Debiaser Beware: Pitfalls of Centering Regularized Transport Maps

Aram-Alexandre Pooladian, Marco Cuturi, Jonathan Niles-Weed
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:17830-17847, 2022.

Abstract

Estimating optimal transport (OT) maps (a.k.a. Monge maps) between two measures P and Q is a problem fraught with computational and statistical challenges. A promising approach lies in using the dual potential functions obtained when solving an entropy-regularized OT problem between samples P_n and Q_n, which can be used to recover an approximately optimal map. The negentropy penalization in that scheme introduces, however, an estimation bias that grows with the regularization strength. A well-known remedy to debias such estimates, which has gained wide popularity among practitioners of regularized OT, is to center them, by subtracting auxiliary problems involving P_n and itself, as well as Q_n and itself. We do prove that, under favorable conditions on P and Q, debiasing can yield better approximations to the Monge map. However, and perhaps surprisingly, we present a few cases in which debiasing is provably detrimental in a statistical sense, notably when the regularization strength is large or the number of samples is small. These claims are validated experimentally on synthetic and real datasets, and should reopen the debate on whether debiasing is needed when using entropic OT.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-pooladian22a, title = {Debiaser Beware: Pitfalls of Centering Regularized Transport Maps}, author = {Pooladian, Aram-Alexandre and Cuturi, Marco and Niles-Weed, Jonathan}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {17830--17847}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/pooladian22a/pooladian22a.pdf}, url = {https://proceedings.mlr.press/v162/pooladian22a.html}, abstract = {Estimating optimal transport (OT) maps (a.k.a. Monge maps) between two measures P and Q is a problem fraught with computational and statistical challenges. A promising approach lies in using the dual potential functions obtained when solving an entropy-regularized OT problem between samples P_n and Q_n, which can be used to recover an approximately optimal map. The negentropy penalization in that scheme introduces, however, an estimation bias that grows with the regularization strength. A well-known remedy to debias such estimates, which has gained wide popularity among practitioners of regularized OT, is to center them, by subtracting auxiliary problems involving P_n and itself, as well as Q_n and itself. We do prove that, under favorable conditions on P and Q, debiasing can yield better approximations to the Monge map. However, and perhaps surprisingly, we present a few cases in which debiasing is provably detrimental in a statistical sense, notably when the regularization strength is large or the number of samples is small. These claims are validated experimentally on synthetic and real datasets, and should reopen the debate on whether debiasing is needed when using entropic OT.} }
Endnote
%0 Conference Paper %T Debiaser Beware: Pitfalls of Centering Regularized Transport Maps %A Aram-Alexandre Pooladian %A Marco Cuturi %A Jonathan Niles-Weed %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-pooladian22a %I PMLR %P 17830--17847 %U https://proceedings.mlr.press/v162/pooladian22a.html %V 162 %X Estimating optimal transport (OT) maps (a.k.a. Monge maps) between two measures P and Q is a problem fraught with computational and statistical challenges. A promising approach lies in using the dual potential functions obtained when solving an entropy-regularized OT problem between samples P_n and Q_n, which can be used to recover an approximately optimal map. The negentropy penalization in that scheme introduces, however, an estimation bias that grows with the regularization strength. A well-known remedy to debias such estimates, which has gained wide popularity among practitioners of regularized OT, is to center them, by subtracting auxiliary problems involving P_n and itself, as well as Q_n and itself. We do prove that, under favorable conditions on P and Q, debiasing can yield better approximations to the Monge map. However, and perhaps surprisingly, we present a few cases in which debiasing is provably detrimental in a statistical sense, notably when the regularization strength is large or the number of samples is small. These claims are validated experimentally on synthetic and real datasets, and should reopen the debate on whether debiasing is needed when using entropic OT.
APA
Pooladian, A., Cuturi, M. & Niles-Weed, J.. (2022). Debiaser Beware: Pitfalls of Centering Regularized Transport Maps. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:17830-17847 Available from https://proceedings.mlr.press/v162/pooladian22a.html.

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