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 [MATH HERE] poly(n) with high probability finds the optimum of any n-dimensional jump function with jump size [MATH HERE] ln n in [MATH HERE] iterations. Since it is known that the cGA with high probability needs at least [MATH HERE] 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 mo derate-sized valleys of low fitness at no extra cost. Our runtime guarantee improves over the recent upper bound O(µn1.5 log n) valid for µ = Ω(n3.5+ε) of Hasenöhrl and Sutton (GECCO 2018). For the best choice of the hypothetical population size, this result gives a runtime guarantee of O(n5+ε), 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.