On the evolution of monotone conjunctions: Drilling for best approximations

Abstract

We study the evolution of monotone conjunctions using local search; the fitness function that guides the search is correlation with Boolean loss. Building on the work of Diochnos and Turán [6], we generalize Valiant’s algorithm [19] for the evolvability of monotone conjunctions from the uniform distribution Un to binomial distributions Bn. With a drilling technique, for a frontier q, we exploit a structure theorem for best q-approximations. We study the algorithm using hypotheses from their natural representation (H = C), as well as when hypotheses contain at most q variables (H = C≤q). Our analysis reveals that Un is a very special case in the analysis of binomial distributions with parameter p, where p ∈ F = {2−1/k| k ∈ ℕ *}. On instances of dimension n, we study approximate learning for (Formula presented) when H = C and for (Formula presented) when H = C≤q. Thus, in either case, approximate learning can be achieved for any 0 < p < 1, for sufficiently large n.