A Tight Runtime Analysis for the cGA on Jump Functions---EDAs Can Cross Fitness Valleys at No Extra Cost

We prove that the compact genetic algorithm (cGA) with hypothetical population size μ = Omega(√ n log n) cap poly(n) with high probability finds the optimum of any n-dimensional jump function with jump size k < frac 1 20 ln n in O(μ √ n) iterations. Since it is known that the cGA with high probability needs at least Omega(μ √ n + n log n) iterations to optimize the unimodal OneMax function, our result shows that the cGA in contrast to most classic evolutionary algorithms here is able to cross moderate-sized valleys of low fitness at no extra cost. Our runtime guarantee improves over the recent upper bound O(μ n1.5 log n) valid for μ = Omega(n3.5+varepsilon) of Hasen"ohrl and Sutton (GECCO 2018). For the best choice of the hypothetical population size, this result gives a runtime guarantee of O(n5+varepsilon), whereas ours gives O(n log n). We also provide a simple general method based on parallel runs that, under mild conditions, (i)~overcomes the need to specify a suitable population size, but gives a performance close to the one stemming from the best-possible population size, and (ii)~transforms EDAs with high-probability performance guarantees into EDAs with similar bounds on the expected runtime.