Self-adjusting offspring population sizes outperform fixed parameters on the cliff function

In the discrete domain, self-adjusting parameters of evolutionary algorithms (EAs) has emerged as a fruitful research area with many runtime analyses showing that self-adjusting parameters can out-perform the best fixed parameters. Most existing runtime analyses focus on elitist EAs on simple problems, for which moderate performance gains were shown. Here we consider a much more challenging scenario: the multimodal function Cliff, defined as an example where a (1, λ) EA is effective, and for which the best known upper runtime bound for standard EAs is O(n²⁵).

We prove that a (1, λ) EA self-adjusting the offspring population size λ using success-based rules optimises Cliff in O(n) expected generations and O(n log n) expected evaluations. Along the way, we prove tight upper and lower bounds on the runtime for fixed λ (up to a logarithmic factor) and identify the runtime for the best fixed λ as n^η for η ≈ 3.9767 (up to sub-polynomial factors). Hence, the self-adjusting (1, λ) EA outperforms the best fixed parameter by a factor of at least n^2.9767 (up to sub-polynomial factors).