Analysis and Optimization of Probabilities of Beneficial Mutation and Crossover Recombination in a Hamming Space

TL;DR

This paper analyzes the probabilities of beneficial mutations and crossover recombination in a Hamming space, deriving optimal parameters and highlighting their different roles in evolutionary search processes.

Contribution

It provides closed-form expressions for transition probabilities and optimality conditions for mutation and crossover in a Hamming space, advancing understanding of their roles in evolution.

Findings

Optimal mutation radius decreases near the optimum.

Crossover probabilities are balanced and invariant, complementing mutation.

Mutation slows down near the optimum, while crossover can boost evolution.

Abstract

Inspired by Fisher's geometric approach to study beneficial mutations, we analyse probabilities of beneficial mutation and crossover recombination of strings in a general Hamming space with arbitrary finite alphabet. Mutations and recombinations that reduce the distance to an optimum are considered as beneficial. Geometric and combinatorial analysis is used to derive closed-form expressions for transition probabilities between spheres around an optimum giving a complete description of Markov evolution of distances from an optimum over multiple generations. This paves the way for optimization of parameters of mutation and recombination operators. Here we derive optimality conditions for mutation and recombination radii maximizing the probabilities of mutation and crossover into the optimum. The analysis highlights important differences between these evolutionary operators. While mutation can potentially reach any part of the search space, the probability of beneficial mutation decreases with distance to an optimum, and the optimal mutation radius or rate should also decrease resulting in a slow-down of evolution near the optimum. Crossover recombination, on the other hand, acts in a subspace of the search space defined by the current population of strings. However, probabilities of beneficial and deleterious crossover are balanced, and their characteristics, such as variance, are translation invariant in a Hamming space, suggesting that recombination may complement mutation and boost the rate of evolution near the optimum.