Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients
This provides a more efficient solver for optimal transport problems, particularly in weak-regularization regimes, which is incremental but offers strong performance gains for researchers and practitioners in machine learning and optimization.
The paper tackles the problem of solving discrete optimal transport with high precision by proposing MDOT-PNCG, a method that unifies temperature annealing with mirror descent and uses a GPU-parallel conjugate gradients algorithm, achieving faster wall-clock times and operations than existing solvers, with empirical convergence rates ranging from O(n^2 ε^{-1/4}) to O(n^2 ε^{-1}) and runtime scaling up to O(n^{5/2}) for high precision.
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.