Illuminating the Diversity-Fitness Trade-Off in Black-Box Optimization

TL;DR

This work reframes black-box optimization as selecting a batch of $k$ solutions with $d_{ ext{min}}$-level diversity while maximizing average fitness, rather than returning a single optimum. It investigates the utility of off-the-shelf trajectories (random, Sobol', CMA-ES, MMO methods) by extracting high-quality diverse batches via a greedy subset selection from a portfolio of $T$ evaluated points, benchmarking on the 24 BBOB functions across $D=2,5,10$. Key findings show that a naive uniform random baseline is surprisingly strong, that greedy extraction closely approximates optimal subsets, and that MMO and CMA-ES offer benefits mainly at small $d_{ ext{min}}$ while random/Sobol can outperform them as diversity constraints tighten; these results motivate developing algorithms that actively enforce diversity without sacrificing quality. The study provides a diagnostic framework for understanding the trade-off between diversity and fitness and highlights the need for diversity-aware optimization methods in real-world design problems where post-hoc qualitative criteria matter. Overall, the work underlines the practical value of generating diverse, high-quality solution batches and points to future directions for integrating diversity mechanisms into global optimizers.

Abstract

In real-world applications, users often favor structurally diverse design choices over one high-quality solution. It is hence important to consider more solutions that decision makers can compare and further explore based on additional criteria. Alongside the existing approaches of evolutionary diversity optimization, quality diversity, and multimodal optimization, this paper presents a fresh perspective on this challenge by considering the problem of identifying a fixed number of solutions with a pairwise distance above a specified threshold while maximizing their average quality. We obtain first insight into these objectives by performing a subset selection on the search trajectories of different well-established search heuristics, whether they have been specifically designed with diversity in mind or not. We emphasize that the main goal of our work is not to present a new algorithm but to understand the capability of off-the-shelf algorithms to quantify the trade-off between the minimum pairwise distance within batches of solutions and their average quality. We also analyze how this trade-off depends on the properties of the underlying optimization problem. A possibly surprising outcome of our empirical study is the observation that naive uniform random sampling establishes a very strong baseline for our problem, hardly ever outperformed by the search trajectories of the considered heuristics. We interpret these results as a motivation to develop algorithms tailored to produce diverse solutions of high average quality.

Figures (12)

• Figure 1: $D = 2$. Isocontours of the 24 BBOB functions with $k = 5$ Gurobi solutions, obtained from an initial set of $T = 10\,000$ points sampled uniformly at random. Showing results for $d_{\text{min}}$ = 1, 3, 5 and the batch of solutions with no distance constraint. Lighter colors correspond to worse solutions, the optima are hence in regions surrounded by dark lines.
• Figure 2: $D = 2$. Pairwise distances between the $k = 5$ points selected by Gurobi for the 24 BBOB functions, initiated from random samples of size $T = 10\,000$.
• Figure 3: One run of the greedy approach, applied to an initial set of $T = 10\,000$ points sampled uniformly at random and evaluated on the 24 BBOB functions in $D=2$ (a) and $D=10$ (b). Results are for batches of $k=5$ solutions. The red dots show the results obtained with Gurobi, applied to the same initial sets and different distances, $d_{\text{min}}$ = 1, 3, 5 (a) and $d_{\text{min}}$ = 6, 10, 12 (b) (distance 6 is included where feasible).
• Figure 4: $D = 2.$ Impact of the size $T$ of the initial set on the distance-quality trade-off. Results are for the 24 BBOB functions, using a batch size of $k=5$, and using initial sets that are sampled uniformly at random. 10 and 20 independent repetitions for $T = 10\,000$ and $T = 100\,000$, respectively, and one run for an initial set of $T = 1\,000\,000$ points.
• Figure 5: Evolution of the loss, plotted against minimum distance in the optimal batch for the 24 BBOB functions for $D=2$. Curves are generated by extracting the lower envelope from 10 independent runs for $k = 2, 7$ and 20 runs for $k = 5, 10$ of the greedy approach, initiated from a random sample of size $T = 100\,000$.