Conditional flow-matching batch gradients and exact-flow output law

Fix $k$ and a history for which ${\theta }_k=\theta \in \mathrm{\Theta }$ and both losses at $\theta$ are finite. Choose a compact parameter box $H\subset \mathrm{\Theta }$ with $\theta$ in its interior. Changing coordinates one at a time, the mean value theorem [F14] and the derivative majorants in [F2] bound the differences between each loss integrand at ${\theta }^{\mathrm{\prime }}$ and at $\theta$ by the corresponding sums of $\mathrm{\mid }{\theta }_j^{\mathrm{\prime }}-{\theta }_j\mathrm{\mid }{G}_{H,j}^C$ and $\mathrm{\mid }{\theta }_j^{\mathrm{\prime }}-{\theta }_j\mathrm{\mid }{G}_{H,j}^M$. Integrating these bounds proves local finiteness of both losses throughout the interior of $H$, so [F2] applies there and gives ${\mathrm{\nabla }}_{\theta }{\mathcal{L}}_C(\theta )={\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M(\theta )$. For fixed $w=(t,x_1,x)$, put $a_{\theta }:=v(t,x;\theta )-u_t^c(x\mid x_1)$. Bilinearity and symmetry [F5] give $\mathrm{\parallel }{a}_{\theta +\epsilon e_j}{\mathrm{\parallel }}_2^2-\mathrm{\parallel }{a}_{\theta }{\mathrm{\parallel }}_2^2=⟨a_{\theta +\epsilon e_j}-a_{\theta },a_{\theta +\epsilon e_j}+a_{\theta }⟩\mathrm{.}$ Dividing by $\epsilon$ and taking the limit using differentiability of $v$ shows that the $j$th coordinate of the sample gradient is $2⟨{\mathrm{\partial }}_{{\theta }_j}v,v-u^c⟩$, exactly the $j$th coordinate of $\zeta (\theta ,w)$. Since the triple has law $Q$, the expectation change-of-variables formula [F10] identifies its $Q$-mean with ${\mathrm{\nabla }}_{\theta }{\mathcal{L}}_C(\theta )$. Thus its $Q$-mean is ${\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M(\theta )$.

Each ${\zeta }_{k,b}$, and hence $g_k$, is integrable by the explicit batch-integrability assumption. For fixed $\theta$, put $m(\theta ):={\int }_Q\zeta (\theta ,w)Q(dw)$. Joint measurability, Tonelli [F13], and the local derivative majorants make $m$ measurable and finite; at each visited parameter step 1.1 gives $m({\theta }_k)={\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M({\theta }_k)$. Fubini [F11] and the integral triangle inequality [F12] show that each coordinate of $m({\theta }_k)$ is integrable, with its absolute expectation bounded by that of the corresponding sample-gradient coordinate. For every $A\in {\mathcal{F}}_k$, independence and the product-law fact [F9], the change-of-variables formula [F10], and Fubini [F11] give $\mathbb{E}[{\mathbf{1}}_A{g}_k]=\mathbb{E}\text{\hspace{0.17em}⁣}[{\mathbf{1}}_A{\int }_{Q^{\otimes B}}\frac{1}{B}{\sum }_{b=1}^B\zeta ({\theta }_k,w_b)Q^{\otimes B}(d(w_1,\dots ,w_B))]=\mathbb{E}[{\mathbf{1}}_A{\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M({\theta }_k)],$ where step 1.1 identifies each fixed-parameter $Q$-mean and [F6] gives linearity. Thus [F8] makes ${\mathrm{\nabla }}_{\theta }{\mathcal{L}}_M({\theta }_k)$ a version of $\mathbb{E}[g_k\mid {\mathcal{F}}_k]$; the batch estimate is unbiased conditionally on its pre-batch history. The arithmetic update is defined for every nonnegative step size when ${\theta }_k\in \mathrm{\Theta }$.

For the realized ${\theta }_N$ satisfying the exact-flow bounds, [F3] identifies the sampling equation as a parameterized ODE and [F4] gives its unique solution on the full interval and joint continuity of $(x,\theta )\mapsto {\mathrm{\Phi }}_{1,0}^{\theta }(x)$. The terminal output is therefore measurable as a function of the independent initial draw and the training history. Conditional on ${\mathcal{F}}_N$, ${\theta }_N$ is fixed, so [F7] identifies the conditional output law by ${\mu }_0(\{x:{\mathrm{\Phi }}_{1,0}^{{\theta }_N}(x)\in A\})$ for each Borel set $A$. This verifies that the exact-flow output is defined; it makes no claim that the learned field realizes the target path.