Expected Batch Optimal Transport Plans and Consequences for Flow Matching

TL;DR

This paper analyzes the expected batch optimal transport plan in large-scale learning, establishing its properties, convergence, and implications for flow matching and generative modeling.

Contribution

It formalizes the expected batch OT plan, proves its large-batch consistency, and derives convergence rates, enhancing understanding of minibatch OT in flow matching.

Findings

Expected batch OT plan converges to the population OT plan as batch size increases.

Derived rates for transport-cost bias and plan convergence in the semidiscrete case.

Quantified the interaction between batch size and numerical integration in experiments.

Abstract

Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan $\overline{\pi}_{k}$, obtained by averaging empirical OT plans over independent minibatches of size $k$. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of $\overline{\pi}_{k}$ to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments.