@inproceedings{
rioux2024multivariate,
title={Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking},
author={Gabriel Rioux and Apoorva Nitsure and Mattia Rigotti and Kristjan Greenewald and Youssef Mroueh},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=NCX3Kgb1nh}
}
Stochastic dominance is an important concept in probability theory, econometrics and social choice theory for robustly modeling agents' preferences between random outcomes.  While many works have been dedicated to the univariate case,
little has been done in the multivariate scenario, wherein an agent has to decide between different multivariate outcomes. By exploiting a  characterization  of  multivariate first stochastic dominance in terms of couplings, we  introduce  a statistic that assesses multivariate almost stochastic dominance under the framework of Optimal Transport  with a smooth cost. Further, we introduce an entropic regularization of this statistic, and establish a central limit theorem (CLT) and consistency of the bootstrap procedure for the empirical statistic. Armed with this CLT, we propose a hypothesis testing framework as well as an efficient implementation using the Sinkhorn algorithm. We showcase our  method in comparing and benchmarking Large Language Models that are evaluated on multiple metrics. Our multivariate stochastic dominance test allows us to capture the dependencies between the metrics in order to make an informed and statistically significant  decision on the relative performance of the models.
Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking
Gabriel Rioux Apoorva Nitsure Mattia Rigotti Kristjan Greenewald Youssef Mroueh