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

  • Pavel DvurechenskyEmail author
  • Alexander Gasnikov
  • Sergey Omelchenko
  • Alexander Tiurin
Conference paper
  • 219 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Smooth convex optimization Linear constraints First-order methods Accelerated gradient descent Algorithm complexity Entropy-linear programming Dual problem Primal-dual method Sinkhorn’s fixed point algorithm Entropy-regularized optimal transport Traffic demand matrix estimation 

Supplementary material

495544_1_En_28_MOESM1_ESM.pdf (149 kb)
Supplementary material 1 (pdf 149 KB)

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Moscow Institute of Physics and TechnologyMoscowRussia
  3. 3.Institute for Information Transmission Problems RASMoscowRussia
  4. 4.National Research University Higher School of EconomicsMoscowRussia

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