A Tight Runtime Analysis for the cGA on Jump Functions---EDAs Can Cross Fitness Valleys at No Extra Cost
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We prove that the compact genetic algorithm (cGA) with hypothetical population size μ = Ω(√(n) n) ∩poly(n) with high probability finds the optimum of any n-dimensional jump function with jump size k < 1/20 n in O(μ√(n)) iterations. Since it is known that the cGA with high probability needs at least Ω(μ√(n) + n 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 n^1.5 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(n^5+ε), whereas ours gives O(n 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.
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