Self-Adjusting Mutation Rates with Provably Optimal Success Rules

The one-fifth success rule is one of the best-known and most widely accepted techniques to control the parameters of evolutionary algorithms. While it is often applied in the literal sense, a common interpretation sees the one-fifth success rule as a family of success-based updated rules that are determined by an update strength $F$ and a success rate $s$. We analyze in this work how the performance of the (1+1) Evolutionary Algorithm (EA) on LeadingOnes depends on these two hyper-parameters. Our main result shows that the best performance is obtained for small update strengths $F=1+o(1)$ and success rate $1/e$. We also prove that the running time obtained by this parameter setting is asymptotically optimal among all dynamic choices of the mutation rate for the (1+1) EA, up to lower order error terms. We show similar results for the resampling variant of the (1+1) EA, which enforces to flip at least one bit per iteration.