function has a plateau of second-best fitness in a ball of radius $k$ around the optimum. As evolutionary algorithm, we regard the $(1 + 1)$ EA using an arbitrary unbiased mutation operator. Denoting by $\alpha$ the random number of bits flipped in an application of this operator and assuming that $\Pr[\alpha = 1]$ has at least some small sub-constant value, we show the surprising result that for all constant $k \ge 2$, the runtime $T$ follows a distribution close to the geometric one with success probability equal to the probability to flip between $1$ and $k$ bits divided by the size of the plateau. Consequently, the expected runtime is the inverse of this number, and thus only depends on the probability to flip between $1$ and $k$ bits, but not on other characteristics of the mutation operator. Our result also implies that the optimal mutation rate for standard bit mutation here is approximately $k/(en)$. Our main analysis tool is a combined analysis of the Markov chains on the search point space and on the Hamming level space, an approach that promises to be useful also for other plateau problems.