A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning

Abstract

We provide a central limit theorem for the Monge–Kantorovich distance between two empirical distributions with sizes $n$ and $m$, $\mathcal{W}_p(P_n,Q_m), \ p\geqslant 1,$ for observations on the real line. In the case $p>1$ our assumptions are sharp in terms of moments and smoothness. We prove results dealing with the choice of centring constants. We provide a consistent estimate of the asymptotic variance, which enables to build two sample tests and confidence intervals to certify the similarity between two distributions. These are then used to assess a new criterion of data set fairness in classification.

Central limit theorems for empirical transportation cost in general dimension

“…We refer to [19] for a more detailed account about the history of the problem. The problem has received a renewed interest in the last few years, both in the setup P = Q (see [3,25,39]) or for general P and Q (see [36] and [41] for finitely and countably supported probabilities, [19] for the case p = 2 and general probabilities and dimension and [18,6] for dimension d = 1 and general costs).…”

Fair Learning: an optimal transport based approach

“…These have then been used to assess a new criterion of dataset fairness in classification. These contributions correspond to the publication del Barrio et al [2019b]. Additionally, we have provided a moderate deviation principle for the empirical transportation cost in general dimension.…”

“…Yet all other fairness criteria should be given with the calculation of a confidence interval. For instance in del Barrio et al [2019b] we propose confidence intervals for Wasserstein distance which is used in many methods in fair learning.…”

“…This chapter corresponds to the publication del Barrio et al [2019b]. We provide a Central Limit Theorem for the Monge-Kantorovich distance between two empirical distributions with sizes n and m, W p (P n , Q m ), p ≥ 1, for observations on the real line.…”

“…Note that → w denotes weak convergence in probabilities. Extensions of (6.1.2) to general distances W p , p ≥ 1, in the one-dimensional case, as well as results dealing with the choice of centering constants were provided in del Barrio et al [2019b]. The proposed approach in del is based on the analysis of the optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem.…”