Towards a Theory of Population-Based Incremental Learning

Probabilistic Incremental Learning (PBIL) has been proposed as a model for binary coded evolutionary algorithms where the population is represented by its mean vector which is updated in an autoregressive manner. In this paper we prove that under the PBIL update rule each component of the population vector converges with probability one to 0 (I), whenever a 0 (1) in the associated bit consistently yields better fitness values than the opposite setting. As a corollary, we obtain global convergence of PBIL for linear pseudoboolean functions including the commonly investigated Counting-Ones problem.