🤖 AI Summary
This work addresses the challenge of efficiently approximating the Pareto front in multi-objective optimization (MOO). We propose a novel set-based optimization framework grounded in the R2 utility function, which reformulates MOO as a single-objective optimization problem over solution sets. We theoretically establish that the R2 utility is both monotonic and submodular—properties enabling a greedy algorithm with a guaranteed (1−1/e) approximation ratio. Integrating this with Bayesian optimization, our approach achieves efficient, high-coverage approximation of the Pareto front. Methodologically, it unifies scalarization, submodular optimization, and Bayesian optimization within a coherent framework. Empirical evaluation on multi-objective Bayesian optimization benchmarks demonstrates significant improvements in convergence speed and Pareto front quality, validating both its practical efficacy and theoretical advantages.
📝 Abstract
The goal of multi-objective optimisation is to identify a collection of points which describe the best possible trade-offs between the multiple objectives. In order to solve this vector-valued optimisation problem, practitioners often appeal to the use of scalarisation functions in order to transform the multi-objective problem into a collection of single-objective problems. This set of scalarised problems can then be solved using traditional single-objective optimisation techniques. In this work, we formalise this convention into a general mathematical framework. We show how this strategy effectively recasts the original multi-objective optimisation problem into a single-objective optimisation problem defined over sets. An appropriate class of objective functions for this new problem are the R2 utilities, which are utility functions that are defined as a weighted integral over the scalarised optimisation problems. As part of our work, we show that these utilities are monotone and submodular set functions which can be optimised effectively using greedy optimisation algorithms. We then analyse the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focusses on Bayesian optimisation, which is a popular probabilistic framework for black-box optimisation.