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

TL;DR

This paper introduces a new stochastic gradient descent algorithm with adaptive entropic regularization for semi-discrete optimal transport, achieving nearly minimax optimal convergence rates and improving upon previous bounds.

Contribution

It proves an $ ext{O}(t^{-1})$ lower bound for the OT map in the one-sample setting and develops an efficient SGD-based method with adaptive regularization to nearly attain this rate.

Findings

The proposed algorithm achieves fast convergence rates in numerical experiments.

An $ ext{O}(t^{-1})$ lower bound rate is established for the OT map.

The method is computationally efficient, matching vanilla SGD in resource usage.

Abstract

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.