Offspring Population Size Matters when Comparing Evolutionary Algorithms with Self-Adjusting Mutation Rates

We analyze the performance of the 2-rate $(1+\lambda)$ Evolutionary Algorithm (EA) with self-adjusting mutation rate control, its 3-rate counterpart, and a $(1+\lambda)$~EA variant using multiplicative update rules on the OneMax problem. We compare their efficiency for offspring population sizes ranging up to $\lambda=3,200$ and problem sizes up to $n=100,000$. Our empirical results show that the ranking of the algorithms is very consistent across all tested dimensions, but strongly depends on the population size. While for small values of $\lambda$ the 2-rate EA performs best, the multiplicative updates become superior for starting for some threshold value of $\lambda$ between 50 and 100. Interestingly, for population sizes around 50, the $(1+\lambda)$~EA with static mutation rates performs on par with the best of the self-adjusting algorithms. We also consider how the lower bound $p_{\min}$ for the mutation rate influences the efficiency of the algorithms. We observe that for the 2-rate EA and the EA with multiplicative update rules the more generous bound $p_{\min}=1/n^2$ gives better results than $p_{\min}=1/n$ when $\lambda$ is small. For both algorithms the situation reverses for large~$\lambda$.Comment: To appear at Genetic and Evolutionary Computation Conference (GECCO'19). v2: minor language revisio

When Does Hillclimbing Fail on Monotone Functions: An entropy compression argument

Hillclimbing is an essential part of any optimization algorithm. An important benchmark for hillclimbing algorithms on pseudo-Boolean functions $f: \{0,1\}^n \to \mathbb{R}$ are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the $(1+1)$-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability $c/n$ in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any monotone pseudo-boolean function in quasilinear time if $c<1$, but there are monotone functions for which the algorithm needs exponential time if $c>2.2$. But so far it was unclear whether the threshold is at $c=1$. In this paper we show how Moser's entropy compression argument can be adapted to this situation, that is, we show that a long runtime would allow us to encode the random steps of the algorithm with less bits than their entropy. Thus there exists a $c_0 > 1$ such that for all $0<c\le c_0$ the $(1+1)$-Evolutionary Algorithm with rate $c/n$ finds the optimum in $O(n \log^2 n)$ steps in expectation.Comment: 14 pages, no figure

Analysing Equilibrium States for Population Diversity

Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how population diversity of $(\mu+1)$ algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.Comment: To appear at GECCO 202

Self-Adjusting Evolutionary Algorithms for Multimodal Optimization

Recent theoretical research has shown that self-adjusting and self-adaptive mechanisms can provably outperform static settings in evolutionary algorithms for binary search spaces. However, the vast majority of these studies focuses on unimodal functions which do not require the algorithm to flip several bits simultaneously to make progress. In fact, existing self-adjusting algorithms are not designed to detect local optima and do not have any obvious benefit to cross large Hamming gaps. We suggest a mechanism called stagnation detection that can be added as a module to existing evolutionary algorithms (both with and without prior self-adjusting algorithms). Added to a simple (1+1) EA, we prove an expected runtime on the well-known Jump benchmark that corresponds to an asymptotically optimal parameter setting and outperforms other mechanisms for multimodal optimization like heavy-tailed mutation. We also investigate the module in the context of a self-adjusting (1+$\lambda$) EA and show that it combines the previous benefits of this algorithm on unimodal problems with more efficient multimodal optimization. To explore the limitations of the approach, we additionally present an example where both self-adjusting mechanisms, including stagnation detection, do not help to find a beneficial setting of the mutation rate. Finally, we investigate our module for stagnation detection experimentally.Comment: 26 pages. Full version of a paper appearing at GECCO 202