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International Conference on Mathematical Optimization Theory and Operations Research

MOTOR 2020: Mathematical Optimization Theory and Operations Research pp 406–423Cite as

A Stable Alternative to Sinkhorn’s Algorithm for Regularized Optimal Transport

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12095))

Abstract

In this paper, we are motivated by two important applications: entropy-regularized optimal transport problem and road or IP traffic demand matrix estimation by entropy model. Both of them include solving a special type of optimization problem with linear equality constraints and objective given as a sum of an entropy regularizer and a linear function. It is known that the state-of-the-art solvers for this problem, which are based on Sinkhorn’s method (also known as RSA or balancing method), can fail to work, when the entropy-regularization parameter is small. We consider the above optimization problem as a particular instance of a general strongly convex optimization problem with linear constraints. We propose a new algorithm to solve this general class of problems. Our approach is based on the transition to the dual problem. First, we introduce a new accelerated gradient method with adaptive choice of gradient’s Lipschitz constant. Then, we apply this method to the dual problem and show, how to reconstruct an approximate solution to the primal problem with provable convergence rate. We prove the rate \(O(1/k^2)\), k being the iteration counter, both for the absolute value of the primal objective residual and constraints infeasibility. Our method has similar to Sinkhorn’s method complexity of each iteration, but is faster and more stable numerically, when the regularization parameter is small. We illustrate the advantage of our method by numerical experiments for the two mentioned applications. We show that there exists a threshold, such that, when the regularization parameter is smaller than this threshold, our method outperforms the Sinkhorn’s method in terms of computation time.

Submitted to the editors DATE. This research was funded by Russian Science Foundation (project 18-71-10108).

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Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Pavel Dvurechensky

  2. Moscow Institute of Physics and Technology, Moscow, Russia

    Alexander Gasnikov & Sergey Omelchenko

  3. Institute for Information Transmission Problems RAS, Moscow, Russia

    Pavel Dvurechensky & Alexander Gasnikov

  4. National Research University Higher School of Economics, Moscow, Russia

    Alexander Tiurin

Authors
  1. Pavel Dvurechensky
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  2. Alexander Gasnikov
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  3. Sergey Omelchenko
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  4. Alexander Tiurin
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Correspondence to Pavel Dvurechensky .

Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Alexander Kononov

  2. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Valery A Kalyagin

  4. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Dvurechensky, P., Gasnikov, A., Omelchenko, S., Tiurin, A. (2020). A Stable Alternative to Sinkhorn’s Algorithm for Regularized Optimal Transport. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_28

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