General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems

Abstract

We study empirical divergence-regularized optimal transport on arbitrary Polish spaces. For bounded continuous costs and dual-regular conjugates $\psi \in C^1(\mathbb{R})$, we establish a dimension-free $n^{-1/2}$ bound for the empirical transport value using dual interpolation, Hoeffding projection, and bounded-difference arguments. For the asymptotic theory, we isolate the population identifiability condition needed for stability and give sufficient conditions for population uniqueness. In particular, active-set identifiability implies uniqueness of the canonical population potentials, and connectedness of either population support is sufficient. This condition requires neither compactness nor Euclidean structure and covers quadratic regularization, whose conjugate is $C^1$ but not $C^2$. We also give finite-support and large-regularization criteria. Under a bounded Lipschitz cost, the empirical potentials are stable, the transport value is asymptotically linear, and one- and two-sample central limit theorems hold with population centering. This includes noncompact and infinite-dimensional Polish state spaces.