Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Henry Bambury, Antoine Bultel, Benjamin Doerr

TL;DR

The paper generalizes the classic Jump_k benchmark to Jump_{k,δ}, introducing a valley of width δ at Hamming distance k from the optimum and investigating how randomized search heuristics navigate it. Through rigorous runtime analyses, it shows that in the standard regime the asymptotically optimal fixed mutation rate is p = δ/n and that a fast (1+1) EA gains a factor roughly binom{k}{δ} over the classic rate, while some existing results (notably SD-RLS^*) do not uniformly extend to Jump_{k,δ}. It also extends the framework to heavy-tailed mutation (FEA_β) with only polynomial overhead relative to the optimal fixed-rate run, and provides experimental evidence that corroborates the theory while highlighting regime-dependent behaviors. Overall, Jump_{k,δ} provides a richer, more representative benchmark for understanding how EAs escape local optima, informing mutation strategies and stagnation-detection approaches in multimodal landscapes.

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 $\textsc{Jump}_{k,δ}$ of jump functions that contain a valley of low fitness of width $δ$ 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 $δ< k$, the optimal mutation rate for the $(1+1)$~EA is $\fracδ{n}$, and the fast $(1+1)$~EA runs faster than the classical $(1+1)$~EA by a factor super-exponential in $δ$. 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 $\textsc{Jump}_{k,δ}$ instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.

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

TL;DR

The paper generalizes the classic Jump_k benchmark to Jump_{k,δ}, introducing a valley of width δ at Hamming distance k from the optimum and investigating how randomized search heuristics navigate it. Through rigorous runtime analyses, it shows that in the standard regime the asymptotically optimal fixed mutation rate is p = δ/n and that a fast (1+1) EA gains a factor roughly binom{k}{δ} over the classic rate, while some existing results (notably SD-RLS^*) do not uniformly extend to Jump_{k,δ}. It also extends the framework to heavy-tailed mutation (FEA_β) with only polynomial overhead relative to the optimal fixed-rate run, and provides experimental evidence that corroborates the theory while highlighting regime-dependent behaviors. Overall, Jump_{k,δ} provides a richer, more representative benchmark for understanding how EAs escape local optima, informing mutation strategies and stagnation-detection approaches in multimodal landscapes.

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 of jump functions that contain a valley of low fitness of width starting at distance from the global optimum. We prove that several previous results extend to this more general class: for all {} and , the optimal mutation rate for the ~EA is , and the fast ~EA runs faster than the classical ~EA by a factor super-exponential in . 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 ~EA by a factor polynomial in on , is slower by a factor polynomial in on some instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.

Paper Structure

This paper contains 12 sections, 18 theorems, 53 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. State of the Art
  3. Preliminaries
  4. Definition of the $\textsc{Jump}\xspace_{k,\delta}$ Function
  5. Stochastic Domination
  6. The $(1 + 1)$ EA with Fixed Mutation Rate
  7. General Upper and Lower Bounds on the Expected Runtime
  8. Optimal Mutation Rate in the Standard Regime
  9. Heavy-tailed Mutation
  10. Stagnation Detection
  11. Experiments
  12. Conclusion

Key Result

Lemma 2

The following two conditions are equivalent.

Figures (7)

  • Figure 1: Profile of the $\textsc{Jump}\xspace_6$ function.
  • Figure 2: Profile of the $\textsc{Jump}\xspace_{16,6}$ function.
  • Figure 3: Optimization times of different algorithms on the classic jump function $\textsc{Jump}\xspace_k = \textsc{Jump}\xspace_{k,\delta}$ with $k=\delta=4$.
  • Figure 4: Optimization times on $\textsc{Jump}\xspace_{k,\delta}$ with $k=6$, $\delta = 4$.
  • Figure 5: Optimization times on $\textsc{Jump}\xspace_{k,\delta}$ with $k=3\ln(n)$, $\delta=\frac{k}{2}$.
  • ...and 2 more figures

Theorems & Definitions (33)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Definition 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • proof
  • ...and 23 more