Asymptotical Analysis of the $(1+(λ,λ))$ GA Escape Time from Local Optima on Jump Functions

Abstract

The paper develops the approach to the runtime analysis of evolutionary algorithms on the basis of limit theorems from probability theory. We consider the family of Jump$_k$ benchmark functions, defined on the search space of binary strings of length $n$, parametrized by the integer $k$, which have a plateau of multiple local optima at the Hamming distance $k$ from a unique global optimum. In this work, we consider the genetic algorithm $(1+(λ,λ)) GA$ from (Doerr, Doerr and Ebel, 2015) with tunable parameters of the mutation rate $p$, crossover bias $c$, and two intermediate population sizes $λ_M$ and $λ_C$. We study the time it escapes from the plateau of local optima and reaches the global optimum in the case of Jump$_k$ fitness function and tighten the upper bounds on the expected escape time, known from the work of Antipov, Doerr and Karavaev (2022). The obtained bounds also apply to a wider range of algorithmic parameters. The main result of this work applies to the case when $k\to \infty$ as $n \to \infty.$ The case of finite $k$ is investigated quite simply and considered tangentially.