Weak Evolvability Equals Strong Evolvability

This paper provides an answer to the open problem raised by Feldman and Valiant at COLT'08. That is, it is known that concepts classes of conjunctions, disjunctions, linear threshold functions and $k$-CNF/DNF are evolvable over fixed distributions, but it is unknown whether they are evolvable distribution-independently. Or in other words, whether \textit{weak evolvability} is equivalent to \textit{strong evolvability}. The \textit{evolvability} was defined by Valiant in a way similar to the learnability, which was proposed to understand the mysteries underlying natural evolution. In this paper, the equivalence is proved by constructing a boosting algorithm which uses correlational statistical queries (CSQs) only. It is known that evolvability is equivalent to CSQ learnability, and therefore our construction connects weak evolvability with strong evolvability. This result leads to a complete characterization of the learning power of evolution.