EDAs cannot be Balanced and Stable

Abstract

Estimation of Distribution Algorithms (EDAs) work by it-eratively 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(nlogn).