On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport
This work addresses a computational bottleneck in fair division and multi-agent transport problems, offering incremental improvements to existing methods.
The paper tackles the convergence analysis of the projected alternating maximization algorithm for solving the equitable and optimal transport problem, providing the first theoretical convergence guarantees and proposing a variant with extrapolation that improves numerical performance.
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.