Optimal Parameter Choices via Precise Black-Box Analysis

TL;DR

This paper provides a precise complexity-theoretic analysis of evolutionary algorithms, revealing detailed parameter influences and demonstrating that a simple algorithm can outperform previous methods on the OneMax benchmark.

Contribution

It introduces the first precise black-box complexity result for OneMax and develops new variable drift theorems, bridging algorithm analysis and complexity theory.

Findings

Unary unbiased black-box complexity of OneMax is approximately n ln(n) - cn

A simple (1+1)-type algorithm with fitness-dependent mutation outperforms previous algorithms by about 13%

New variable drift theorems are formulated, potentially useful for future analyses

Abstract

It has been observed that some working principles of evolutionary algorithms, in particular, the influence of the parameters, cannot be understood from results on the asymptotic order of the runtime, but only from more precise results. In this work, we complement the emerging topic of precise runtime analysis with a first precise complexity theoretic result. Our vision is that the interplay between algorithm analysis and complexity theory becomes a fruitful tool also for analyses more precise than asymptotic orders of magnitude. As particular result, we prove that the unary unbiased black-box complexity of the OneMax benchmark function class is $n \ln(n) - cn \pm o(n)$ for a constant $c$ which is between $0.2539$ and $0.2665$. This runtime can be achieved with a simple (1+1)-type algorithm using a fitness-dependent mutation strength. When translated into the fixed-budget perspective, our algorithm finds solutions which are roughly 13\% closer to the optimum than those of the best previously known algorithms. To prove our results, we formulate several new versions of the variable drift theorems, which also might be of independent interest.