🤖 AI Summary
This paper investigates the runtime theoretical performance of Evolutionary Diversity Optimization (EDO) on the multi-objective LOTZₖ problem. We analyze the well-known GSEMO algorithm and its diversity-enhanced variant GSEMO$_D$ via stochastic process modeling and expected runtime analysis. Our main contribution is the first rigorous proof that GSEMO$_D$ achieves optimal diversity—under two standard diversity measures—in $O(kn^2log n)$ and $O(k^2n^3log n)$ expected time, respectively—strictly before the Pareto front fully converges. This demonstrates that diversity optimization can precede convergence, challenging the conventional wisdom that diversity must be traded off against convergence. The theoretical bounds are empirically validated with high accuracy. To the best of our knowledge, this work provides the first provably efficient guarantee for simultaneously optimizing solution quality and diversity in multi-objective evolutionary algorithms.
📝 Abstract
The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ$_k$, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn^3)$ expected iterations. We also analyze the runtime of the GSEMO$_D$ (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO$_D$ optimizes it asymptotically faster than it finds a Pareto-optimal population, in $O(kn^2log(n))$ expected iterations, and for the second measure we show an upper bound of $O(k^2n^3log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.