Detecting the epistatic structure of generalized embedded landscape

Abstract

Working under the premise that most optimizable functions are of bounded epistasis, this paper addresses the problem of discovering the linkage structure of a black-box function with a domain of arbitrary-cardinality under the assumption of bounded epistasis. To model functions of bounded epistasis, we develop a generalization of the mathematical model of "embedded landscapes" over domains of cardinality M. We then generalize the Walsh transform as a discrete Fourier transform, and develop algorithms for linkage learning of epistatically bounded GELs. We propose Generalized Embedding Theorem that models the relationship between the underlying decomposable structure of GEL and its Fourier coefficients. We give a deterministic algorithm to exactly calculate the Fourier coefficients of GEL with bounded epistasis. Complexity analysis shows that the epistatic structure of epistatically bounded GEL can be obtained after a polynomial number of function evaluations. Finally, an example experiment of the algorithm is presented.