Log-Averaged Mirror Prox for Fast, Large-Scale Optimal Transport in Linear Space

We propose Log-Averaged Mirror Prox (LAMP), a linear-space primal-dual method for large-scale optimal transport. LAMP implements primal mirror prox updates by tracking an averaged dual sequence, reducing storage complexity from O(nm) to O(n+m) while preserving dense, GPU-friendly reductions. Consequently, LAMP preserves the last-iterate widetildeO( nmvarepsilon-1) arithmetic complexity of conservatively parameterized primal-dual mirror prox. We further analyze LAMP as a direct optimal transport solver in a more performant parameter regime, providing a last-iterate sub-optimality certificate dependent on infeasibility and an explicit O(1/t) term. Moreover, we give a computable sufficient condition for best-iterate convergence to a saddle-point. Numerical experiments with an optimized CUDA implementation show that LAMP outperforms first-order baselines in several high-accuracy (entropic) optimal transport problems. LAMP is further shown to scale up to problems with n=m=218 marginal supports, which were previously beyond the reach of primal-dual first-order methods.