Runtime Analysis of Quality Diversity Algorithms
Abstract
Quality diversity~(QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the ``number of ones'' feature space, where the th cell stores the best solution amongst those with a number of ones in . Here is a granularity parameter . We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all and analyse the expected optimisation time of QD on \textsc{OneMax} and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a -approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of a connected graph, we show that QD finds a minimum spanning tree in expected polynomial time.
Cite
@article{arxiv.2305.18966,
title = {Runtime Analysis of Quality Diversity Algorithms},
author = {Jakob Bossek and Dirk Sudholt},
journal= {arXiv preprint arXiv:2305.18966},
year = {2023}
}
Comments
This version fixes a bug in Theorem 6.1 of the paper published at GECCO 2023. It claims an upper bound of $O(nm \log n)$ for the expected time to locate the empty edge set, but the proof arguments were insufficient (thanks to Frank Neumann for spotting this). We now give a bound of $O(nm \log(n w_{\max}))$ for integer edge weights where $w_{\max}$ is the largest edge weight