General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems
Jiaping Yang, Yunxin Zhang
TL;DR
The paper addresses statistical properties of general divergence-regularized OT (DOT), extending beyond entropic regularization to a broad class of $f$-divergences under a dual-regularity framework. It proves a dimension-free parametric convergence rate of $n^{-1/2}$ for the empirical DOT cost and establishes central limit theorems for both one-sample and two-sample settings, using a dual formulation, bias-variance decomposition, Hoeffding, and Efron-Stein techniques. The results demonstrate that general DOT can avoid the curse of dimensionality in sample complexity and provide rigorous distributional limits, enabling inference tasks such as confidence intervals and hypothesis testing for high-dimensional OT problems. The findings are significant for statistical applications where the cost function is bounded and the regularizer satisfies dual regularity, with extensions to quadratic and other differentiable regularizers under suitable uniqueness conditions. Future work includes relaxing the bounded-cost assumption and analyzing convergence properties of transport plans and potentials beyond the cost itself.
Abstract
Optimal transport has emerged as a fundamental methodology with applications spanning multiple research areas in recent years. However, the convergence rate of the empirical estimator to its population counterpart suffers from the curse of dimensionality, which prevents its application in high-dimensional spaces. While entropic regularization has been proven to effectively mitigate the curse of dimensionality and achieve a parametric convergence rate under mild conditions, these statistical guarantees have not been extended to general regularizers. Our work bridges this gap by establishing analogous results for a broader family of regularizers. Specifically, under boundedness constraints, we prove a convergence rate of order $n^{-1/2} with respect to sample size n. Furthermore, we derive several central limit theorems for divergence regularized optimal transport.