Improved Runtime Guarantees for the SPEA2 Multi-Objective Optimizer

TL;DR

This paper provides new runtime guarantees for the SPEA2 multi-objective optimizer, showing it can efficiently compute Pareto fronts with less dependence on population size, making parameter tuning easier.

Contribution

It offers the first runtime bounds for SPEA2 that are less dependent on population size, improving understanding of its efficiency and parameter selection.

Findings

SPEA2 has different population dynamics than NSGA-II.

Runtime guarantees depend less on population size for certain benchmarks.

Efficient Pareto front computation with arbitrary population parameters.

Abstract

Together with the NSGA-II, the SPEA2 is one of the most widely used domination-based multi-objective evolutionary algorithms. For both algorithms, the known runtime guarantees are linear in the population size; for the NSGA-II, matching lower bounds exist. With a careful study of the more complex selection mechanism of the SPEA2, we show that it has very different population dynamics. From these, we prove runtime guarantees for the OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks that depend less on the population size. For example, we show that the SPEA2 with parent population size $\mu \ge n - 2k + 3$ and offspring population size $\lambda$ computes the Pareto front of the OneJumpZeroJump benchmark with gap size $k$ in an expected number of $O( (\lambda+\mu)n + n^{k+1})$ function evaluations. This shows that the best runtime guarantee of $O(n^{k+1})$ is not only achieved for $\mu = \Theta(n)$ and $\lambda = O(n)$ but for arbitrary $\mu, \lambda = O(n^k)$. Thus, choosing suitable parameters -- a key challenge in using heuristic algorithms -- is much easier for the SPEA2 than the NSGA-II.