揭示小批量最优传输的平均效应，为生成模型中的流匹配提供理论支撑
Expected Batch Optimal Transport Plans and Consequences for Flow Matching
- 提出期望批次最优传输计划，分析小批量方法在大样本下的收敛特性
- 证明在半离散情形下，批次越大，运输成本偏差越小，耦合趋于真实最优传输
- 给出理论与实验结合的指导：批大小与数值积分精度的权衡关系
在大规模学习中，对随机小批量求解最优传输（OT）是精确OT的常见替代方案。在流匹配（FM）中，该替代方法用于获得类OT耦合，以拉直概率路径并降低数值积分成本。然而，反复使用小批量OT所诱导的总体耦合仍不完全清楚。本文将此耦合形式化为期望批次OT计划 $\overlineπ_{k}$，即对独立小批量的实证OT计划进行平均所得。我们建立了其大批次一致性，并在生成建模相关的半离散情形下，推导出运输成本偏差和 $\overlineπ_{k}$ 收敛到最优传输计划的速率。对于流匹配，这得到了一个足够规则的诱导速度场，可从源分布唯一定义到离散目标分布的流。最后，我们在一个可解析的两原子模型以及合成与图像实验中量化了OT批大小与数值积分之间的交互作用。
原文摘要 · Abstract (English)
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 $\overlineπ_{k}$, 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 $\overlineπ_{k}$ 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.
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