On the Private Estimation of Smooth Transport Maps

TL;DR

This paper develops a differentially private estimator for smooth optimal transport maps, achieving near-optimal error rates depending on sample size, privacy level, smoothness, and dimension, with a proven lower bound.

Contribution

It introduces a novel differentially private estimator for Brenier potentials in optimal transport, with theoretical error bounds and a matching lower bound for the problem.

Findings

Achieves $L^2$ error bounds depending on sample size, privacy, smoothness, and dimension.

Provides a lower bound demonstrating near-optimality of the proposed estimator.

Extends optimal transport estimation to privacy-preserving settings.

Abstract

Estimating optimal transport maps between two distributions from respective samples is an important element for many machine learning methods. To do so, rather than extending discrete transport maps, it has been shown that estimating the Brenier potential of the transport problem and obtaining a transport map through its gradient is near minimax optimal for smooth problems. In this paper, we investigate the private estimation of such potentials and transport maps with respect to the distribution samples.We propose a differentially private transport map estimator achieving an $L^2$ error of at most $n^{-1} \vee n^{-\frac{2 \alpha}{2 \alpha - 2 + d}} \vee (n\epsilon)^{-\frac{2 \alpha}{2 \alpha + d}} $ up to poly-logarithmic terms where $n$ is the sample size, $\epsilon$ is the desired level of privacy, $\alpha$ is the smoothness of the true transport map, and $d$ is the dimension of the feature space. We also provide a lower bound for the problem.