Self-adjusting evolutionary algorithms for multimodal optimization

Abstract

Recent theoretical research has shown that self-adjusting and selfadaptive 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 schemes). 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+ ) 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.

Evolutionary Algorithms with Self-adjusting Asymmetric Mutation

“…A promising setting for this bias is learned through a self-adjusting scheme being similar in style with the 1/5-rule [2] and related techniques: in a certain observation phase of length N two different parameter values are each tried N/2 times and the value that is relatively more successful is used in the next phase. Hence, this approach is in line with a recent line of theoretical research of self-adjusting algorithms where the concrete implementation of self-adjustment is an ongoing debate [4,5,7,[9][10][11]15,23,24]. See also the recent survey article [3] for an in-depth coverage of parameter control, self-adjusting algorithms, and theoretical runtime results.…”

Automatic adaptation of hypermutation rates for multimodal optimisation

“…A value of 𝛽 = 1 is the best possible if more than 𝑛 𝜖 bits, for any arbitrarily small constant 𝜖 > 0, have to be flipped to escape from the local optima. We point out that the best possible expected runtime achievable with SBM is 𝑂 ((𝑒𝑛/𝑘) 𝑘 which is matched by the stagnation detection adaptive (1+1) EA [35].…”

“…Future work should further evaluate experimentally and theoretically the performance of the proposed algorithms on classical combinatorial optimization problems and real-world applications. We note that recently a different self-adjusting mechanism has been proposed in the literature with the aim of increasing the standard bit mutation rate of evolutionary algorithms when detecting local optima [35]. While the mechanism was not designed with the aim of enhancing the non-elitist performance of search heuristics as well as the mutation operator in the presence of local optima, a comparison with our own proposed mechanism should be performed in the near future.…”

“…Recently a stagnation detection mechanism was introduced in the literature for steady-state EAs to identify whether they are on a local optima, and consequently increase the mutation rate [35]. Apart from the different motivations behind our work and theirs, one crucial difference between the update mechanisms, is that stagnation detection only increases the standard bit mutation rate above the standard 1/n rate, when with high probability no neighbouring improvements exist, while our proposed mechanism increases the rate after each unsuccessful hypermutation (at a rate than depends on parameter 𝜌).…”

Self-Adjusting Evolutionary Algorithms for Multimodal Optimization

“…We finish with some conclusions. Due to space restrictions, several proofs had to be omitted from this paper and have been replaced by proof sketches, but note that these proofs can be found in the preprint [26].…”