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Mathematics > Optimization and Control

arXiv:2109.15030 (math)
[Submitted on 29 Sep 2021 (v1), last revised 1 Oct 2021 (this version, v2)]

Title:On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport

Authors:Minhui Huang, Shiqian Ma, Lifeng Lai
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Abstract:This paper studies the equitable and optimal transport (EOT) problem, which has many applications such as fair division problems and optimal transport with multiple agents etc. In the discrete distributions case, the EOT problem can be formulated as a linear program (LP). Since this LP is prohibitively large for general LP solvers, Scetbon \etal \cite{scetbon2021equitable} suggests to perturb the problem by adding an entropy regularization. They proposed a projected alternating maximization algorithm (PAM) to solve the dual of the entropy regularized EOT. In this paper, we provide the first convergence analysis of PAM. A novel rounding procedure is proposed to help construct the primal solution for the original EOT problem. We also propose a variant of PAM by incorporating the extrapolation technique that can numerically improve the performance of PAM. Results in this paper may shed lights on block coordinate (gradient) descent methods for general optimization problems.
Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
Cite as: arXiv:2109.15030 [math.OC]
  (or arXiv:2109.15030v2 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2109.15030

Submission history

From: Minhui Huang [view email]
[v1] Wed, 29 Sep 2021 04:32:06 UTC (382 KB)
[v2] Fri, 1 Oct 2021 01:00:36 UTC (382 KB)
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