Evolutionary Optimization Through PAC Learning

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Abstract

Strategies for evolutionary optimization (EO) typically experience a convergence phenomenon that involves a steady increase in the frequency of particular allelic combinations. When some allele is consistent throughout the population we essentially have a reduction in the dimension of the binary string space that is the objective function domain of feasible solutions. In this paper we consider dimension reduction to be the most salient feature of evolutionary optimization and present the theoretical setting for a novel algorithm that manages this reduction in a very controlled albeit stochastic manner. The ”Rising Tide Algorithm” facilitates dimension reductions through the discovery of bit interdependencies that are expressible as ring-sum-expansions (linear combinations in GF2). When suitable constraints are placed on the objective function these interdependencies are generated by algorithms involving approximations to the discrete Fourier transform (or Walsh Transform). Based on analytic techniques that are now used by researchers in PAC learning, the Rising Tide Algorithm attempts to capitalize on the intrinsic binary nature of the fitness function deriving from it a representation that is highly amenable to a theoretical analysis. Our overall objective is to describe certain algorithms for evolutionary optimization as heuristic techniques that work within the analytically rich environment of PAC learning. We also contend that formulation of these algorithms and empirical demonstrations of their success should give EO practitioners new insights into the current traditional strategies.

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