Improved runtime bounds for the univariate marginal distribution algorithm via anti-concentration

Abstract

Unlike traditional evolutionary algorithms which produce offspring via genetic operators, Estimation of Distribution Algorithms (EDAs) sample solutions from probabilistic models which are learned from selected individuals. It is hoped that EDAs may improve optimisation performance on epistatic fitness landscapes by learning variable interactions. However, hardly any rigorous results are available to support claims about the performance of EDAs, even for fitness functions without epistasis. The expected runtime of the Univariate Marginal Distribution Algorithm (UMDA) on OneMax was recently shown to be in O (nλ log λ) [8]. Later, Krejca and Witt [15] proved the lower bound Ω (λ √ n + n log n) via an involved drift analysis . We prove a O (nλ) bound, given some restrictions on the population size. This implies the tight bound Θ (n log n) when λ = O (log n), matching the runtime of classical EAs. Our analysis uses the levelbased theorem and anti-concentration properties of the Poisson-binomial distribution. We expect that these generic methods will facilitate further analysis of EDAs.

On the limitations of the univariate marginal distribution algorithm to deception and where bivariate EDAs might help

“…While rigorous runtime analyses provide deep insights into the performance of randomised search heuristics, it is highly challenging even for simple algorithms on toy functions. Most current runtime results merely concern univariate EDAs on functions like OneMax [32,51,36,53,40], LeadingOnes [15,22,37,53,38], BinVal [52,37] and Jump [26,11,12], hoping that this provides valuable insights into the development of new techniques for analysing multivariate variants of EDAs and the behaviour of such algorithms on easy parts of more complex problem spaces [13]. There are two main reasons accounted for this.…”

Theory of Estimation-of-Distribution Algorithms

“…The ONEMAX function class is used to analyze how well an EDA performs as a hill climber. The usual expected runtime of an EDA on this function is Θ(n log n) [36,38,63,66].…”

“…Recently, two independent improvements of the bound were presented. The first one due to Lehre and Nguyen [38] builds on a refinement of the level-based analysis, carefully using properties of the Poisson-binomial distribution, and is summarized by the following theorem. We emphasize that UMDA always refers to the algorithm with borders 1/n and 1 − 1/n on the frequencies, i. e., Algorithm 2 extended by a step that narrows all frequencies down to the interval [1/n, 1 − 1/n].…”

“…Also, the theorem includes a limit on µ, which is exactly in the regime of the phase transition. For greater values of µ and λ , Witt [66] independently derived runtime bounds, see the following theorem; it also includes the regime covered by Lehre and Nguyen [38], albeit with an assumption on the ratio λ /µ. Theorem 7 ([66]).…”

“…EDAs have been used very successfully in real-world applications [29,37,54,55] and recently gathered momentum in the theory community analyzing EAs [10,20,22,36,38,63,66]. The aim of theoretically analyzing EAs is to provide guarantees for the algorithms and to gain insights into their behavior in order to optimize the algorithms themselves.…”