Abstract
Explaining to what extent the real power of genetic algorithms (GAs) lies in the ability of crossover to recombine individuals into higher quality solutions is an important problem in evolutionary computation. In this paper we show how the interplay between mutation and crossover can make GAs hillclimb faster than their mutation-only counterparts. We devise a Markov chain framework that allows to rigorously prove an upper bound on the runtime of standard steady state GAs to hillclimb the OneMax function. The bound establishes that the steady-state GAs are 25% faster than all standard bit mutation-only evolutionary algorithms with static mutation rate up to lower order terms for moderate population sizes. The analysis also suggests that larger populations may be faster than populations of size 2. We present a lower bound for a greedy (2 + 1) GA that matches the upper bound for populations larger than 2, rigorously proving that two individuals cannot outperform larger population sizes under greedy selection and greedy crossover up to lower order terms. In complementary experiments the best population size is greater than 2 and the greedy GAs are faster than standard ones, further suggesting that the derived lower bound also holds for the standard steady state (2 + 1) GA.
Highlights
G ENETIC algorithms (GAs) rely on a population of individuals that simultaneously explore the search space
A runtime of O(nk−1) may be achieved for any sublinear jump length k > 2 versus the required by standard bit mutation-only evolutionary algorithms (EAs). We show that this interplay between mutation and crossover may speed-up the hillclimbing capabilities of steady state genetic algorithms (GAs) without the need of enforcing diversity artificially
Today it has been rigorously proved that the standard steady state (μ+1)GA with realistic parameter settings does not require artificial diversity enforcement to outperform its standard bit mutation-only counterpart to escape the plateau of local optima of the JUMP function [5]
Summary
G ENETIC algorithms (GAs) rely on a population of individuals that simultaneously explore the search space. A runtime of O(nk−1) may be achieved for any sublinear jump length k > 2 versus the (nk) required by standard bit mutation-only EAs. In this paper, we show that this interplay between mutation and crossover may speed-up the hillclimbing capabilities of steady state GAs without the need of enforcing diversity artificially. Better runtimes are achieved for mutation rates that are even larger than the ones that minimize our theoretical upper bound (i.e., c/n with 1.5 ≤ c ≤ 1.6 versus the c =1.3 we have derived mathematically; interestingly this experimental rate is similar to the optimal mutation rate for OneMax of the algorithm analyzed in [39]) These theoretical and experimental results seem to be in line with those recently presented for the same steady state GAs for the JUMP function [5], [6]: higher mutation rates than 1/n are more effective on JUMP. Separate supplementary files contain an Appendix of omitted proofs due to space constraints and a complete version of this paper including all the proofs
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