Neural Estimation of Entropic Optimal Transport

Optimal transport (OT) serves as a natural frame-work for comparing probability measures, with applications in statistics, machine learning, and applied mathematics. Alas, statistical estimation and exact computation of the OT distances suffer from the curse of dimensionality. To circumvent these issues, entropic regularization has emerged as a remedy that enables parametric estimation rates via plug-in and efficient computation using Sinkhorn iterations. Motivated by further scaling up entropic OT (EOT) to data dimensions and sample sizes that appear in modern machine learning applications, we propose a novel neural estimation approach. Our estimator parametrizes a semi-dual representation of the EOT distance by a neural network, approximates expectations by sample means, and optimizes the resulting empirical objective over parameter space. We establish non-asymptotic error bounds on the EOT neural estimator of the cost and optimal plan. Our bounds characterize the effective error in terms of neural network size and the number of samples, revealing optimal scaling laws that guarantee parametric convergence. The bounds hold for compactly supported distributions, and imply that the proposed estimator is minimax-rate optimal over that class. Numerical experiments validating our theory are also provided.