Sample Complexity of Sinkhorn divergences

TL;DR

This paper investigates the sample complexity of Sinkhorn divergences, a regularized optimal transport measure, providing bounds on approximation error, optimizer boundedness, and the first sample complexity rate, bridging OT and MMD.

Contribution

It derives a new sample complexity bound for Sinkhorn divergences, connecting OT and MMD, and analyzes their approximation error and optimizer properties.

Findings

Sample complexity of SDs scales as 1/√n, similar to MMD.

Bound on approximation error of SDs relative to OT depending on regularization.

Optimizers of regularized OT are bounded in a Sobolev (RKHS) ball, independent of measures.

Abstract

Optimal transport (OT) and maximum mean discrepancies (MMD) are now routinely used in machine learning to compare probability measures. We focus in this paper on \emph{Sinkhorn divergences} (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength $\varepsilon$, between OT ($\varepsilon=0$) and MMD ($\varepsilon=\infty$). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively $O(n^3\log n)$, $O(n^2)$ and $n^2$ operations given a sample size $n$), much less is known in terms of their \emph{sample complexity}, namely the gap between these quantities, when evaluated using finite samples \emph{vs.} their respective densities. Indeed, while the sample complexity of OT and MMD stand at two extremes, $1/n^{1/d}$ for OT in dimension $d$ and $1/\sqrt{n}$ for MMD, that for SDs has only been studied empirically. In this paper, we \emph{(i)} derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, \emph{(ii)} prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and \emph{(iii)} provide the first sample complexity bound for SDs, obtained,by reformulating SDs as a maximization problem in a RKHS. We thus obtain a scaling in $1/\sqrt{n}$ (as in MMD), with a constant that depends however on $\varepsilon$, making the bridge between OT and MMD complete.