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 = Ω(n log n) ∩ poly(n) with high probability finds the optimum of any n-dimensional jump function with jump size k < ₂₀¹ ln n in O(µn) iterations. Since it is known that the cGA with high probability needs at least Ω(µn + 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(µn^1.5 log n) valid for µ = Ω(n^3.5+^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.