An Extended Jump Functions Benchmark for the Analysis of Randomized Search Heuristics

TL;DR

This paper introduces an extended class of jump functions with wider valleys to analyze how randomized search heuristics perform in more complex landscapes, revealing both extended and limited applicability of previous results.

Contribution

It proposes the $ extsc{Jump}_{k, extdelta}$ class, generalizing jump functions, and extends theoretical analyses of mutation rates and algorithm speeds to this broader setting.

Findings

Optimal mutation rate for $(1+1)$ EA is $rac{ extdelta}{n}$ for certain parameters.

Fast $(1+1)$ EA outperforms classical $(1+1)$ EA exponentially faster on the new class.

Some known results do not extend, showing limitations of existing algorithms on wider valleys.

Abstract

Jump functions are the {most-studied} non-unimodal benchmark in the theory of randomized search heuristics, in particular, evolutionary algorithms (EAs). They have significantly improved our understanding of how EAs escape from local optima. However, their particular structure -- to leave the local optimum one can only jump directly to the global optimum -- raises the question of how representative such results are. For this reason, we propose an extended class $\textsc{Jump}_{k,\delta}$ of jump functions that contain a valley of low fitness of width $\delta$ starting at distance $k$ from the global optimum. We prove that several previous results extend to this more general class: for all {$k \le \frac{n^{1/3}}{\ln{n}}$} and $\delta < k$, the optimal mutation rate for the $(1+1)$~EA is $\frac{\delta}{n}$, and the fast $(1+1)$~EA runs faster than the classical $(1+1)$~EA by a factor super-exponential in $\delta$. However, we also observe that some known results do not generalize: the randomized local search algorithm with stagnation detection, which is faster than the fast $(1+1)$~EA by a factor polynomial in $k$ on $\textsc{Jump}_k$, is slower by a factor polynomial in $n$ on some $\textsc{Jump}_{k,\delta}$ instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.