On the Evolvability of Monotone Conjunctions with an Evolutionary Mutation Mechanism

Abstract

A Bernoulli(p)ⁿ distribution B_n,p over {0, 1}ⁿ is a product distribution where each variable is satisfied with the same constant probability p. Diochnos (2016) showed that Valiant's swapping algorithm for monotone conjunctions converges efficiently under B_n,p distributions over {0, 1}ⁿ for any 0 < p < 1. We continue the study of monotone conjunctions in Valiant's framework of evolvability. In particular, we prove that given a B_n,p distribution characterized by some p ∈ (0, 1/3] ∪ {1/2}, then an evolutionary mechanism that relies on the basic mutation mechanism of a (1+1) evolutionary algorithm converges efficiently, with high probability, to an ε-optimal hypothesis. Furthermore, for 0 < α ≤ 3/13, a slight modification of the algorithm, with a uniform setup this time, evolves with high probability an ε-optimal hypothesis, for every B_n,p distribution such that p ∈ [α, 1/3 - 4α/9] ∪ {1/3} ∪ {1/2}.