Semidiscrete optimal transport with unknown costs

TL;DR

This paper introduces a novel approach to semidiscrete optimal transport with unknown costs, combining online learning and stochastic approximation to efficiently learn cost functions and achieve optimal convergence.

Contribution

It presents a new variant of semidiscrete optimal transport where cost functions are unknown and can only be learned through noisy, sequential sampling.

Findings

Achieves optimal convergence rates in learning unknown costs.

Handles non-smooth stochastic gradients effectively.

Provides theoretical guarantees despite lack of strong concavity.

Abstract

Semidiscrete optimal transport is a challenging generalization of the classical transportation problem in linear programming. The goal is to design a joint distribution for two random variables (one continuous, one discrete) with fixed marginals, in a way that minimizes expected cost. We formulate a novel variant of this problem in which the cost functions are unknown, but can be learned through noisy observations; however, only one function can be sampled at a time. We develop a semi-myopic algorithm that couples online learning with stochastic approximation, and prove that it achieves optimal convergence rates, despite the non-smoothness of the stochastic gradient and the lack of strong concavity in the objective function.