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 ${\mathcal U}_n$ to binomial distributions ${\mathcal B}_n$.

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 ($\mathcal H = \mathcal C$), as well as when hypotheses contain at most q variables ($\mathcal H = \mathcal C _{\le q}$). Our analysis reveals that ${\mathcal U}_n$ is a very special case in the analysis of binomial distributions with parameter p, where $p\in \mathcal {F} = \{2^{-1/k} \ | \ k\in \mathbb N ^*\}$. On instances of dimension n, we study approximate learning for $0< p < 2^{-\frac{1}{n-1}}$ when $\mathcal H = \mathcal C$ and for $0< p < \root n-1 \of {2/3}$ when $\mathcal H = \mathcal C _{\le q}$. Thus, in either case, approximate learning can be achieved for any $0< p < 1$, for sufficiently large n.