%0 Conference Paper %1 pmlr-v89-genevay19a %A Genevay, Aude %A Chizat, Lénaïc %A Bach, Francis %A Cuturi, Marco %A Peyré, Gabriel %B Proceedings of Machine Learning Research %D 2019 %E Chaudhuri, Kamalika %E Sugiyama, Masashi %I PMLR %K divergences information readings theory %P 1574--1583 %T Sample Complexity of Sinkhorn Divergences %U http://proceedings.mlr.press/v89/genevay19a.html %V 89 %X 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 Sinkhorn divergences (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength $\varepsilon$, between OT ($\varepsilon=0$) and MMD ($\varepsilon=ınfty$). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively $O(n^3n)$, $O(n^2)$ and $n^2$ operations given a sample size $n$), much less is known in terms of their sample complexity, namely the gap between these quantities, when evaluated using finite samples 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 $1/n$ for MMD, that for SDs has only been studied empirically. In this paper, we (i) derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, (ii) prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and (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 $1/n$ (as in MMD), with a constant that depends however on $\varepsilon$, making the bridge between OT and MMD complete.

@inproceedings{pmlr-v89-genevay19a, 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 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 sample complexity, namely the gap between these quantities, when evaluated using finite samples 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 (i) derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, (ii) prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and (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.}, added-at = {2019-12-11T13:47:25.000+0100}, author = {Genevay, Aude and Chizat, L\'{e}na\"{i}c and Bach, Francis and Cuturi, Marco and Peyr\'{e}, Gabriel}, biburl = {https://www.bibsonomy.org/bibtex/28369f2e483f25cf003f8e70c2360a7b1/kirk86}, booktitle = {Proceedings of Machine Learning Research}, description = {Sample Complexity of Sinkhorn Divergences}, editor = {Chaudhuri, Kamalika and Sugiyama, Masashi}, interhash = {98579e3e62330f895b0c1d84db79e335}, intrahash = {8369f2e483f25cf003f8e70c2360a7b1}, keywords = {divergences information readings theory}, month = {16--18 Apr}, pages = {1574--1583}, pdf = {http://proceedings.mlr.press/v89/genevay19a/genevay19a.pdf}, publisher = {PMLR}, series = {Proceedings of Machine Learning Research}, timestamp = {2019-12-11T13:47:25.000+0100}, title = {Sample Complexity of Sinkhorn Divergences}, url = {http://proceedings.mlr.press/v89/genevay19a.html}, volume = 89, year = 2019 }