The limitations of distribution sampling for linkage learning

This paper investigates the performance of estimation of distribution algorithms (EDAs) over binary test problems containing parity functions. We describe two test problems; the concatenated parity function (CPF), and the concatenated parity/trap function (CP/TF). Although these functions are separable, with bounded complexity and uniformly scaled sub-function contributions, the hierarchical Bayesian optimization algorithm (hBOA) scales exponentially on both. hBOA is able to solve large CPFs with small population sizes when it is unable to solve them with larger population sizes. We argue that test problems containing parity functions are hard for EDAs because there are no interactions in the contribution to fitness between any strict subset of a parity function´s bits. This means that as population sizes increase the dependency between variable values for any strict subset of a parity function´s bits decreases. Unfortunately most EDAs including hBOA search for their models by looking for dependencies between pairs of variables (at least at first). We make suggestions on how EDAs could be adjusted to handle parity problems, but also comment on the apparently inevitable computational cost.