An Extended Jump Functions Benchmark for the Analysis of Randomized Search Heuristics
This work addresses a theoretical limitation in evolutionary algorithm benchmarking for researchers in optimization and algorithm theory, but it is incremental as it extends known functions with new parameters.
The paper tackles the problem of whether existing jump functions are representative benchmarks for randomized search heuristics by proposing an extended class with a valley of low fitness, proving that some prior results generalize while others do not, such as showing the optimal mutation rate for the (1+1) EA is δ/n and that the fast (1+1) EA runs faster by a super-exponential factor in δ.
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.