Escaping Local Optima using Crossover with Emergent or Reinforced Diversity

TL;DR

This paper uses rigorous analysis to show how crossover and diversity mechanisms in genetic algorithms can significantly improve optimization times on complex functions, surpassing mutation-only methods.

Contribution

It provides the first comprehensive runtime analysis of diversity and crossover effects in GAs with constant crossover probability, comparing multiple diversity mechanisms.

Findings

Crossover and mutation can induce a sudden burst of diversity, speeding up optimization.

Diversity mechanisms in the ($0$+1) GA outperform mutation-only algorithms like (1+1) EA.

Expected runtime improvements include 0(n/1) and 0(n) for certain configurations.

Abstract

Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the ($\mu$+1) GA and the $\text{Jump}_k$ test function. We show that the interplay of crossover and mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order $\Omega(n/\log n)$ compared to mutation-only algorithms like (1+1) EA. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to speedups of order $\Omega(n)$. We also compare seven commonly used diversity mechanisms and evaluate their impact on runtime bounds for the ($\mu$+1) GA. All previous results in this context only hold for unrealistically low crossover probability $p_c=O(k/n)$, while we give analyses for the setting of constant $p_c < 1$ in all but one case. For the typical case of constant $k > 2$ and constant $p_c$, we can compare the resulting expected runtimes for different diversity mechanisms assuming an optimal choice of $\mu$: $O(n^{k-1})$ for duplicate elimination/minim., $O(n^2\log n)$ for maximising the convex hull, $O(n\log n)$ for deterministic crowding (assuming $p_c = k/n$), $O(n\log n)$ for maximising Hamming distance, $O(n\log n)$ for fitness sharing, $O(n\log n)$ for single-receiver island model. This proves a sizeable advantage of all variants of the ($\mu$+1) GA compared to (1+1) EA, which requires time $\Theta(n^k)$. Experiments complement our theoretical findings and further highlight the benefits of crossover and diversity on $\text{Jump}_k$.