Expected Batch Optimal Transport Plans and Consequences for Flow Matching
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 π̅k, 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 π̅k 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.
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BibTeX
@misc{boite2026expected,
title = {{Expected Batch Optimal Transport Plans and Consequences for Flow Matching}},
author = {Samuel Boïté and Julie Delon and Kimia Nadjahi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.12174},
eprint = {2605.12174},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12174},
}
RIS
TY - UNPB
TI - Expected Batch Optimal Transport Plans and Consequences for Flow Matching
AU - Samuel Boïté
AU - Julie Delon
AU - Kimia Nadjahi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2605.12174
UR - https://arxiv.org/abs/2605.12174
AB - 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 π̅_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 π̅_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.
ER -