Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularization

Optimal Transport (OT) based distances are powerful tools for machine learning to compare probability measures and manipulate them using OT maps. In this field, a setting of interest is semi-discrete OT, where the source measure $\mu$ is continuous, while the target $\nu$ is discrete. Recent works have shown that the minimax rate for the OT map is $\mathcal{O}(t^{-1/2})$ when using $t$ i.i.d. subsamples from each measure (two-sample setting). An open question is whether a better convergence rate can be achieved when the full information of the discrete measure $\nu$ is known (one-sample setting). In this work, we answer positively to this question by (i) proving an $\mathcal{O}(t^{-1})$ lower bound rate for the OT map, using the similarity between Laguerre cells estimation and density support estimation, and (ii) proposing a Stochastic Gradient Descent (SGD) algorithm with adaptive entropic regularization and averaging acceleration. To nearly achieve the desired fast rate, characteristic of non-regular parametric problems, we design an entropic regularization scheme decreasing with the number of samples. Another key step in our algorithm consists of using a projection step that permits to leverage the local strong convexity of the regularized OT problem. Our convergence analysis integrates online convex optimization and stochastic gradient techniques, complemented by the specificities of the OT semi-dual. Moreover, while being as computationally and memory efficient as vanilla SGD, our algorithm achieves the unusual fast rates of our theory in numerical experiments.