Multi-objective optimisation via the R2 utilities

TL;DR

This paper introduces a unified mathematical framework for multi-objective optimisation using R2 utilities, demonstrating their properties and effective greedy algorithms within Bayesian optimisation.

Contribution

It formalizes the use of R2 utilities as a set function for multi-objective problems and analyzes their optimization via greedy algorithms both theoretically and empirically.

Findings

R2 utilities are monotone and submodular functions.

Greedy algorithms effectively optimize R2 utilities.

Empirical results show improved trade-off identification in Bayesian optimisation.

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.