Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients (2307.08507v4)

Abstract: We propose Mirror Descent Optimal Transport (MDOT), a novel method for solving discrete optimal transport (OT) problems with high precision, by unifying temperature annealing in entropic-regularized OT (EOT) with mirror descent techniques. In this framework, temperature annealing produces a sequence of EOT dual problems, whose solution gradually gets closer to the solution of the original OT problem. We solve each problem efficiently using a GPU-parallel nonlinear conjugate gradients algorithm (PNCG) that outperforms traditional Sinkhorn iterations under weak regularization. Moreover, our investigation also reveals that the theoretical convergence rate of Sinkhorn iterations can exceed existing non-asymptotic bounds when its stopping criterion is tuned in a manner analogous to MDOT. Our comprehensive ablation studies of MDOT-PNCG affirm its robustness across a wide range of algorithmic parameters. Benchmarking on 24 problem sets of size $n=4096$ in a GPU environment demonstrate that our method attains high-precision, feasible solutions significantly faster than a representative set of existing OT solvers, including accelerated gradient methods and advanced Sinkhorn variants, in both wall-clock time and number of operations. Empirical convergence rates range between $O(n^2 \varepsilon^{-1/4})$ and $O(n^2 \varepsilon^{-1})$, where $\varepsilon$ is the optimality gap. For problem sizes up to $n=16384$, the empirical runtime scales as $O(n^2)$ for moderate precision and as $O(n^{5/2})$ at worst for high precision. These findings establish MDOT-PNCG as a compelling alternative to current OT solvers, particularly in challenging weak-regularization regimes.