The Interplay of Population Size and Mutation Probability in the ( $$1+\lambda $$ 1 + λ ) EA on OneMax

Abstract

The ($$1+\lambda $$1+ź) EA with mutation probability c / n, where $$c>0$$c>0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and $$\lambda $$ź. It turns out that 1 / n is the only optimal mutation probability if $$\lambda =o(\ln n\ln \ln n/\ln \ln \ln n)$$ź=o(lnnlnlnn/lnlnlnn), which is the cut-off point for linear speed-up. However, if $$\lambda $$ź is above this cut-off point then the standard mutation probability 1 / n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater. The theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes.