---
title: Expected Batch OT Plan Explained
url: https://www.emergentmind.com/topics/expected-batch-ot-plan
type: topic
---

# Expected Batch OT Plan Explained

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" [2605.12174], 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 [2410.12176], [2408.02701], [2606.06272].

## 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 $\hat\sigma$ [2605.12174].

The expected batch OT plan $\bar\pi_k\in P(\mathbb{R}^d\times\mathbb{R}^d)$ is the unique coupling satisfying, for every bounded Borel $g:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}$,
$$
\int g\,d\bar\pi_k
=
\mathbb{E}_{(\vec X_k,\vec Y_k)\sim \mu^{\otimes k}\otimes \nu^{\otimes k}}
\left[\int g\,d\hat\pi_{\vec X_k,\vec Y_k}\right].
$$
An equivalent characterization is operational: if one computes minibatch OT and then picks one of the $k$ matched pairs uniformly at random, the resulting law is $\bar\pi_k$ [2605.12174].

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 $\bar\pi_k$ to the expected empirical OT cost:
$$
\int \|x-y\|^2\,d\bar\pi_k(x,y)
=
\mathbb{E}\big[W_2^2(\hat\mu_k,\hat\nu_k)\big].
$$
This is Proposition 3.1 in [2605.12174]. 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
$$
k\mapsto \mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]
$$
is nonincreasing and satisfies
$$
\mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]\ge W_2^2(\mu,\nu).
$$
Thus finite-batch OT is systematically biased upward in cost, and larger batches reduce that bias [2605.12174].

In the semidiscrete setting, more explicit rates are available. If $\mu$ has compact support, $\nu$ is uniform on $M$ atoms, and the semidiscrete OT dual is nondegenerate, then
$$
\mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]-W_2^2(\mu,\nu)=O(k^{-1}).
$$
If instead $\mu\in P_q(\mathbb{R}^d)$ for some $q>4$ and $\nu$ is uniform on $M$ atoms, then
$$
\mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]-W_2^2(\mu,\nu)=O(k^{-1/2}).
$$
The $O(k^{-1})$ result is obtained by showing that the semidiscrete dual $H(\lambda)$ has a unique quadratic maximizer $\lambda^*$ and that the dual excess
$$
\mathbb{E}\big[\sup_{\lambda\in\Lambda}(H_k(\lambda)-H(\lambda^*))\big]
$$
is $O(1/k)$ [2605.12174].

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 $k$; 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 $\mu\ll\mathrm{Leb}$, so that the OT plan $\pi^*$ is unique. In this setting, Proposition 3.3 of [2605.12174] proves that
$$
\bar\pi_k \rightharpoonup \pi^*
\qquad\text{weakly as }k\to\infty.
$$
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 $\pi\in\Pi(\mu,\nu)$,
$$
W_2(\pi,\pi^*) \ge \frac{1}{\sqrt{2}}
\left|
\left(\int \|x-y\|^2\,d\pi\right)^{1/2}
-
W_2(\mu,\nu)
\right|.
$$
This inequality shows that plan discrepancy controls cost discrepancy from below, but not conversely in full generality [2605.12174].

A sharper upper bound is available in the Gaussian-to-discrete case. If $\mu=N(0,I_d)$ and $\nu$ is uniform on $M$ atoms, then there exists $A_\nu>0$ such that
$$
W_2^2(\bar\pi_k,\pi^*)
\le
A_\nu\cdot
\big[
\mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]-W_2^2(\mu,\nu)
\big]^{1/2}
=
O(k^{-1/4}).
$$
This yields an explicit convergence rate for the plan itself in a semidiscrete generative-modeling regime [2605.12174].

The paper also emphasizes a limitation: without additional structure, no modulus $\omega$ exists with
$$
W_2(\pi,\pi^*) \le \omega\!\left(\int \|x-y\|^2\,d\pi - W_2^2(\mu,\nu)\right)\to 0.
$$
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 [2605.12174] is flow matching in the Gaussian-to-discrete setting
$$
\mu=N(0,I_d),\qquad \nu=\mathrm{Unif}\{v_1,\dots,v_M\}.
$$
Any coupling $\pi$ induces the velocity field
$$
u_t^\pi(x)=\mathbb{E}[X_1-X_0\mid (1-t)X_0+tX_1=x],
$$
and the associated ODE flow $\dot x(t)=u_t(x(t))$.

Lemma 4.1 gives the posterior form
$$
u_t^\pi(x)=\frac{m_t^\pi(x)-x}{1-t},
\qquad
m_t^\pi(x)=\sum_{j=1}^M v_j\cdot P_\pi(X_1=v_j\mid Z_t=x),
$$
where $Z_t=(1-t)X_0+tX_1$. If $f_\pi(x_0,v_j)$ is the density of $\pi$ with respect to $\mathrm{Leb}\otimes\mathrm{counting}$, then
$$
P(X_1=v_j\mid Z_t=x)
=
\frac{
f_\pi\!\left(\frac{x-tv_j}{1-t},v_j\right)
}{
\sum_\ell
f_\pi\!\left(\frac{x-tv_\ell}{1-t},v_\ell\right)
}.
$$
This expresses the flow field through posterior assignment probabilities under the coupling [2605.12174].

For the expected batch plan $\bar\pi_k$, Proposition 4.2 establishes rectifiability. For each fixed $k$:

- $(t,x)\mapsto u_t^{\bar\pi_k}(x)$ is jointly continuous on $[0,\tau]\times K$ for any compact $K\subset\mathbb{R}^d$ and $\tau<1$, and locally Lipschitz in $x$, uniformly in $t\in[0,\tau]$.
- For each $x_0$, the ODE $\dot x(t)=u_t^{\bar\pi_k}(x(t))$, $x(0)=x_0$, has a unique solution $\phi_t^{\bar\pi_k}(x_0)$ on $t\in[0,1)$, and $\phi_t^{\bar\pi_k}\sharp\mu=\rho_t^{\bar\pi_k}$.
- There is a Borel map $\phi^{\bar\pi_k}:\mathbb{R}^d\to\{v_j\}$ with $\phi^{\bar\pi_k}(x_0)=\lim_{t\to 1^-}\phi_t^{\bar\pi_k}(x_0)$ almost surely, and $\phi^{\bar\pi_k}\sharp\mu=\nu$.

The proof sketch relies on posterior weights $a_j(x)=P[Y=v_j\mid X=x]$ that are globally Lipschitz and bounded away from $0$, together with Gaussian density bounds, Picard–Lindelöf for the ODE, and a trapping argument near atoms [2605.12174].

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
$$
\mathcal{E}_{n,k}
=
\mathbb{E}_X\big[\|f_n^{\bar\pi_k}(X)-\phi^{\bar\pi_k}(X)\|\big],
$$
where $f_n$ uses $n$ Euler steps. In the two-atom model with $\mu=N(0,1)$ and $\nu=\mathrm{Unif}\{-1,+1\}$, Proposition 4.3 shows that, as $n\to\infty$,
$$
\log \mathcal{E}_{n,1}\sim -\frac{2\pi}{\sqrt{3}}\,n^{2/3},
$$
and as $k\to\infty$,
$$
\log \mathcal{E}_{1,k}\sim -\frac{1}{2}\log k.
$$
Hence increasing NFE yields a stretched-exponential decay in error $\sim \exp(-c\,n^{2/3})$, whereas increasing OT batch size gives only a polynomial $k^{-1/2}$ decay [2605.12174].

Synthetic and image experiments reinforce this distinction. In Gaussian-to-discrete synthetic runs, the cost-bias decays $O(k^{-1})$ and the plan-distance proxy decays $O(k^{-1/2})$. The posterior concentration
$$
P\!\left[\max_j P(Y=v_j\mid Z_t)\right]
$$
approaches $1$ faster with larger $k$, and also faster with higher $d$ when $k=1$. In real-data FM on CIFAR-10 and SVHN,
$$
\mathbb{E}[W_2^2(\hat\mu_k,\hat\nu_k)]
$$
steadily decreases up to $k=8192$ with no saturation, but increasing $k$ improves sample FID only when using very coarse integration; at high NFE the benefit of larger $k$ vanishes or even reverses [2605.12174].

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 $k$ 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 $\bar\pi_k$ | Independent minibatches of size $k$ | Population coupling |
| Expected sliced transport plan $\Pi_{\mathrm{EST}}$ | Random slices $\theta\sim U(S^{d-1})$ | Averaged lifted coupling |
| HFPD-OT expected plan $\hat P$ | Samples from a hyperprior over plans | Posterior mean plan |
| GFlowNet empirical batch plan $\hat P_N$ | Sampled trajectories under learned policy | Empirical coupling |

Expected sliced transport, introduced in "Expected Sliced Transport Plans" [2410.12176], starts from discrete measures
$$
\mu^1=\sum_{i=1}^n p_i\delta_{x_i},
\qquad
\mu^2=\sum_{j=1}^m q_j\delta_{y_j}
$$
in $\mathbb{R}^d$. For each $\theta\in S^{d-1}$, one computes the one-dimensional OT plan $\pi_\theta$ between the projected measures, lifts it back to $\mathbb{R}^d\times\mathbb{R}^d$ via $L(\pi_\theta)=\gamma_\theta$, and averages:
$$
\Pi_{\mathrm{EST}}
=
\mathbb{E}_{\theta\sim U(S^{d-1})}[\gamma_\theta].
$$
Using $\Pi_{\mathrm{EST}}$ to weight the Euclidean $p$-cost defines
$$
D_p(\mu^1,\mu^2)
=
\left(
\sum_{i=1}^n\sum_{j=1}^m
\|x_i-y_j\|^p\,\Pi_{\mathrm{EST}}(x_i,y_j)
\right)^{1/p}.
$$
The paper proves that $D_p$ is a valid metric on input discrete probability measures, gives an algorithm with total complexity $O(L(n\log n+m\log m+nm))$ and memory $O(nm)$, and introduces temperature weighting $\sigma_\tau(\theta)\propto \exp[-\tau D_p^p(\mu^1,\mu^2;\theta)]$, which recovers EST as $\tau\to 0$ and degenerates to the "min-SWGG" transport map as $\tau\to\infty$ [2410.12176].

HFPD-OT, developed in "Randomized Transport Plans via Hierarchical Fully Probabilistic Design" [2408.02701], treats the transport plan itself as a random matrix $P=[\pi_{ij}]$ in the simplex $\Delta_{mn-1}$ and defines a Gibbs-form hyperprior $S^0(P\mid K)$ subject to marginal moment constraints
$$
E_S[D_{KL}(\mu\|\mu_0)]\le \eta,
\qquad
E_S[D_{KL}(\nu\|\nu_0)]\le \zeta.
$$
The expected plan is
$$
\hat P = E_{S^0}[P].
$$
Although no closed form exists in general, the limit $\eta,\zeta\to 0$ and $\lambda\to\infty$ converges to the usual entropy-regularized OT plan, whereas $\eta,\zeta\to\infty$ reverts to the unconstrained Gibbs generator. The method further provides entry-wise variances and credible intervals from HMC samples of the hyperprior [2408.02701].

A different notion appears in "Your GFlowNet Secretly Learns an Optimal Transport Plan" [2606.06272]. 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 $N$ trajectories yields the empirical coupling
$$
\hat P_N(u,x)=\frac1N\sum_{n=1}^N
\mathbf1\{\tau_n\text{ starts at }u,\ \tau_n\text{ ends at }x\},
$$
which converges almost surely to the optimal coupling entrywise, with Hoeffding concentration
$$
\Pr\bigl[|\hat P_N(u,x)-P^*(u,x)|\ge \varepsilon\bigr]
\le 2\exp(-2N\varepsilon^2).
$$
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 [2606.06272].

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 $\bar\pi_k$ 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 [2605.12174].

Source: https://www.emergentmind.com/topics/expected-batch-ot-plan