First Steps Towards a Runtime Analysis When Starting With a Good Solution

TL;DR

This paper initiates a mathematical analysis of evolutionary algorithms starting from good solutions, revealing how initialization quality affects performance and how adaptive parameters can improve outcomes.

Contribution

It introduces the first formal analysis of evolutionary algorithms with non-random initial solutions, highlighting the impact on algorithm performance and parameter tuning.

Findings

Better initial solutions can significantly improve algorithm performance.

Adaptive parameter strategies like self-adjusting and heavy-tailed choices are beneficial.

There is a gap between current algorithms and the optimal exploitation of good initial solutions.

Abstract

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of the algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the $(1+1)$ evolutionary algorithm and the static, self-adjusting, and heavy-tailed $(1 + (\lambda,\lambda))$ GA on the OneMax benchmark. We are optimistic that the question how to profit from good initial solutions is interesting beyond these first examples.