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

TL;DR

The paper addresses the challenge of comparing stochastic multi-objective distributions without losing information from scalarization. It introduces center-outward $q$-dominance, a sample-computable proxy for strong first-order stochastic dominance that maps each distribution to a common uniform reference via center-outward quantile maps, and proves that $P_1 \succeq_q P_2$ for all $q \in [0,1)$ implies $P_1 \succeq_1 P_2$. A finite-sample test with explicit $n^*(\delta)$ is derived, along with a practical $q$-dominance sorting algorithm. Empirical validation on multi-objective hyperparameter optimization and noise-augmented NSGA-II demonstrates that $q$-dominance provides stable rankings when traditional metrics blur together and can accelerate convergence under uncertainty. The approach offers a scalable, tunable-free foundation for robust decision-making in stochastic MO problems, grounded in optimal transport and center-outward quantiles.

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, $n^*(δ)$, 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.

Figures (6)

• Figure 1: Samples of a bi-variate distribution (left) and the points on the augmented grid (right). Selected center-outward quantile contours are shown in blue and signs in red.
• Figure 2: Results of YAHPO-MO benchmarks based on $q$-dominance (left) and HVI (right).
• Figure 3: Left: pooled samples from the final stochastic Pareto sets for MIES and ParEGO on lcbench 167152 (marker size $\propto \text{quantile}^{-1}$). Right: histogram of the HVI values for those same Pareto sets.
• Figure 4: The difference in HV ($\Delta HV$) between the deterministic Pareto front and the expected HV of the solutions at a specific budget used. ZDT5 is excluded as it is a Boolean optimization problem.
• Figure 5: Scatter plots of the noise-augmented ZDT benchmarks, evaluated at the deterministic optimum, shown for several noise levels $\sigma$ (standard deviation of the input perturbation).

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3: Weak FSD
• Definition 4: Strong FSD
• Proposition 1
• proof
• Theorem 1
• proof
• Example 1
• Theorem 2