On the Convergence of Projected Alternating Maximization for Equitable and Optimal TransportDownload PDF

Published: 28 Jan 2022, Last Modified: 13 Feb 2023ICLR 2022 SubmittedReaders: Everyone
Keywords: Equitable and Optimal Transport, Fairness, Saddle Point Problem, Projected Alternating Maximization, Block Coordinate Descent, Acceleration, Rounding
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 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.
One-sentence Summary: Convergence rate analysis for Equitable and Optimal Transport with Entropy regularization.
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