On the Private Estimation of Smooth Transport Maps
Clément Lalanne, Franck Iutzeler, Jean-Michel Loubes, Julien Chhor
TL;DR
This paper addresses the problem of privately estimating smooth optimal transport maps between distributions from samples by leveraging Brenier potentials and their gradients. It develops a differentially private estimator built on the empirical semi-dual and a wavelet-based subspace approximation, yielding an $L^2$ error rate of $n^{-1} \vee n^{- rac{2\alpha}{2\alpha-2+d}} \vee (n\epsilon)^{- rac{2\alpha}{2\alpha+d}}$ (up to polylog terms) and provides a matching (up to constants) lower bound under DP. A practical discretization approach is proposed to implement the estimator, and experiments on a Gaussian attraction/repulsion model illustrate the method's behavior under privacy constraints. The work highlights the privacy-utility trade-off in private OT estimation and opens avenues for further improvements via regularization and scalable algorithms.
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 α}{2 α- 2 + d}} \vee (nε)^{-\frac{2 α}{2 α+ d}} $ up to poly-logarithmic terms where $n$ is the sample size, $ε$ is the desired level of privacy, $α$ 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.