004 Datenverarbeitung; Informatik
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A simplified run time analysis of the univariate marginal distribution algorithm on LeadingOnes
(2021)
With elementary means, we prove a stronger run time guarantee for the univariate marginal distribution algorithm (UMDA) optimizing the LEADINGONES benchmark function in the desirable regime with low genetic drift. If the population size is at least quasilinear, then, with high probability, the UMDA samples the optimum in a number of iterations that is linear in the problem size divided by the logarithm of the UMDA's selection rate. This improves over the previous guarantee, obtained by Dang and Lehre (2015) via the deep level-based population method, both in terms of the run time and by demonstrating further run time gains from small selection rates. Under similar assumptions, we prove a lower bound that matches our upper bound up to constant factors.
Multiplicative Up-Drift
(2020)
Drift analysis aims at translating the expected progress of an evolutionary algorithm (or more generally, a random process) into a probabilistic guarantee on its run time (hitting time). So far, drift arguments have been successfully employed in the rigorous analysis of evolutionary algorithms, however, only for the situation that the progress is constant or becomes weaker when approaching the target. Motivated by questions like how fast fit individuals take over a population, we analyze random processes exhibiting a (1+delta)-multiplicative growth in expectation. We prove a drift theorem translating this expected progress into a hitting time. This drift theorem gives a simple and insightful proof of the level-based theorem first proposed by Lehre (2011). Our version of this theorem has, for the first time, the best-possible near-linear dependence on 1/delta} (the previous results had an at least near-quadratic dependence), and it only requires a population size near-linear in delta (this was super-quadratic in previous results). These improvements immediately lead to stronger run time guarantees for a number of applications. We also discuss the case of large delta and show stronger results for this setting.
We contribute to the theoretical understanding of randomized search heuristics by investigating their optimization behavior on satisfiable random k-satisfiability instances both in the planted solution model and the uniform model conditional on satisfiability. Denoting the number of variables by n, our main technical result is that the simple () evolutionary algorithm with high probability finds a satisfying assignment in time when the clause-variable density is at least logarithmic. For low density instances, evolutionary algorithms seem to be less effective, and all we can show is a subexponential upper bound on the runtime for densities below . We complement these mathematical results with numerical experiments on a broader density spectrum. They indicate that, indeed, the () EA is less efficient on lower densities. Our experiments also suggest that the implicit constants hidden in our main runtime guarantee are low. Our main result extends and considerably improves the result obtained by Sutton and Neumann (Lect Notes Comput Sci 8672:942-951, 2014) in terms of runtime, minimum density, and clause length. These improvements are made possible by establishing a close fitness-distance correlation in certain parts of the search space. This approach might be of independent interest and could be useful for other average-case analyses of randomized search heuristics. While the notion of a fitness-distance correlation has been around for a long time, to the best of our knowledge, this is the first time that fitness-distance correlation is explicitly used to rigorously prove a performance statement for an evolutionary algorithm.