# Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking

arXiv: https://arxiv.org/abs/2406.06425 · HTML: https://mlsanity.com/paper/2406.06425/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 10 Jun 2024.

## Builds on (most cited first, 13 of 13)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Survey on Evaluation of Large Language Models](https://mlsanity.com/paper/2307.03109/) (2023)
- [LLM-Blender: Ensembling Large Language Models with Pairwise Ranking and Generative Fusion](https://mlsanity.com/paper/2306.02561/) (2023)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [TrustGPT: A Benchmark for Trustworthy and Responsible Large Language Models](https://mlsanity.com/paper/2306.11507/) (2023)
- [An optimal transportation approach for assessing almost stochastic order](https://mlsanity.com/paper/1705.01788/) (2017)
- [deep-significance - Easy and Meaningful Statistical Significance Testing in the Age of Neural Networks](https://mlsanity.com/paper/2204.06815/) (2022)
- [Inherent Trade-Offs between Diversity and Stability in Multi-Task Benchmarks](https://mlsanity.com/paper/2405.01719/) (2024)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)

## Built on by (most cited first, 2 of 2)

- [Center-Outward q-Dominance: A Sample-Computable Proxy for Strong Stochastic Dominance in Multi-Objective Optimisation](https://mlsanity.com/paper/2511.12545/) (2025)
- [Fidelity-Diversity Metrics for Text](https://mlsanity.com/paper/2607.04563/) (2026)

## Abstract (arXiv)

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.