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Debiaser Beware: Pitfalls of Centering Regularized Transport Maps
Aram-Alexandre Pooladian · Marco Cuturi · Jonathan Niles-Weed

Thu Jul 21 03:00 PM -- 05:00 PM (PDT) @ Hall E #731

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 Pn and Qn, 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 Pn and itself, as well as Qn 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.

Author Information

Aram-Alexandre Pooladian (New York University)
Marco Cuturi (Apple and ENSAE/CREST)
Marco Cuturi

Marco is a researcher in machine learning at Apple, working since Jan. 2022 in the Machine Learning Research team led by Samy Bengio. Marco has also been affiliated with the ENSAE / IP Paris school since 2016, working there part-time from 2018. Marco also worked at Google Brain (2018~2022), Kyoto University (2010~2016), Princeton University (2009~2010), the financial industry (2007~2008) and the Institute of Statistical Mathematics (Tokyo, 2006~2007). Marco received his Ph.D. in 2005 from Ecole des Mines de Paris. Marco's research interests cover differentiable optimization, time series, optimal transport theory and its application to ML.

Jonathan Niles-Weed (NYU)

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