Center-Outward q-Dominance: A Sample-Computable Proxy for Strong Stochastic Dominance in Multi-Objective Optimisation

TL;DR

This paper introduces a new q-dominance relation based on optimal transport theory for ranking multivariate distributions in stochastic multi-objective optimization, providing a reliable alternative to scalarization.

Contribution

It develops a sample-computable proxy for strong stochastic dominance using q-dominance, with an empirical test and sample size threshold, applicable in hyperparameter tuning and multi-objective algorithms.

Findings

Effective ranking of hyperparameter tuners using q-dominance.

Improved convergence rate of NSGA-II with q-dominance.

Validation on benchmark problems demonstrating superiority.

Abstract

Stochastic multi-objective optimization (SMOOP) requires ranking multivariate distributions; yet, most empirical studies perform scalarization, which loses information and is unreliable. Based on the optimal transport theory, we introduce the center-outward q-dominance relation and prove it implies strong first-order stochastic dominance (FSD). Also, we develop an empirical test procedure based on q-dominance, and derive an explicit sample size threshold, $n^*(\delta)$, to control the Type I error. We verify the usefulness of our approach in two scenarios: (1) as a ranking method in hyperparameter tuning; (2) as a selection method in multi-objective optimization algorithms. For the former, we analyze the final stochastic Pareto sets of seven multi-objective hyperparameter tuners on the YAHPO-MO benchmark tasks with q-dominance, which allows us to compare these tuners when the expected hypervolume indicator (HVI, the most common performance metric) of the Pareto sets becomes indistinguishable. For the latter, we replace the mean value-based selection in the NSGA-II algorithm with $q$-dominance, which shows a superior convergence rate on noise-augmented ZDT benchmark problems. These results establish center-outward q-dominance as a principled, tractable foundation for seeking truly stochastically dominant solutions for SMOOPs.