Rethinking Initialization of the Sinkhorn Algorithm

James Thornton, Marco Cuturi
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:8682-8698, 2023.

Abstract

While the optimal transport (OT) problem was originally formulated as a linear program, the addition of entropic regularization has proven beneficial both computationally and statistically, for many applications. The Sinkhorn fixed-point algorithm is the most popular approach to solve this regularized problem, and, as a result, multiple attempts have been made to reduce its runtime using, e.g., annealing in the regularization parameter, momentum or acceleration. The premise of this work is that initialization of the Sinkhorn algorithm has received comparatively little attention, possibly due to two preconceptions: since the regularized OT problem is convex, it may not be worth crafting a good initialization, since any is guaranteed to work; secondly, because the outputs of the Sinkhorn algorithm are often unrolled in end-to-end pipelines, a data-dependent initialization would bias Jacobian computations. We challenge this conventional wisdom, and show that data-dependent initializers result in dramatic speed-ups, with no effect on differentiability as long as implicit differentiation is used. Our initializations rely on closed-forms for exact or approximate OT solutions that are known in the 1D, Gaussian or GMM settings. They can be used with minimal tuning, and result in consistent speed-ups for a wide variety of OT problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-thornton23a, title = {Rethinking Initialization of the Sinkhorn Algorithm}, author = {Thornton, James and Cuturi, Marco}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {8682--8698}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/thornton23a/thornton23a.pdf}, url = {https://proceedings.mlr.press/v206/thornton23a.html}, abstract = {While the optimal transport (OT) problem was originally formulated as a linear program, the addition of entropic regularization has proven beneficial both computationally and statistically, for many applications. The Sinkhorn fixed-point algorithm is the most popular approach to solve this regularized problem, and, as a result, multiple attempts have been made to reduce its runtime using, e.g., annealing in the regularization parameter, momentum or acceleration. The premise of this work is that initialization of the Sinkhorn algorithm has received comparatively little attention, possibly due to two preconceptions: since the regularized OT problem is convex, it may not be worth crafting a good initialization, since any is guaranteed to work; secondly, because the outputs of the Sinkhorn algorithm are often unrolled in end-to-end pipelines, a data-dependent initialization would bias Jacobian computations. We challenge this conventional wisdom, and show that data-dependent initializers result in dramatic speed-ups, with no effect on differentiability as long as implicit differentiation is used. Our initializations rely on closed-forms for exact or approximate OT solutions that are known in the 1D, Gaussian or GMM settings. They can be used with minimal tuning, and result in consistent speed-ups for a wide variety of OT problems.} }
Endnote
%0 Conference Paper %T Rethinking Initialization of the Sinkhorn Algorithm %A James Thornton %A Marco Cuturi %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-thornton23a %I PMLR %P 8682--8698 %U https://proceedings.mlr.press/v206/thornton23a.html %V 206 %X While the optimal transport (OT) problem was originally formulated as a linear program, the addition of entropic regularization has proven beneficial both computationally and statistically, for many applications. The Sinkhorn fixed-point algorithm is the most popular approach to solve this regularized problem, and, as a result, multiple attempts have been made to reduce its runtime using, e.g., annealing in the regularization parameter, momentum or acceleration. The premise of this work is that initialization of the Sinkhorn algorithm has received comparatively little attention, possibly due to two preconceptions: since the regularized OT problem is convex, it may not be worth crafting a good initialization, since any is guaranteed to work; secondly, because the outputs of the Sinkhorn algorithm are often unrolled in end-to-end pipelines, a data-dependent initialization would bias Jacobian computations. We challenge this conventional wisdom, and show that data-dependent initializers result in dramatic speed-ups, with no effect on differentiability as long as implicit differentiation is used. Our initializations rely on closed-forms for exact or approximate OT solutions that are known in the 1D, Gaussian or GMM settings. They can be used with minimal tuning, and result in consistent speed-ups for a wide variety of OT problems.
APA
Thornton, J. & Cuturi, M.. (2023). Rethinking Initialization of the Sinkhorn Algorithm. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:8682-8698 Available from https://proceedings.mlr.press/v206/thornton23a.html.

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