Revisiting Norm Optimization for Multi-Objective Black-Box Problems: A Finite-Time Analysis

TL;DR

This paper provides a finite-time analysis of the Tchebycheff weighted method for multi-objective black-box problems, establishing a link between weighted sum approaches and quality indicators with theoretical guarantees.

Contribution

It introduces a finite-time bound on the Pareto-compliant additive epsilon-indicator for the Tchebycheff weighted method within a hierarchical bandits framework, connecting weighted sum methods with quality indicators.

Findings

Established a finite-time bound on the epsilon-indicator.

Linked weighted sum methods with quality indicators in finite time.

Provided theoretical guarantees for Pareto front approximation.

Abstract

The complexity of Pareto fronts imposes a great challenge on the convergence analysis of multi-objective optimization methods. While most theoretical convergence studies have addressed finite-set and/or discrete problems, others have provided probabilistic guarantees, assumed a total order on the solutions, or studied their asymptotic behaviour. In this paper, we revisit the Tchebycheff weighted method in a hierarchical bandits setting and provide a finite-time bound on the Pareto-compliant additive $\epsilon$-indicator. To the best of our knowledge, this paper is one of few that establish a link between weighted sum methods and quality indicators in finite time.