Estimating Quality in Multi-Objective Bandits Optimization

TL;DR

This paper introduces the concept of preference radius to assess the robustness of preference functions in multi-objective bandit optimization, providing theoretical analysis and experimental validation for decision-making accuracy.

Contribution

It formalizes the preference radius concept, linking it to the optimal gap, and analyzes Thompson sampling's effectiveness in multi-objective bandit problems.

Findings

Preference radius characterizes robustness of preference functions.

Thompson sampling performance depends on the preference radius.

Scalarizing multi-objective problems can be misleading.

Abstract

Many real-world applications are characterized by a number of conflicting performance measures. As optimizing in a multi-objective setting leads to a set of non-dominated solutions, a preference function is required for selecting the solution with the appropriate trade-off between the objectives. The question is: how good do estimations of these objectives have to be in order for the solution maximizing the preference function to remain unchanged? In this paper, we introduce the concept of preference radius to characterize the robustness of the preference function and provide guidelines for controlling the quality of estimations in the multi-objective setting. More specifically, we provide a general formulation of multi-objective optimization under the bandits setting. We show how the preference radius relates to the optimal gap and we use this concept to provide a theoretical analysis of the Thompson sampling algorithm from multivariate normal priors. We finally present experiments to support the theoretical results and highlight the fact that one cannot simply scalarize multi-objective problems into single-objective problems.