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 < 1 20 ln n in O(µ √ n) iterations. Since it is known that the cGA with high probability needs at least Ω(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(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(n 5+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.
We refer to Section 2.3 of REF for a detailed discussion of the relation of expectations and tail bounds for runtimes of EDAs, including a method to transform EDAs with with-high-probability guarantees into those with guarantees on the expected runtime).
A Tight Runtime Analysis for the cGA on Jump Functions---EDAs Can Cross Fitness Valleys at No Extra Cost
GECCO '19
Computer Science