Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the

Estimation of Distribution Algorithms (EDAs) work by iteratively updating a distribution over the search space with the help of samples from each iteration. Up to now, theoretical analyses of EDAs are scarce and present run time results for specific EDAs. We propose a new framework for EDAs that captures the idea of several known optimizers, including PBIL, UMDA, λ-MMASIB, cGA, and (1, λ)-EA. Our focus is on analyzing two core features of EDAs: a balanced EDA is sensitive to signals in the fitness; a stable EDA remains uncommitted under a biasless fitness function. We prove that no EDA can be both balanced and stable. The LeadingOnes function is a prime example where, at the beginning of the optimization, the fitness function shows no bias for many bits. Since many well-known EDAs are balanced and thus not stable, they are not well-suited to optimize LeadingOnes. We give a stable EDA which optimizes LeadingOnes within a time of O(n log n).

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Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that maintain a probabilistic model of the solution space. This model is updated from iteration to iteration, based on the quality of the solutions sampled according to the model. As previous works show, this short-term perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. Such frequencies take long to be moved back to the middle range, leading to significant performance losses.In order to overcome this problem, we propose a new EDA based on the classic compact genetic algorithm (cGA) that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based compact genetic algorithm (sig-cGA) optimizes the commonly regarded benchmark functions OneMax, LeadingOnes, and BinVal all in O(n log n) time, a result shown for no other EDA or evolutionary algorithm so far.For the recently proposed scGA -an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model -we prove that it optimizes OneMax only in a time exponential in the hypothetical population size 1/ρ. Similarly, we show that the convex search algorithm cannot optimize OneMax in polynomial time.

The benefit of sexual recombination is one of the most fundamental questions both in population genetics and evolutionary computation. It is widely believed that recombination helps solving difficult optimization problems. We present the first result, which rigorously proves that it is beneficial to use sexual recombination in an uncertain environment with a noisy fitness function. For this, we model sexual recombination with a simple estimation of distribution algorithm called the Compact Genetic Algorithm (cGA), which we compare with the classical µ + 1 EA. For a simple noisy fitness function with additive Gaussian posterior noise N (0, σ 2 ), we prove that the mutation-only µ + 1 EA typically cannot handle noise in polynomial time for σ 2 large enough while the cGA runs in polynomial time as long as the population size is not too small. This shows that in this uncertain environment sexual recombination is provably beneficial. We observe the same behavior in a small empirical study.