We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring.We analyze how the (1+λ) evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the (1+λ) EA finds the optimum in an expected optimization time (number of fitness evaluations) of O(nλ/ log λ + n log n). This time is asymptotically smaller than the optimization time of the classic (1 + λ) EA. Previous work shows that this performance is best-possible among all λ-parallel mutation-based unbiased black-box algorithms.This result shows that the new way of adjusting the mutation rate can find optimal dynamic parameter values on the fly. Since our adjustment mechanism is simpler than the ones previously used for adjusting the mutation rate and does not have parameters itself, we are optimistic that it will find other applications.
We consider stochastic versions of OneMax and LeadingOnes and analyze the performance of evolutionary algorithms with and without populations on these problems. It is known that the (1+1) EA on OneMax performs well in the presence of very small noise, but poorly for higher noise levels. We extend these results to LeadingOnes and to many different noise models, showing how the application of drift theory can significantly simplify and generalize previous analyses.Most surprisingly, even small populations (of size Θ(log n)) can make evolutionary algorithms perform well for high noise levels, well outside the abilities of the (1+1) EA! Larger population sizes are even more beneficial; we consider both parent and offspring populations. In this sense, populations are robust in these stochastic settings.
The (1+λ) EA with mutation probability c/n, where 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 λ. It turns out that 1/n is the only optimal mutation probability if λ = o(ln n ln ln n/ln ln ln n), which is the cut-off point for linear speed-up. However, if λ 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 results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds.
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