Expected Batch OT Plan Explained

• Expected Batch OT Plans are population-level couplings derived by averaging optimal transport plans computed on random minibatches, formalizing a precise stochastic framework.
• They leverage cost identities and asymptotic bias analysis to bridge empirical OT with true optimal transport, showing how bias decreases as batch size increases.
• In flow matching, the averaged plans induce well-posed, Lipschitz velocity fields that guarantee unique ODE solutions for transporting distributions between Gaussian and discrete measures.

Searching arXiv for the cited work and closely related OT-plan literature. Expected batch OT plans are population-level couplings obtained by averaging optimal transport plans computed on random minibatches. In the formulation introduced in "Expected Batch Optimal Transport Plans and Consequences for Flow Matching" (Boïté et al., 12 May 2026), the object of interest is the coupling $\bar\pi_k$ induced by repeated OT on independent empirical measures of fixed batch size $k$. This construction makes precise the law implicitly generated by minibatch OT surrogates in large-scale learning, especially in settings such as flow matching where minibatch couplings are used as substitutes for exact OT. Closely related work studies other expectation-based transport plans, including averages over one-dimensional sliced couplings, posterior means under probabilistic hyperpriors, and empirical couplings recovered from sampled GFlowNet trajectories (Liu et al., 2024, Y. et al., 2024, Maksimov et al., 4 Jun 2026).

1. Formal definition

Let $\mu,\nu\in P_2(\mathbb{R}^d)$ and fix $k\in\mathbb{N}_+$. Draw independent minibatches

$$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$

Let $\hat\pi_{\vec X_k,\vec Y_k}$ be any measurable choice of an optimal transport plan between the empirical measures

$$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$

When $\hat\mu,\hat\nu$ are uniform on $k$ points, $\hat\pi_{\vec X_k,\vec Y_k}$ can equivalently be represented by an optimal permutation $k$0 (Boïté et al., 12 May 2026).

The expected batch OT plan $k$1 is the unique coupling satisfying, for every bounded Borel $k$2,

$k$3

An equivalent characterization is operational: if one computes minibatch OT and then picks one of the $k$4 matched pairs uniformly at random, the resulting law is $k$5 (Boïté et al., 12 May 2026).

This definition isolates a population object from a stochastic training heuristic. Rather than viewing minibatch OT as a sequence of unrelated empirical plans, the construction treats repeated minibatch matching as a random coupling mechanism with a well-defined expectation.

2. Cost identity, monotonicity, and asymptotic bias

A central identity links the transport cost of $k$6 to the expected empirical OT cost:

$k$7

This is Proposition 3.1 in (Boïté et al., 12 May 2026). It shows that the cost of the expected batch plan is exactly the average minibatch Wasserstein cost, not merely an upper or lower approximation.

The same work proves that the map

$k$8

is nonincreasing and satisfies

$k$9

Thus finite-batch OT is systematically biased upward in cost, and larger batches reduce that bias (Boïté et al., 12 May 2026).

In the semidiscrete setting, more explicit rates are available. If $\mu,\nu\in P_2(\mathbb{R}^d)$0 has compact support, $\mu,\nu\in P_2(\mathbb{R}^d)$1 is uniform on $\mu,\nu\in P_2(\mathbb{R}^d)$2 atoms, and the semidiscrete OT dual is nondegenerate, then

$\mu,\nu\in P_2(\mathbb{R}^d)$3

If instead $\mu,\nu\in P_2(\mathbb{R}^d)$4 for some $\mu,\nu\in P_2(\mathbb{R}^d)$5 and $\mu,\nu\in P_2(\mathbb{R}^d)$6 is uniform on $\mu,\nu\in P_2(\mathbb{R}^d)$7 atoms, then

$\mu,\nu\in P_2(\mathbb{R}^d)$8

The $\mu,\nu\in P_2(\mathbb{R}^d)$9 result is obtained by showing that the semidiscrete dual $k\in\mathbb{N}_+$0 has a unique quadratic maximizer $k\in\mathbb{N}_+$1 and that the dual excess

$k\in\mathbb{N}_+$2

is $k\in\mathbb{N}_+$3 (Boïté et al., 12 May 2026).

These statements delimit what minibatch OT approximates at population level. A common misconception is that averaging minibatch couplings necessarily recovers the exact OT plan at moderate $k\in\mathbb{N}_+$4; the established result is weaker and more precise: the expected batch cost decreases monotonically toward the OT cost, with rates that depend on the semidiscrete structure and moment assumptions.

3. Plan consistency in the semidiscrete regime

Assume $k\in\mathbb{N}_+$5, so that the OT plan $k\in\mathbb{N}_+$6 is unique. In this setting, Proposition 3.3 of (Boïté et al., 12 May 2026) proves that

$k\in\mathbb{N}_+$7

This is a direct plan-consistency statement: the expected batch OT plan converges to the true OT plan in the large-batch limit.

The same proposition gives a cost-to-plan lower bound. For any $k\in\mathbb{N}_+$8,

$k\in\mathbb{N}_+$9

This inequality shows that plan discrepancy controls cost discrepancy from below, but not conversely in full generality (Boïté et al., 12 May 2026).

A sharper upper bound is available in the Gaussian-to-discrete case. If $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$0 and $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$1 is uniform on $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$2 atoms, then there exists $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$3 such that

$$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$4

This yields an explicit convergence rate for the plan itself in a semidiscrete generative-modeling regime (Boïté et al., 12 May 2026).

The paper also emphasizes a limitation: without additional structure, no modulus $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$5 exists with

$$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$6

This rules out a general principle that small excess cost alone forces plan proximity. A plausible implication is that semidiscrete regularity assumptions are not merely technical conveniences but are essential to obtain quantitative plan-recovery guarantees.

4. Consequences for flow matching

The main application developed in (Boïté et al., 12 May 2026) is flow matching in the Gaussian-to-discrete setting

$$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$7

Any coupling $$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$8 induces the velocity field

$$\vec X_k=(X_1,\dots,X_k)\sim \mu^{\otimes k},\qquad \vec Y_k=(Y_1,\dots,Y_k)\sim \nu^{\otimes k}.$$9

and the associated ODE flow $\hat\pi_{\vec X_k,\vec Y_k}$0.

Lemma 4.1 gives the posterior form

$\hat\pi_{\vec X_k,\vec Y_k}$1

where $\hat\pi_{\vec X_k,\vec Y_k}$2. If $\hat\pi_{\vec X_k,\vec Y_k}$3 is the density of $\hat\pi_{\vec X_k,\vec Y_k}$4 with respect to $\hat\pi_{\vec X_k,\vec Y_k}$5, then

$\hat\pi_{\vec X_k,\vec Y_k}$6

This expresses the flow field through posterior assignment probabilities under the coupling (Boïté et al., 12 May 2026).

For the expected batch plan $\hat\pi_{\vec X_k,\vec Y_k}$7, Proposition 4.2 establishes rectifiability. For each fixed $\hat\pi_{\vec X_k,\vec Y_k}$8:

• $\hat\pi_{\vec X_k,\vec Y_k}$9 is jointly continuous on $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$0 for any compact $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$1 and $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$2, and locally Lipschitz in $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$3, uniformly in $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$4.
• For each $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$5, the ODE $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$6, $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$7, has a unique solution $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$8 on $$\hat\mu=\frac{1}{k}\sum_i \delta_{X_i},\qquad \hat\nu=\frac{1}{k}\sum_j \delta_{Y_j}.$$9, and $\hat\mu,\hat\nu$0.
• There is a Borel map $\hat\mu,\hat\nu$1 with $\hat\mu,\hat\nu$2 almost surely, and $\hat\mu,\hat\nu$3.

The proof sketch relies on posterior weights $\hat\mu,\hat\nu$4 that are globally Lipschitz and bounded away from $\hat\mu,\hat\nu$5, together with Gaussian density bounds, Picard–Lindelöf for the ODE, and a trapping argument near atoms (Boïté et al., 12 May 2026).

These results distinguish expected batch OT plans from ad hoc minibatch pairings. In this regime, the averaged coupling does not merely reduce variance; it induces a well-posed, locally Lipschitz FM velocity field and therefore a unique flow from the source to the discrete target.

5. Numerical tradeoffs and computational interpretation

The expected batch OT framework also quantifies the interaction between OT batch size and numerical integration. Define the Euler-map error

$\hat\mu,\hat\nu$6

where $\hat\mu,\hat\nu$7 uses $\hat\mu,\hat\nu$8 Euler steps. In the two-atom model with $\hat\mu,\hat\nu$9 and $k$0, Proposition 4.3 shows that, as $k$1,

$k$2

and as $k$3,

$k$4

Hence increasing NFE yields a stretched-exponential decay in error $k$5, whereas increasing OT batch size gives only a polynomial $k$6 decay (Boïté et al., 12 May 2026).

Synthetic and image experiments reinforce this distinction. In Gaussian-to-discrete synthetic runs, the cost-bias decays $k$7 and the plan-distance proxy decays $k$8. The posterior concentration

$k$9

approaches $\hat\pi_{\vec X_k,\vec Y_k}$0 faster with larger $\hat\pi_{\vec X_k,\vec Y_k}$1, and also faster with higher $\hat\pi_{\vec X_k,\vec Y_k}$2 when $\hat\pi_{\vec X_k,\vec Y_k}$3. In real-data FM on CIFAR-10 and SVHN,

$\hat\pi_{\vec X_k,\vec Y_k}$4

steadily decreases up to $\hat\pi_{\vec X_k,\vec Y_k}$5 with no saturation, but increasing $\hat\pi_{\vec X_k,\vec Y_k}$6 improves sample FID only when using very coarse integration; at high NFE the benefit of larger $\hat\pi_{\vec X_k,\vec Y_k}$7 vanishes or even reverses (Boïté et al., 12 May 2026).

The practical interpretation is narrow but important. Expected batch OT plans justify minibatch OT as a population coupling and show large-batch consistency, but they do not imply that arbitrarily large OT batches are the most efficient route to better inference. The stated asymptotics indicate that, in the analyzed setting, moderate $\hat\pi_{\vec X_k,\vec Y_k}$8 and higher NFE can be the more effective training–inference tradeoff.

6. Related expected-plan constructions

The expected batch OT plan is one instance of a broader family of transport plans obtained by averaging over a random mechanism. The following constructions are closely related in spirit but differ in what is randomized and what guarantees are proved.

Construction	Averaging mechanism	Resulting object
Expected batch OT plan $\hat\pi_{\vec X_k,\vec Y_k}$9	Independent minibatches of size $k$00	Population coupling
Expected sliced transport plan $k$01	Random slices $k$02	Averaged lifted coupling
HFPD-OT expected plan $k$03	Samples from a hyperprior over plans	Posterior mean plan
GFlowNet empirical batch plan $k$04	Sampled trajectories under learned policy	Empirical coupling

Expected sliced transport, introduced in "Expected Sliced Transport Plans" (Liu et al., 2024), starts from discrete measures

$k$05

in $k$06. For each $k$07, one computes the one-dimensional OT plan $k$08 between the projected measures, lifts it back to $k$09 via $k$10, and averages:

$k$11

Using $k$12 to weight the Euclidean $k$13-cost defines

$k$14

The paper proves that $k$15 is a valid metric on input discrete probability measures, gives an algorithm with total complexity $k$16 and memory $k$17, and introduces temperature weighting $k$18, which recovers EST as $k$19 and degenerates to the "min-SWGG" transport map as $k$20 (Liu et al., 2024).

HFPD-OT, developed in "Randomized Transport Plans via Hierarchical Fully Probabilistic Design" (Y. et al., 2024), treats the transport plan itself as a random matrix $k$21 in the simplex $k$22 and defines a Gibbs-form hyperprior $k$23 subject to marginal moment constraints

$k$24

The expected plan is

$k$25

Although no closed form exists in general, the limit $k$26 and $k$27 converges to the usual entropy-regularized OT plan, whereas $k$28 reverts to the unconstrained Gibbs generator. The method further provides entry-wise variances and credible intervals from HMC samples of the hyperprior (Y. et al., 2024).

A different notion appears in "Your GFlowNet Secretly Learns an Optimal Transport Plan" (Maksimov et al., 4 Jun 2026). There, a minimum-flow non-acyclic GFlowNet with fixed source and sink flows is shown to be equivalent to a Kantorovich OT problem on a graph with graph-induced shortest-path cost. The optimal edge-flow induces a stochastic policy, and sampling $k$29 trajectories yields the empirical coupling

$k$30

which converges almost surely to the optimal coupling entrywise, with Hoeffding concentration

$k$31

This construction is empirical rather than population-averaged, but it also produces a batch OT plan in the sense of a coupling estimated from repeated randomized transport trajectories (Maksimov et al., 4 Jun 2026).

Taken together, these constructions show that an "expected" or "batch" OT plan need not refer to a single mechanism. It may denote expectation over minibatches, over slices, over posterior samples of plans, or over sampled transport trajectories. The expected batch OT plan $k$32 is distinguished by formalizing the exact population coupling induced by repeated minibatch OT and by connecting that coupling to consistency, semidiscrete rates, and well-posed flow matching dynamics (Boïté et al., 12 May 2026).