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Counterfactual Generation via Flow Matching: Coupling-Sensitive End-to-End Rates
Yunrui Guan, Krishnakumar Balasubramanian, Shiva Prasad Kasiviswanathan
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%
Last extracted: 10/3/2026, 2:22:19 AM
Summary
This paper introduces a counterfactual generation method using flow matching that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes. The authors propose a score-corrected stochastic sampler based on Gaussian-smoothed interpolation to enable finite-step generation. Theoretical contributions include a coupling-sensitive Kullback-Leibler (KL) bound for constant-step Euler discretization, where error is controlled by moments of the source-target displacement rather than global regularity, and finite-sample non-parametric guarantees for the learned fields. Experiments on synthetic and semi-synthetic image benchmarks demonstrate that the stochastic sampler can outperform deterministic ODE samplers at finite discretization budgets.
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Shiva Prasad Kasiviswanathan → affiliatedwith → Amazon
confidence 99% · Affiliation: Amazon
Yunrui Guan → affiliatedwith → Johns Hopkins University
confidence 99% · Affiliation: Department of Applied Mathematics and Statistics, Johns Hopkins University.
Krishnakumar Balasubramanian → affiliatedwith → University of California Davis
confidence 99% · Affiliation: Department of Statistics, University of California, Davis.
Krishnakumar Balasubramanian → affiliatedwith → Amazon
confidence 99% · Affiliation: Amazon
Counterfactual Generation → isaddressedby → Flow Matching
confidence 98% · Counterfactual generation seeks to sample outcomes... We develop a flow-matching approach...
Flow Matching → uses → Doubly Robust Training
confidence 96% · We develop a flow-matching approach that combines a sample-split, doubly robust training objective...
Yunrui Guan → affiliatedwith → Amazon
confidence 95% · This work was done during an internship at Amazon.
Score-Corrected Stochastic Sampler → uses → Gaussian Smoothing
confidence 95% · ...leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation.
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Abstract
Abstract:Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source--target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.
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- Source: https://arxiv.org/abs/2610.01193v1
- Canonical: https://arxiv.org/abs/2610.01193v1
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Counterfactual Generation via Flow Matching: Coupling-Sensitive End-to-End Rates Yunrui Guan †thanks: This work was done during an internship at Amazon. Email: yguan34@jh.edu Affiliation: Department of Applied Mathematics and Statistics, Johns Hopkins University. Krishnakumar Balasubramanian Affiliation: Department of Statistics, University of California, Davis. Affiliation: Amazon Shiva Prasad Kasiviswanathan Affiliation: Amazon Abstract Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source–target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source–target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler. Contents 1 Introduction 2 Problem Setup and Method 2.1 Gaussian-Smoothed Flow Matching 2.2 Sample Splitting and Learned Coupling 2.3 Doubly Robust Field Training 2.4 Score-Corrected SDE Sampler 3 Coupling-Dependent Sampling and End-to-End Guarantees 3.1 Coupling-Dependent Finite-Step Sampling 3.2 Finite-Sample Field Estimation and End-to-End Guarantee 4 Experiments 4.1 Angular Mode Reallocation 4.2 Image Benchmarks 5 Conclusion References A Additional Experiment Details A.1 Angular-Mode Experiment Details A.2 Image-Benchmark Details and Additional Results B Additional Related-Work Details B.1 Comparison With Dance et al. (2026) C Reader’s Guide and Proof Roadmap C.1 Summary of Notation D Algorithms and Auxiliary-Model Details D.1 Prediction-Oracle Accuracy D.2 Score-Field Training Objective D.3 Algorithms E Proof of Example F Proof of Example G Proof of Theorem G.1 Path-Space Comparison via Girsanov G.2 Posterior Derivative Identities G.3 Integrated Derivative Bounds G.4 Euler Increment Bounds G.5 Completion of the KL Sampling Bound G.6 Wasserstein Conversion to the Unsmoothed Target H Coupling Construction and Replacement H.1 The Learned EOT Kernel H.2 Finite-Sample Target Approximation I Proof of Proposition I.1 Velocity-Field Bound I.1.1 Velocity Network Class and Approximation I.1.2 Completion of the Velocity Bound I.2 Score-Field Bound I.2.1 Score Loss and Exact Risk Identity I.2.2 Score Network Class and Approximation I.2.3 Completion of the Score Bound J Proof of Theorem 1 Introduction Counterfactual generation seeks to model the distribution of outcomes that would arise if a treatment were assigned throughout a target population. The full distribution reveals heterogeneity, variability, and tail behavior that an average alone cannot capture. With observational data, however, treatment assignment may depend on background covariates. Individuals who receive treatment a can therefore differ systematically from the target population, so a generator trained directly on their outcomes may reproduce this selection bias. Concretely, we observe data (C,A,Y)(C,A,Y), where C denotes background covariates, A is the treatment, and Y∈ℝdY ^d is the outcome. Fix a treatment value a. The observed and target outcome distributions average the same conditional outcome law over different covariate distributions: ℙY|A=a=∫ℙY∣C=c,A=adℙC|A=a(c)⏟Observed outcome distributionand∫ℙY∣C=c,A=adℙC(c)⏟Target outcome distribution. P_Y A=a= _Y C=c,A=a\,dP_C A=a(c)_Observed outcome distribution _Y C=c,A=a\,dP_C(c)_Target outcome distribution. The observed law weights covariates as they occur among individuals who received treatment a; the target instead retains the covariate distribution of the full population. Under overlap, this target is identified from the observed-data law. If consistency and conditional exchangeability also hold, it coincides with the interventional outcome distribution under treatment a (Hernán and Robins, 2020; Dance et al., 2026). This structure suggests transporting observed outcomes to the target rather than generating the target from an unrelated reference distribution. Given a coupled source–target pair (Y0,Y1)(Y_0,Y_1), flow matching learns a time-dependent velocity field by regressing the displacement Y1−Y0Y_1-Y_0 along paths between the paired endpoints. The coupling determines which source and target outcomes are connected. Structured choices, such as optimal-transport couplings, can shorten these displacements and facilitate flow learning (Pooladian et al., 2023; Tong et al., 2024; Liu et al., 2023). Counterfactual flow matching nevertheless poses three linked problems: the target-dependent training loss must be estimated from observational data, the source–target coupling must itself be learned, and the learned dynamics must be simulated accurately with finitely many steps. Recent work primarily addresses the first problem through doubly robust training. DoubleGen (Luedtke and Fukumizu, 2025) combines a prediction oracle ℙ^Y∣C=c,A=a P_Y C=c,A=a for ℙY∣C=c,A=aP_Y C=c,A=a with an estimate of the inverse propensity αa(c)=1/ℙ(A=a∣C=c) _a(c)=1/P(A=a C=c). Oracle samples provide conditional training losses, while observed treatment-a outcomes provide inverse-propensity-weighted residual corrections. This construction trains a generator directly for the counterfactual target. In a complementary approach, Dance et al. (2026) use ℙY|A=aP_Y A=a as the flow source and derive a debiased objective that accounts for its dependence on the observed-data law. They also extend the construction to learned optimal-transport couplings. Appendix B.1 gives a detailed comparison with their method. Our focus is the joint effect of coupling estimation, field estimation, and finite-step simulation. Existing theory relates flow straightness to finite-step Wasserstein error (Bansal et al., 2026; Liu et al., 2023), but straightness is a property of the induced flow and cannot be read directly from the selected coupling. Other analyses rely on global Lipschitz bounds (Zhou and Liu, 2025; Liu et al., 2026; Guan et al., 2026b), which reflect worst-case regularity over the state space. Neither approach yields a guarantee controlled directly by the source–target displacement under the chosen coupling. Appendix B provides a broader comparison with related finite-step sampling results. Examples 1 and 2 in Section 2.4 isolate the two resulting obstacles. Even with fixed Gaussian smoothing, Example 1 constructs couplings whose displacement approaches zero while the velocity derivative diverges and the Euler ODE sampler has infinite KL error. Without smoothing, Example 2 shows that arbitrarily small displacement does not control the integrated squared derivative of the score. Together, the examples motivate stochastic sampling, positive smoothing, and regularity control averaged along the interpolation rather than uniformly over the state space. We address these obstacles with a Gaussian-smoothed interpolation and a score-corrected SDE. With exact velocity and score fields, the SDE preserves the interpolation marginals and reaches the smoothed target. We estimate the propensity model, prediction oracle, and entropic optimal-transport coupling on an auxiliary data split, then train the velocity and score fields on an independent split using doubly robust objectives. Our analysis combines displacement-dependent discretization control with finite-sample field bounds, separating smoothing, coupling, estimation, and numerical errors in an end-to-end guarantee. Our main contributions are as follows. • Score-Corrected Stochastic Sampling. We analyze a score-corrected SDE for a Gaussian-smoothed interpolation with constant smoothing. The two counterexamples delineate the respective roles of stochasticity and positive smoothing in obtaining coupling-dependent finite-step control. • Coupling-Sensitive Discretization Theory. For constant-step Euler discretization, we prove a KL error bound controlled by D5=(‖Y1−Y0‖5)1/5, D_5= (E\|Y_1-Y_0\|^5 )^1/5, (1) rather than by global uniform regularity. The fifth moment is the highest displacement moment required by our derivative bounds; its empirical analogue is a coupling diagnostic. With exact fields and the prescribed noise tuning, Euler-step complexity is near-linear in dimension for fixed smoothing, D5D_5, and target accuracy. • Finite-Sample Learning and End-to-End Guarantees. We derive finite-sample bounds for sample-split, doubly robust velocity and score estimators when the conditional outcome model, propensity model, and source–target coupling are learned from data. The bounds separate neural approximation, coupling replacement, nuisance estimation, generalization, and Monte Carlo errors. Combined with the sampling analysis, they yield an end-to-end W1W_1 guarantee relative to the unsmoothed counterfactual target. • Empirical Validation. Experiments on synthetic and semi-synthetic image benchmarks support the predicted dependence on coupling quality and show that, at finite discretization budgets, the score-corrected SDE can outperform the corresponding deterministic ODE sampler. 2 Problem Setup and Method This section develops our counterfactual flow-matching procedure. We first define the Gaussian-smoothed interpolation and its population fields, then construct the learned source–target coupling and doubly robust field objectives, and finally introduce the score-corrected SDE sampler. 2.1 Gaussian-Smoothed Flow Matching Given a coupled source–target pair (Y0,Y1)(Y_0,Y_1), let IdI_d denote the d×d× d identity matrix, and let G∼(0,Id)G (0,I_d) be independent of (Y0,Y1)(Y_0,Y_1). For t∈[0,1]t∈[0,1], define St=Y0+t(Y1−Y0),Xt=St+σG,S_t=Y_0+t(Y_1-Y_0), X_t=S_t+σ G, where σ>0σ>0 is constant in time. This Gaussian-smoothed interpolation is a special case of the stochastic interpolant framework (Albergo et al., 2025; Liu et al., 2026). Write qtq_t for the density of XtX_t. Its endpoint distributions are those of Y0+σGY_0+σ G and Y1+σGY_1+σ G; thus, the terminal population law is a Gaussian-smoothed approximation to the target. For t∼Unif[0,1]t [0,1], independently of (Y0,Y1,G)(Y_0,Y_1,G), the population flow-matching loss and its minimizer are ℒFM(v)=[‖v(Xt,t)−(Y1−Y0)‖2],vt(x):=v⋆(x,t)=[Y1−Y0∣Xt=x].L_FM(v)=E\! [ \|v(X_t,t)-(Y_1-Y_0) \|^2 ], v_t(x):=v (x,t)=E[Y_1-Y_0 X_t=x]. Gaussian smoothing gives the score as the population minimizer of the score-matching loss: ℒscore(s)=[‖s(Xt,t)+Gσ‖2],st(x):=s⋆(x,t)=∇xlogqt(x)=[−Gσ|Xt=x].L_score(s)=E\! [ \|s(X_t,t)+ Gσ \|^2 ], s_t(x):=s (x,t)= _x q_t(x)=E\! [- Gσ\, |\,X_t=x ]. Appendix D.2 verifies the score identity. Under suitable regularity conditions, the ODE with velocity vtv_t, initialized from q0q_0, has marginal density qtq_t. Section 2.4 introduces stochastic dynamics that preserve the same marginal path. 2.2 Sample Splitting and Learned Coupling For a fixed treatment value a, we observe 2n2n independent samples from the observed-data law and split them into tr=(Ci,Ai,Yi)i=1n,D_tr=\(C_i,A_i,Y_i)\_i=1^n, and aux=(Ci,Ai,Yi)i=n+12n.D_aux=\(C_i,A_i,Y_i)\_i=n+1^2n. The auxiliary split is used to estimate the propensity ℙ^(A=a∣C=c) P(A=a C=c), prediction oracle ℙ^Y∣C=c,A=a P_Y C=c,A=a, and coupling kernel Q^n(dy0∣y1) Q_n(dy_0 y_1). The velocity and score fields are then trained on trD_tr. Conditioning on the auxiliary split fixes these estimated objects during the training analysis. The empirical source distribution is ℙ^Y|A=aaux=∑i=n+12nAi=aδYi∑i=n+12nAi=a. P_Y A=a^aux= _i=n+1^2n1\A_i=a\ _Y_i _i=n+1^2n1\A_i=a\. This distribution is defined when the auxiliary split contains at least one treatment-a observation. The sampler is initialized from its Gaussian convolution: ℙ^Y|A=aaux∗(0,σ2Id). P_Y A=a^aux*N(0,σ^2I_d). We approximate the target distribution using ℙ^Y∣C=c,A=a P_Y C=c,A=a, implemented as a conditional flow trained on auxD_aux; see Appendix D.1. For each auxiliary covariate CiC_i, we draw MQM_Q independent oracle samples Y^i,1Q,…,Y^i,MQQ Y_i,1^Q,…, Y_i,M_Q^Q, yielding ℙ^Y1Q=1nMQ∑i=n+12n∑r=1MQδY^i,rQ. P_Y_1^Q= 1nM_Q _i=n+1^2n _r=1^M_Q _ Y_i,r^Q. We construct an entropic optimal-transport (EOT) coupling with regularization parameter εOT>0 _OT>0 between ℙ^Y|A=aaux P_Y A=a^aux and ℙ^Y1Q P_Y_1^Q. Disintegration with respect to the target coordinate gives the kernel Q^n(dy0∣y1) Q_n(dy_0 y_1); its extension to target queries outside the finite Monte Carlo sample is described in Appendix H. The prediction-based target is used only to construct this coupling kernel. The field objectives use the doubly robust correction described next. 2.3 Doubly Robust Field Training With the auxiliary objects fixed, a training loss is constructed by pairing each target endpoint y1y_1 with y0∼Q^n(⋅∣y1)y_0 Q_n(· y_1), and then sampling the interpolation time and Gaussian noise. For each training covariate, let M≥1M≥ 1 be the number of independent prediction-oracle draws used to approximate the plug-in loss. Let ℓ^iaux(v) _i^aux(v) denote the average flow-matching loss using oracle-generated outcomes at CiC_i, and let ℓ^iobs(v) _i^obs(v) denote the corresponding loss using the observed outcome YiY_i when Ai=aA_i=a. For a minibatch ℬ⊆1,…,nB \1,…,n\, the loss is L^ℬ,DR(v)=1|ℬ|∑i∈ℬ[ℓ^iaux(v)+Ai=aℙ^(A=a∣C=Ci)ℓ^iobs(v)−ℓ^iaux(v)]. L_B,DR(v)= 1|B| _i [ _i^aux(v)+ 1\A_i=a\ P(A=a C=C_i) \ _i^obs(v)- _i^aux(v) \ ]. The first term is the prediction-oracle plug-in loss. Observed treatment-a outcomes provide an inverse-propensity-weighted correction to its residual error. We train the score using the same construction with ℒFML_FM replaced by ℒscoreL_score. Algorithm 2 gives the complete minibatch procedure. 2.4 Score-Corrected SDE Sampler Let N≥1N≥ 1, h=1/Nh=1/N, and recall the coupling-displacement moment Equation (1). We seek a finite-step error bound controlled by D5D_5, rather than by global uniform regularity, which need not follow from small coupling displacement (Zhou and Liu, 2025; Liu et al., 2026; Guan et al., 2026b; Guan et al., 2026a; Stéphanovitch, 2026). Examples 1 and 2 isolate the two resulting obstacles: stochasticity addresses the deterministic Euler obstruction, whereas positive smoothing provides the score regularity needed for displacement-dependent control. Let X^NODE X_N^ODE denote the terminal constant-step Euler iterate using the exact velocity vtv_t, initialized from q0q_0. The following example shows that arbitrarily small D5D_5 guarantees neither small uniform velocity regularity nor finite KL error for deterministic Euler sampling. Example 1. Let p∈(0,1)p∈(0,1), and let B∼Bernoulli(p)B (p) and U∼(0,1)U (0,1) be independent. Define (Y0,Y1)=(U,U),B=0,(2U,2(1−λ)U),B=1,λ=4+σ24h.(Y_0,Y_1)= cases(U,U),&B=0,\\ (2U,2(1-λ)U),&B=1, cases λ= 4+σ^24h. As p→0p→ 0, D5→0,supt∈[0,1],x∈ℝ|∂xvt(x)|→∞,but∫01|∂xvt(Xt)|2dt→0.D_5→ 0, _t∈[0,1],\,x | _xv_t(x)|→∞, _0^1E| _xv_t(X_t)|^2\,dt→ 0. Moreover, for every p∈(0,1)p∈(0,1), we have KL(ℒ(X1)∥ℒ(X^NODE))=+∞.KL(L(X_1)\|L( X_N^ODE))=+∞. Appendix E proves the claim. It motivates controlling derivatives averaged along the interpolation rather than uniformly over the state space. Following the Itô-based approach of Xu et al. (2026); Benton et al. (2024), we therefore use a stochastic sampler for which this integrated regularity is sufficient. For τ>0τ>0, the exact-field sampling process is dZt=(vt(Zt)+τ22st(Zt))dt+τdBt,Z0∼q0,dZ_t= (v_t(Z_t)+ τ^22s_t(Z_t) )dt+τ\,dB_t, Z_0 q_0, where BtB_t is a standard Brownian motion. Under the conditions of Lemma 1, ZtZ_t has density qtq_t for every t∈[0,1]t∈[0,1]. Thus the score correction preserves the interpolation’s marginal path while introducing stochasticity. Applying Itô’s formula to vtv_t and sts_t along ZtZ_t reduces the analysis to integrated derivative bounds, which Gaussian smoothing controls through moments of Y1−Y0Y_1-Y_0. Initialize the learned sampler according to Z^0∼ℙ^Y|A=aaux∗(0,σ2Id). Z_0 P_Y A=a^aux*N(0,σ^2I_d). For tk=kht_k=kh, the learned Euler sampler is Z^k+1=Z^k+h(v^(Z^k,tk)+τ22s^(Z^k,tk))+τhGk,Gk∼(0,Id), Z_k+1= Z_k+h ( v( Z_k,t_k)+ τ^22 s( Z_k,t_k) )+τ h\,G_k, G_k (0,I_d), where the GkG_k are mutually independent and independent of the initialization. Since tN=Nh=1t_N=Nh=1, the terminal Euler output Z^N Z_N is compared with ZtN=Z1Z_t_N=Z_1. Algorithm 1 gives the complete procedure. Setting τ=0τ=0 recovers deterministic Euler sampling, although our KL analysis requires τ>0τ>0. Appendix B discusses related stochastic samplers and finite-step comparisons. Gaussian smoothing is essential for controlling the required score regularity through D5D_5. The following example shows that this control fails in general when σ=0σ=0. Example 2. Fix ε>0 >0, set δ=21/5εδ=2^1/5 , and let Y0∼(0,1)Y_0 (0,1). For each integer m≥0m≥ 0, define Km=(2m+1)πδ,pm(x)=14+18sin(Kmx).K_m= (2m+1)πδ, p_m(x)= 14+ 18 (K_mx). Conditional on Y0=xY_0=x, let Y1=x+δ,with probability pm(x),x−δ,with probability pm(x),x,with probability 1−2pm(x).Y_1= casesx+δ,&with probability p_m(x),\\ x-δ,&with probability p_m(x),\\ x,&with probability 1-2p_m(x). cases For the unsmoothed interpolation Yt=(1−t)Y0+tY1Y_t=(1-t)Y_0+tY_1, let qt(m)q_t^(m) be its density and st(m)=∂xlogqt(m)s_t^(m)= _x q_t^(m); the dependence of YtY_t on m is suppressed in the notation.. Every coupling in this sequence satisfies D5=εD_5= , but ∫01[|∂xst(m)(Yt)|2]t≥3532768Km4(1−e−2Km2)4⟶+∞. _0^1E\! [| _xs_t^(m)(Y_t)|^2 ]dt≥ 3532768K_m^4 (1-e^-2K_m^2 )^4 +∞. Thus, even arbitrarily small D5D_5 does not uniformly control the integrated score regularity without smoothing. 3 Coupling-Dependent Sampling and End-to-End Guarantees This section proves a coupling-dependent finite-step sampling bound, derives finite-sample guarantees for the doubly robust velocity and score estimators, and combines them into an end-to-end sample-complexity result. Unless otherwise stated, ≲ suppresses universal numerical constants and may also hide logarithmic factors in the displayed parameters. A complete notation list is provided in Appendix C.1. 3.1 Coupling-Dependent Finite-Step Sampling For tk=kht_k=kh, where h=1/Nh=1/N (so tN=1t_N=1), write v^tk(x)=v^(x,tk) v_t_k(x)= v(x,t_k) and s^tk(x)=s^(x,tk) s_t_k(x)= s(x,t_k). We use the Standard Sampling Conditions in Appendix G. For σ,τ>0σ,τ>0 and D5<∞D_5<∞, Lemma 1 provides a jointly continuous exact drift and a weak solution Z with ℒ(Zt)=qtL(Z_t)=q_t; throughout the sampling analysis, we use this solution. The following Assumption 1 defines εvel2 _vel^2 and εscore2 _score^2 as the grid-averaged mean-squared differences between the learned and population fields evaluated at ZtkZ_t_k, and assumes both are finite. This is standard in related sampling analyses (Liu et al., 2025; Liu et al., 2026; Li et al., 2025; Benton et al., 2024). Section 3.2 later provides finite-sample conditions that verify this assumption. Assumption 1 (Integrated Estimation Error). Assume εvel2:=h∑k=0N−1[‖vtk(Ztk)−v^tk(Ztk)‖2]<∞,εscore2:=h∑k=0N−1[‖stk(Ztk)−s^tk(Ztk)‖2]<∞. _vel^2:=h _k=0^N-1E\! [\|v_t_k(Z_t_k)- v_t_k(Z_t_k)\|^2 ]<∞, _score^2:=h _k=0^N-1E\! [\|s_t_k(Z_t_k)- s_t_k(Z_t_k)\|^2 ]<∞. Theorem 1 (KL Sampling Bound). Let N,d≥1N,d≥ 1, h=1/Nh=1/N, σ,τ>0σ,τ>0, and D5∈(0,∞)D_5∈(0,∞). Under Standard Sampling Conditions and Assumption 1, define Ah A_h =4(9d+51)3h2D54σ2+2εvel2,Bh=1243h2(d+6)3D52σ4+12εscore2. = 4(9d+51)3 h^2D_5^4σ^2+2 _vel^2, B_h= 1243 h^2(d+6)^3D_5^2σ^4+ 12 _score^2. Then KL(ℒ(ZtN)∥ℒ(Z^N))≤Ahτ2+Bhτ2+hd(d+3)σ4τ4+68h2(d+6)33σ6τ6+2h(d+1)D52σ2. (L(Z_t_N)\,\|\,L( Z_N) )≤ A_hτ^2+B_hτ^2+ hd(d+3)σ^4τ^4+ 68h^2(d+6)^33σ^6τ^6+ 2h(d+1)D_5^2σ^2. (2) The first two terms are balanced by τ=(Ah/Bh)1/4τ=(A_h/B_h)^1/4. Remark 1 (Coupling-Dependent Scaling). In the exact-field case εvel=εscore=0 _vel= _score=0, choosing τ2=σD5/dτ^2=σ D_5/d gives KL(ℒ(ZtN)∥ℒ(Z^N))≲hdσ2D52+h2d2σ3D53. (L(Z_t_N)\,\|\,L( Z_N) ) hdσ^2D_5^2+ h^2d^2σ^3D_5^3. (3) Thus, for any εtarget∈(0,1) _target∈(0,1), h≲1dD52/(εtarget2σ2)+dD5/σ h 1dD_5^2/( _target^2σ^2)+dD_5/σ gives KL error of order εtarget2 _target^2, corresponding to ~(d[D52εtarget2σ2+D5σ]) O (d [ D_5^2 _target^2σ^2+ D_5σ ] ) Euler steps. Recall that Example 2 motivates the importance of smoothing. Compared with bounds for constant-step Euler discretizations of stochastic-interpolant SDE and ODE samplers in (Liu et al., 2025; Liu et al., 2026), our result improves the dimension dependence from quadratic to near-linear; Appendix B gives a detailed comparison. This dimension comparison holds with smoothing, displacement moment, and target accuracy fixed. 3.2 Finite-Sample Field Estimation and End-to-End Guarantee We now give sufficient conditions for Assumption 1. Below, d is fixed, so implicit constants may depend on d in addition to the logarithmic factors suppressed by ≲ . Assumption 2 (Clipping, Overlap, and Bounded Support). We work with outcomes clipped to [−1,1]d[-1,1]^d and assume that, for every c∈supp(ℙC)c (P_C), ℙY∣C=c,A=a([−1,1]d)=ℙ^Y∣C=c,A=a([−1,1]d)=1,ℙ(A=a∣C=c)≥πmin>0,0≤α^(c)≤2πmin−1. gatheredP_Y C=c,A=a([-1,1]^d)= P_Y C=c,A=a([-1,1]^d)=1,\\[-2.0pt] P(A=a C=c)≥ _ >0, 0≤ α(c)≤ 2 _ ^-1. gathered Here α^(c)=1/ℙ^(A=a∣C=c) α(c)=1/ P(A=a C=c). Thus every endpoint pair satisfies Y0,Y1∈[−1,1]dY_0,Y_1∈[-1,1]^d almost surely, and Q^n([−1,1]d∣y)=1 Q_n([-1,1]^d y)=1 for every y∈ℝdy ^d. Assumption 3 (Propensity Accuracy). Let αa(c)=1/ℙ(A=a∣C=c) _a(c)=1/P(A=a C=c). Conditional on the auxiliary sample used to fit α α, assume ‖α^/αa−1‖L2(ℙC)≤εα. \| α/ _a-1 \|_L^2(P_C)≤ _α. Assumption 2 bounds outcomes, oracle-generated samples, and inverse-propensity weights while enforcing overlap. Assumption 3 requires relative L2(ℙC)L^2(P_C) accuracy of the estimated inverse propensity. Assumption 4 (Prediction-Oracle Accuracy). Define εpred=[(εpred,train)2+εpred,gen2]1/2 _pred=[( _pred,train)^2+ _pred,gen^2]^1/2, and assume [CW1(ℙY∣C=c,A=a,ℙ^Y∣C=c,A=a)2]1/2≤εpred. [E_CW_1\! (P_Y C=c,A=a, P_Y C=c,A=a )^2 ]^1/2≤ _pred. Assumption 4 controls the oracle’s average conditional W1W_1 error over population covariates. Appendix D.1 derives this requirement from empirical-fit and generalization conditions and describes conditional-flow training. Let μ1:=∫ℙY∣C=c,A=adℙC(c). _1:= _Y C=c,A=a\,dP_C(c). For any coupling kernel Q(dy0∣y1)Q(dy_0 y_1), define the joint endpoint law Γμ1,Q(dy0,dy1):=Q(dy0∣y1)μ1(dy1). _ _1,Q(dy_0,dy_1):=Q(dy_0 y_1) _1(dy_1). To emphasize the role of coupling explicitly, under (Y0,Y1)∼Γμ1,Q(Y_0,Y_1) _ _1,Q, we let qt,Qq_t,Q denote the density of Xt=Y0+t(Y1−Y0)+σGX_t=Y_0+t(Y_1-Y_0)+σ G, and define vQ⋆(x,t):=[Y1−Y0∣Xt=x],sQ⋆(x,t):=∇xlogqt,Q(x).v_Q (x,t):=E[Y_1-Y_0 X_t=x], s_Q (x,t):= _x q_t,Q(x). Population Coupling. Let ℰa=∑i=n+12nAi=a≥1E_a=\ _i=n+1^2n1\A_i=a\≥ 1\. On ℰaE_a, optimally couple Y1∼μ1Y_1 _1 and Y^1∼ℙ^Y1Q Y_1 P_Y_1^Q in W1W_1, and then draw Y0∼Q^n(⋅∣Y^1)Y_0 Q_n(· Y_1). We define Q(⋅∣Y1)Q(· Y_1) as the resulting conditional law of Y0Y_0 given Y1Y_1. Thus Γμ1,Q _ _1,Q has target marginal μ1 _1 and source marginal ℙ^Y|A=aaux P_Y A=a^aux. Lemma 10 formalizes this construction and proves the coupling-replacement bound used below. Let nV_n denote the clipped coordinatewise sparse-ReLU velocity-field class with input-clipping radius R and approximation tolerance ε , as defined in Appendix I.1, and let nS_n denote the score-field class induced by clipped sparse-ReLU denoisers through su(x,t)=u(x,t)−x/σ2s_u(x,t)=\u(x,t)-x\/σ^2, as defined in Appendix I.2. The next proposition gives explicit finite-sample bounds for the velocity- and score-field components of Assumption 1. If Assumption 7, which ensures that the auxiliary split contains at least one treatment-a observation, is added to the hypotheses of Proposition 1, then ℰaE_a holds by assumption and the term e−nπmine^-n _ can be omitted from the failure probability. Proposition 1 (Doubly Robust Velocity and Score Field Error Bound). Let n≥2n≥ 2, M,MQ≥1M,M_Q≥ 1, and 0<σ≤10<σ≤ 1. Under Assumptions 2, 3, 4, let Q be the Population Coupling defined above. Let εpred,train≤εpred _pred,train≤ _pred denote the empirical generator accuracy for prediction oracle, formally defined in Assumption 5. Define :=εOT−1σ2[n−1/(d+2)+εpred,train]+1+εOT−1σ2εαεpred.A:= _OT^-1σ^2 [n^-1/(d+2)+ _pred,train ]+ 1+ _OT^-1σ^2 _α _pred. Then, with probability at least 1−(n−1)−e−nπmin1-O(n^-1)-e^-n _ over the data and Monte Carlo draws, ℰaE_a occurs and the following bound holds: 1. For velocity field, Let v^M v_M be the exact empirical minimizer over nV_n, with ε=n−1/(d+5) =n^-1/(d+5) and R=1+σ[2log(32d2/ε2)]1/2R=1+σ[2 (32d^2/ ^2)]^1/2. 1N∑k=0N−1[‖v^M(Xtk,tk)−vQ⋆(Xtk,tk)‖2]≲ 1N _k=0^N-1E\! [\| v_M(X_t_k,t_k)-v_Q (X_t_k,t_k)\|^2 ] +1+2πmin−1σd+1n−2/(d+5). + 1+2 _ ^-1σ^d+1n^-2/(d+5). (4) 2. For score field, let s^M s_M be the exact empirical minimizer over nS_n, with ε=n−1/(d+5) =n^-1/(d+5) and R=1+2σ[log(8d2/(σ4ε2))]1/2R=1+ 2σ[ (8d^2/(σ^4 ^2))]^1/2. 1N∑k=0N−1[‖s^M(Xtk,tk)−sQ⋆(Xtk,tk)‖2]≲ 1N _k=0^N-1E\! [\| s_M(X_t_k,t_k)-s_Q (X_t_k,t_k)\|^2 ] σ−4+1+2πmin−1σ2d+6n−2/(d+5). σ^-4A+ 1+2 _ ^-1σ^2d+6n^-2/(d+5). (5) In Proposition 1, the finite-M and finite-MQM_Q errors are dominated by the displayed n-dependent terms, up to logarithmic factors. Verification of Assumption 1. Set v^tk=v^M(⋅,tk) v_t_k= v_M(·,t_k) and s^tk=s^M(⋅,tk) s_t_k= s_M(·,t_k). For the interpolation induced by Q, vt=vQ⋆(⋅,t)v_t=v_Q (·,t) and st=sQ⋆(⋅,t)s_t=s_Q (·,t), while the interpolation variable XtkX_t_k and the exact SDE state ZtkZ_t_k both have density qtk=qtk,Qq_t_k=q_t_k,Q. Hence the expectations in the two field bounds equal the corresponding expectations in Assumption 1. Since h=1/Nh=1/N, the two displayed grid averages are exactly εvel2 _vel^2 and εscore2 _score^2, respectively. In each field bound, the three terms respectively capture coupling replacement, doubly robust nuisance error, and final-field estimation. The first vanishes for a coupling fixed independently of the prediction-based target, while the second is small when either nuisance model is accurate. Substituting both bounds into Theorem 1 controls finite-step KL error relative to the smoothed target. Corollary 1 then adds the smoothing bias and converts this control into a W1W_1 guarantee for the original target, yielding the following theorem. Theorem 2 (End-to-End Sample Complexity With Negligible Nuisance Error). Fix d≥1d≥ 1, πmin>0 _ >0, and εOT>0 _OT>0. Assume the conditions of Proposition 1. Let μ1=∫ℙY∣C=c,A=adℙC(c) _1= _Y C=c,A=a\,dP_C(c) and let D¯5>0 D_5>0 be a deterministic coupling-displacement envelope such that, almost surely on the event where Q is defined, (Γμ1,Q‖Y1−Y0‖5)1/5≤D¯5. (E_ _ _1,Q\|Y_1-Y_0\|^5 )^1/5≤ D_5. Suppose εpred,train _pred,train and εαεpred _α _pred are negligible relative to the explicit n-dependent terms. For εtarget∈(0,1) _target∈(0,1), choose 0<σ≤10<σ≤ 1 with σ=Θ(εtarget)σ= ( _target), set τ2=σD¯5/dτ^2=σ D_5/d, and choose N≥1N≥ 1 so that h=1/N≤chdD¯52/(εtarget4σ2)+dD¯5/σ, h=1/N≤ c_hd D_5^2/( _target^4σ^2)+d D_5/σ, where ch≤2c_h≤ 2 is a scaling constant. Then, up to logarithmic factors, it suffices to take n≳max n \ [max2+εOT−1D5¯εtarget−7,D5¯εOTεtarget−9]d+2, [ \ 2+ _OT^-1 D_5 _target^-7, D_5 _OT _target^-9 \ ]^d+2, [(1+2πmin−1)maxD5¯−1εtarget−(d+6),D5¯εtarget−(2d+9)](d+5)2. [(1+2 _ ^-1) \ D_5^-1 _target^-(d+6), D_5 _target^-(2d+9) \ ] (d+5)2 \. This yields W1(μ1,ℒ(Z^N))≲εtargetW_1( _1,L( Z_N)) _target with probability at least 1−(n−1)−e−nπmin1-O(n^-1)-e^-n _ . This gives an end-to-end guarantee for counterfactual flow matching. Under Assumption 2, we may clip the final generated sample coordinatewise to [−1,1][-1,1]. This post-processing won’t increase its W1W_1 distance to the target and gives a sharper bound in W1W_1; see Corollary 2. Although Theorem 1 has near-linear dimension dependence, the end-to-end sample complexity still suffers from the curse of dimensionality, as expected under ambient-dimensional nonparametric assumptions and consistent with related flow-matching and diffusion results (Zhou and Liu, 2025; Su et al., 2025; Wibisono et al., 2024; Cai and Li, 2025). 4 Experiments We first isolate the effects of coupling and discretization in two dimensions, then evaluate finite-step generation on two semi-synthetic image benchmarks. 4.1 Angular Mode Reallocation We use a two-dimensional mixture of eight Gaussians arranged on a circle to isolate coupling effects. The observed source places total mass 3/43/4 on the axis-aligned modes and 1/41/4 on the diagonal modes, while the counterfactual target reverses these masses. We compare IID and entropic optimal-transport couplings across regularization levels and use the resulting exact empirical velocity and score fields as references for the learned models. Figures 1(a) and 1(b) show that smaller D5D_5 generally corresponds to smaller field-estimation error. Appendix A gives the complete mixture construction and experimental protocol. The remaining panels isolate finite-step sampling error using exact fields. Figure 1(c) shows how the 88-step KL error varies with D5D_5 when τ=0.5σD5/dτ=0.5 σ D_5/d. For fixed h, σ, and d, Theorem 1 predicts the coupling-dependent order KL(ℒ(ZtN)∥ℒ(Z^N))≲D52+D53KL(L(Z_t_N)\|L( Z_N)) D_5^2+D_5^3, consistent with the empirical trend. Figure 1(d) gives a complementary mode-mass view: with 44 Euler steps, both the ODE and SDE samplers substantially misallocate mass, whereas with 6464 steps both closely match the target and assign little mass outside the mode regions. 4.2 Image Benchmarks We evaluate finite-step counterfactual generation on two semi-synthetic colored-image benchmarks: MNIST digits 11 and 66, adapted from Dance et al. (2026), and Fashion-MNIST sneakers and ankle boots. In both tasks, C∼Unif(0,1)C (0,1) controls image color, A determines the class, and treatment assignment follows ℙ(A=1∣C=c)=exp5(c−12)/(1+exp5(c−12))P(A=1 C=c)= \5(c- 12)\/(1+ \5(c- 12)\). This selection mechanism biases the observed A=1A=1 images toward larger values of C, whereas C remains uniform under do(A=1)do(A=1); the clean counterfactual target can therefore be sampled directly. We vary the SDE noise level τ, compare with the corresponding τ=0τ=0 ODE and a pure σ=0σ=0 ODE baseline, and use 5050 Euler steps throughout. We report sliced Wasserstein-2 (SW2) distance to the clean target. Appendix A.2 gives the complete data construction, training, and evaluation protocols. Digit ColorMNIST. Here A=0A=0 denotes digit 11, A=1A=1 denotes digit 66, and the observational sample size is n=20,000n=20,000. We evaluate smoothing levels σ∈0.01,0.015σ∈\0.01,0.015\; smaller σ reduces smoothing bias and places the smoothed target closer to the clean target. Figure 3 shows that, for both values of σ, some τ>0τ>0 improves on the corresponding τ=0τ=0 ODE, and the best results also outperform the pure σ=0σ=0 baseline. For σ=0.01σ=0.01, performance is best near τ=0.075τ=0.075. Along the corresponding SDE trajectory, the color-distribution KS statistic decreases from 0.2660.266 at t=0t=0 to 0.0790.079 at t=1t=1. Fashion-MNIST. Here A=0A=0 denotes sneakers and A=1A=1 denotes ankle boots. We evaluate σ∈0.005,0.01σ∈\0.005,0.01\. As shown in Figure 3, the benefit of stochastic sampling depends strongly on the smoothing level. For σ=0.005σ=0.005, performance improves with τ, is best near τ=0.02τ=0.02, and clearly outperforms the pure ODE baseline before deteriorating for larger τ. For σ=0.01σ=0.01, varying τ yields no clear improvement over the pure ODE baseline within the reported variation, possibly because of the larger smoothing error. Along the σ=0.005σ=0.005 trajectory, the red–blue color-statistic KS value decreases from 0.2600.260 at t=0t=0 to 0.0630.063 at t=1t=1. Overall, σ=0.005σ=0.005 performs substantially better, consistent with its smaller smoothing error. Qualitative samples are provided in Appendix A.2. (a) εvel2 _vel^2 (b) εscore2 _score^2 (c) KL vs. D5D_5 (d) Mode mass Figure 1: Angular-mode experiment. (a)–(b) Velocity and score errors across couplings (σ=0.12σ=0.12). (c) Exact-field 88-step KL versus D5D_5, using 2,0002,000 samples and τ=0.5σD5/dτ=0.5 σ D_5/d. (d) Mode-mass allocation for 44 and 6464 steps (EOT εOT=0.5 _OT=0.5, σ=0.02σ=0.02): 44-step ODE/SDE misallocate mass, while 6464-step ODE/SDE match the target modes. (a) SW2 (b) t=0t=0 (c) t=0.5t=0.5 (d) t=1t=1 Figure 2: ColorMNIST 1–6. (a) Clean-target SW2 for σ∈0.01,0.015σ∈\0.01,0.015\ (5,0005,000 samples, five seeds; dashed: pure ODE). (b)–(d) Color evolution along the σ=0.01σ=0.01 SDE at t=0,0.5,1t=0,0.5,1 (5,0005,000 samples; dashed: clean target). (a) SW2 (b) t=0t=0 (c) t=0.5t=0.5 (d) t=1t=1 Figure 3: Fashion-MNIST shoes. (a) Clean-target SW2 versus τ for σ∈0.005,0.01σ∈\0.005,0.01\ (5,0005,000 samples, five seeds; dashed: pure ODE). (b)–(d) Color evolution along the σ=0.005σ=0.005 SDE at t=0,0.5,1t=0,0.5,1 (5,0005,000 samples; dashed: clean target). 5 Conclusion We developed a counterfactual flow-matching framework that combines doubly robust field estimation, learned source–target couplings, Gaussian smoothing, and score-corrected stochastic sampling. We prove a coupling-sensitive finite-step guarantee in which the Euler sampling error depends explicitly on the source–target displacement under the chosen coupling. Separately, our finite-sample analysis separates coupling replacement, nuisance estimation, and statistical errors. Together, these results yield an end-to-end guarantee relative to the original counterfactual target. Synthetic and image experiments show that stochastic sampling can improve over the corresponding ODE at fixed step budgets, with performance depending on smoothing and SDE noise. AI Use Statement In this work, we used Generative AI for formulate mathematical claims (e.g., finding counter examples), provide critical ingredients for proving mathematical claims (e.g., carry out some calculation that were checked by the authors), assist in the writing of proofs, design or provide feedback on research methodology or experiments (e.g., asking AI whether a specific theoretical claim can be improved toward a specific direction), implement methods (e.g., write code), literature search (e.g., ask if we missed any existing literature), language editing, proof auditing, and LaTeX assistance. 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Let C∈0,…,7C∈\0,…,7\ index eight Gaussian components equally spaced on a circle, with μc=2(cos(cπ/4),sin(cπ/4))⊤,Pc=(μc,σY2I2),σY=0.08. _c=2( (cπ/4), (cπ/4)) , P_c=N( _c, _Y^2I_2), _Y=0.08. The population assigns weight 1/161/16 to each axis-aligned component c∈0,2,4,6c∈\0,2,4,6\ and 3/163/16 to each diagonal component c∈1,3,5,7c∈\1,3,5,7\. Treatment assignment satisfies ℙ(A=1∣C=c)=0.9P(A=1 C=c)=0.9 for axis-aligned components and 0.10.1 for diagonal components. Bayes’ rule therefore gives conditional weights 3/163/16 and 1/161/16, respectively, among observations with A=1A=1. Taking ℙY∣C=c,A=1=PcP_Y C=c,A=1=P_c, the observed source and counterfactual target are ℙY|A=1 _Y A=1 =∑c∈0,2,4,6316Pc+∑c∈1,3,5,7116Pc, = _c∈\0,2,4,6\ 316P_c+ _c∈\1,3,5,7\ 116P_c, ∫ℙY∣C=c,A=1dℙC(c) _Y C=c,A=1\,dP_C(c) =∑c∈0,2,4,6116Pc+∑c∈1,3,5,7316Pc. = _c∈\0,2,4,6\ 116P_c+ _c∈\1,3,5,7\ 316P_c. Thus, the two distributions share the same components but assign opposite weights to the axis-aligned and diagonal groups. Couplings and Evaluation. For fixed empirical source and target samples, we compare IID and entropic optimal-transport couplings at different regularization levels. Each coupling assigns mass to source–target pairs; after Gaussian smoothing, these weighted pairs determine the interpolation law and the exactly computable empirical fields vt(x)=[Y1−Y0∣Xt=x],st(x)=∇xlogqt(x).v_t(x)=E[Y_1-Y_0 X_t=x], s_t(x)= _x q_t(x). These exact fields serve as references for the learned models. The velocity and score experiments use three-hidden-layer MLPs of width 128128 and σ=0.12σ=0.12. To isolate discretization error, we generate 2,0002,000 exact-field samples with 88 Euler steps and estimate the generated density by kernel density estimation to compute KL divergence. We set τ=0.5σD5/dτ=0.5 σ D_5/d, following the scaling τ≍σD5/dτ σ D_5/d suggested by Theorem 1. The mode-mass experiment uses the closed-form EOT coupling with εOT=0.5 _OT=0.5 and σ=0.02σ=0.02, and compares 44- and 6464-step ODE and SDE samplers. A.2 Image-Benchmark Details and Additional Results Digit ColorMNIST. Following Dance et al. (2026), we draw C∼Unif(0,1)C (0,1) and color each MNIST image using RGB value (C,0,1−C)(C,0,1-C), with the original image intensity controlling brightness. Treatment A=0A=0 selects digit 11, A=1A=1 selects digit 66, and ℙ(A=1∣C=c)=exp5(c−12)1+exp5(c−12).P(A=1 C=c)= \5(c- 12)\1+ \5(c- 12)\. Conditional on A, an image is sampled with replacement from the corresponding MNIST class and colored according to C. We generate n=20,000n=20,000 observational samples and target a=1a=1. Because C remains uniform under do(A=1)do(A=1), the clean counterfactual target can be sampled directly. We train smoothed models with σ∈0.01,0.015σ∈\0.01,0.015\ for 1,0001,000 epochs and evaluate the SDE over a range of τ, with τ=0τ=0 corresponding to the ODE for the same smoothed model. The pure ODE baseline is the same doubly robust flow trained with σ=0σ=0; its best checkpoint through epoch 1,0001,000 occurs at epoch 700700. All samplers use 5050 Euler steps. Generated images are clipped to [0,1][0,1], and SW2 is computed against the clean target using 5,0005,000 samples over five random seeds. The baseline and smoothed models use the same fitted propensity model, conditional-flow prediction oracle, and conditional-sample reservoir. For σ=0.01σ=0.01, the epoch-10001000 model performs best near τ=0.075τ=0.075. Fashion-MNIST. We use C∼Unif(0,1)C (0,1), with A=0A=0 for sneakers and A=1A=1 for ankle boots, and use the same treatment-assignment probability as in the digit experiment. Given C=cC=c, the grayscale image is colored using (0.15+0.75c, 0.10+0.55sin2(πc), 0.85−0.65c).(0.15+0.75c,\;0.10+0.55 ^2(π c),\;0.85-0.65c). Consequently, observed ankle boots tend to have larger C, whereas the clean do(A=1)do(A=1) target contains the full population color range. We train models with σ=0.01σ=0.01 and σ=0.005σ=0.005, together with the pure ODE baseline (σ=0)(σ=0), for 400400 epochs. Checkpoints are saved every 2020 epochs. For σ=0.01σ=0.01, the selected velocity and score checkpoints are epochs 100100 and 220220; for σ=0.005σ=0.005, they are epochs 100100 and 120120. The selected pure ODE checkpoint is epoch 140140. We vary τ, and every evaluation uses 5050 Euler steps, 5,0005,000 generated samples, 256256 random projection directions, and five evaluation seeds. The color-distribution plots use the red–blue statistic x^=R/(R+B) x=R/(R+B). Figure 5 shows images before and after transport. The source contains observed ankle boots with color-selection bias; the generated samples remain realistic ankle boots while covering colors closer to the clean target population. Figures 4 and 6 provide complementary results for the shirt experiment: the former reports sampling performance and color evolution, while the latter shows that the generated samples preserve shirt structure while changing their colors toward the target distribution. (a) Performance evaluation (b) t=0t=0 (c) t=0.5t=0.5 (d) t=1t=1 Figure 4: Fashion-MNIST shirts experiment. (a) Clean-target SW2 as a function of τ for the two smoothing levels; shaded regions show variation across five evaluation seeds, and the dashed line denotes the pure ODE baseline. (b)–(d) Color distributions at t=0,0.5,1t=0,0.5,1 along the σ=0.005σ=0.005 SDE trajectory; dashed distributions denote the clean target. Figure 5: Qualitative examples from the observed source distribution (top), samples generated by the σ=0.005σ=0.005 SDE (bottom). The generated samples preserve the boots shape and structure while showing clear, coherent color changes. Figure 6: Qualitative examples from the observed source distribution (top), samples generated by the σ=0.005σ=0.005 SDE (bottom). The generated samples preserve the shirt shape and structure while showing clear, coherent color changes. Appendix B Additional Related-Work Details The closest comparisons are (Liu et al., 2025; Liu et al., 2026). For stochastic sampling, (Liu et al., 2025) considers a more general stochastic interpolant with time-varying smoothing γt _t. Under a uniform Euler discretization, their complexity has leading dimension dependence ~(d2) O(d^2) and polynomial dependence on the endpoint cutoffs t0t_0 and 1−tN1-t_N. These cutoff parameters are introduced to avoid the singular behavior near the two endpoints and play a role analogous to the fixed smoothing level σ in our analysis. In comparison, our uniform-step result achieves near-linear ~(d) O(d) dimension dependence with leading smoothing dependence of order 1/σ21/σ^2. This comparison holds with displacement moment and target accuracy fixed, using τ2=σD5/dτ^2=σ D_5/d and controlled field error; D5D_5 itself can grow with dimension. Liu et al. (2025) further considers a specially designed nonuniform time schedule, which improves the polynomial endpoint dependence (on 1/t01/t_0 and 1/(1−tN)1/(1-t_N)) to logarithmic dependence while retaining a leading ~(d2) O(d^2) dimension dependence. Thus, compared with their uniform discretization, our result improves the dimension dependence, while their nonuniform schedule achieves sharper endpoint dependence through a more specialized discretization. Although their bounds contain source–target displacement moments, they also retain coupling-independent score terms and do not develop a coupling-quality interpretation analogous to our D5D_5 control. The companion ODE analysis (Liu et al., 2026) likewise gives a ~(d2) O(d^2) dimension dependence for Euler discretization. Diffusion models admit sharper guarantees by exploiting the specific Gaussian noising structure of the diffusion probability path. Under weak distributional assumptions, (Benton et al., 2024) and (Li and Yan, 2025) obtain near-linear dependence on d for reverse-diffusion sampling. More recent results establish sublinear dependence under additional regularity. Jiao et al. (2025) introduce a non-uniform regularity parameter L and obtain dimension dependence ~(dmind,L) O( d \ d,L\) for the standard DDPM sampler. They showed L can be small for some specific class of targets, but in general the bound could still be linear in d. We remark that related works such as (Zhang et al., 2025; Li and Jiao, 2025) obtain sublinear dimension dependence using randomized midpoint schemes, rather than the Euler discretization considered here. Specifically, (Zhang et al., 2025) obtain ~(β~0d+M22) O( β_0 d+M_2^2), which reduces to ~(d) O( d) under bounded regularity constants, while (Li and Jiao, 2025) obtain ~(Ld1/3) O(Ld^1/3). These improved dimension rates depend explicitly on score regularity constants such as β~0 β_0, M2M_2, and L. Table 1 compares our sampling bound with the most relevant existing results, highlighting the dimension dependence, assumptions, and whether the bound reflects the quality of the source–target coupling. Finally, we mention two additional related works. Yang et al. (2025) extends the analysis of (Chen et al., 2023a) to the flow-matching setting using a similar predictor–corrector framework, under assumptions including L2L_2 estimation error, bounded support, and uniform regularity of the learned vector field. Su et al. (2025) also establishes a KL-divergence error bound for flow matching with a deterministic ODE sampler. However, their analysis relies on strong global-in-space regularity assumptions on the density path and velocity fields and does not analyze numerical ODE discretization error. Table 1: Comparison with related finite-step sampling results. Dimension dependence suppresses logarithmic factors and holds accuracy and the other problem parameters fixed; for our result this includes σ,D5σ,D_5, with τ2=σD5/dτ^2=σ D_5/d and controlled field error.“Partial” means that source–target displacement enters the bound, but coupling-independent score terms remain. Work Setting / Method Metric Dim. Assumption Coupling control Stochastic interpolants and flow matching This work Main theorem constant- smoothing SI SDE, Euler KL ~(d) O(d) L2L_2 error D5D_5 (Liu et al., 2025) Thm. 4.3; Sec. 5 SI SDE, Euler KL ~(d2) O(d^2) L2L_2 error Partial (Liu et al., 2026) Thm. 4.5 SI ODE, Euler TV ~(d2) O(d^2) drift/div. errors; unif. reg. No Diffusion-model baselines (Benton et al., 2024) Cor. 1 diffusion exp. integrator KL ~(d) O(d) L2L_2 error – (Li and Yan, 2025) Thm. 1 DDPM TV ~(d) O(d) L2L_2 error – (Jiao et al., 2025) Thm. 1 DDPM KL ~(dCLOSE O( d OPENmind,L) \ d,L\) weighted L2L_2 error; nonuniform reg. – B.1 Comparison With Dance et al. (2026) The main difference from Algorithm 3 of (Dance et al., 2026) is that our training loss is a direct minibatch implementation of the doubly robust loss. That is, for a fixed treatment a, the plug-in term is evaluated for every sample in the minibatch, while the observed correction is included only for samples with Ai=aA_i=a: 1|ℬ|∑i∈ℬ[ℓ^ia(v)+Ai=aπ^a(Ci)ℓiobs,a(v)−ℓ^ia(v)]. 1|B| _i [ _i^a(v)+ 1\A_i=a\ π_a(C_i) \ _i^obs,a(v)- _i^a(v) \ ]. Note that the plug-in term is averaged over the full covariate distribution and the inverse-propensity term corrects only the observed treatment-a samples. In contrast, Algorithm 3 of (Dance et al., 2026) forms, for each treatment a, a subset ℬa=i∈ℬ:Ai=aB_a=\i :A_i=a\ and accumulates a loss only over i∈ℬai _a. This is not a direct implementation of the standard DR loss: the plug-in term is then averaged over the covariate distribution of the observed treatment-a subsample rather than over the full minibatch 11 1 This description refers to Algorithm 3 as stated in the first arXiv version of (Dance et al., 2026); the discrepancy may simply reflect a typo or pseudocode-level issue rather than the intended implementation.. A minor difference is how the coupling is handled. (Dance et al., 2026) recomputes the coupling within each minibatch, whereas our analysis estimates a single kernel Q^n Q_n and uses it for the full dataset of size n. Using one fixed global kernel gives a well-defined empirical loss associated with a specific coupling. Moreover, we remark that (Dance et al., 2026) implements an ODE sampler, whereas we consider an SDE sampler. Appendix C Reader’s Guide and Proof Roadmap This section lists the subclaims needed for the end-to-end result and points to the proof of each one. Appendix Organization. Appendix A provides experimental details and additional results. Appendix B gives additional related-work details first. We then collect the notation and algorithmic details, give the proofs of Examples 1 and 2, and present the proof of Theorem 1, the coupling and field-estimation analyses, and the proof of Theorem 2. Motivating Examples. The proofs of Examples 1 and 2 are given in Appendices E and F, respectively. Sampling Theorem. Theorem 1 follows from the following chain. 1. Marginal Correctness of the Exact SDE. Lemma 1 shows that qtq_t solves the Fokker–Planck equation of the score-corrected SDE and satisfies the integrability and continuity conditions required by the superposition principle. It therefore yields a weak solution Z with Zt∼qtZ_t q_t for every t∈[0,1]t∈[0,1]. 2. Girsanov KL bound for the Euler process. Lemma 2 localizes the exact-to-Euler drift perturbation, applies Girsanov before the stopping time, and passes to the unstopped process by weak convergence and lower semicontinuity of KL. Proposition 3 verifies the pathwise integrability condition when the drifts and sample paths are continuous. 3. Posterior Derivative Formulas. Lemmas 3 and 4 express the score, velocity Jacobian, velocity Hessian, score Jacobian, and score Laplacian as conditional moments of D=Y1−Y0D=Y_1-Y_0 and the Gaussian variable G. 4. Integrated Derivative Bounds. Lemmas 5 and 6 convert the posterior identities into L2(qt)L^2(q_t) bounds using conditional variance contraction, Cauchy–Schwarz, and explicit Gaussian moments. 5. Euler Increment Bounds. Lemmas 7 and 8 apply Itô’s formula along the exact SDE. The score proof uses the continuity equation to cancel DxstvtD_xs_t\,v_t, leaving only the Hessian, mixed velocity–score, and pure score terms controlled by the integrated derivative bounds. 6. Collection of Constants. The proof of Theorem 1 substitutes the Euler increment bounds into the Girsanov reduction and records the explicit coefficients AhA_h and BhB_h, together with the explicit τ4τ^4, τ6τ^6, and displacement terms. Corollary 1 then adds the Gaussian smoothing error and converts KL to W1W_1 using a maximal coupling, Pinsker’s inequality, and a second-moment bound for the Euler recursion. Field-Estimation Propositions. The proof of Proposition 1 first derives Proposition 5 and then verifies its hypotheses for the velocity and score losses. 1. Coupling Construction and Replacement Error. Lemma 9 proves Lipschitz stability of the EOT conditional kernel. Lemma 10 constructs an ideal kernel Q with the required source and target marginals, and Corollary 3 controls the joint-law discrepancy between Q and Q^n Q_n. 2. Exact Excess-Risk Identities. For both fields, conditional projection identifies excess population loss with the grid-averaged squared L2(qtk)L^2(q_t_k) field error. The score proof uses the conditional-mean parametrization su(x,t)=u(x,t)−x/σ2s_u(x,t)=\u(x,t)-x\/σ^2, for which the Gaussian term cancels from the loss. 3. Approximation. Propositions 6 and 7 construct network approximations uniformly on a bounded cube and control the Gaussian tail outside the cube. 4. Loss Perturbations. The coupling-replacement error is controlled using Corollary 3, together with the endpoint Lipschitz bound for the loss. For the doubly robust error, we first write the population bias as C[(1−α^/αa)(mv−m^v)]E_C[(1- α/ _a)(m_v- m_v)]. We then control mv−m^vm_v- m_v through the conditional-generator W1W_1 error and the stability of Q^n Q_n, and apply Cauchy–Schwarz to obtain the final product bound. 5. Statistical Generalization. Lemma 13 controls the empirical-to-population difference for any bounded loss class from a proper finite cover. The two field analyses supply their own network covering numbers and loss-to-network transfer constants. 6. Monte Carlo Approximation. We first decompose the implementation error according to the structure of the doubly robust loss and define separate observed-correction and auxiliary Monte Carlo classes. For each class, the corresponding error can be written as the difference between a conditional expectation and its realized Monte Carlo average. Lemma 13 is then applied conditionally to each array. A key step is to condition on the training observations and fitted auxiliary objects, which gives the conditional independence required in Proposition 5. 7. End-to-End Substitution. Combining the preceding velocity- and score-field steps gives the integrated errors εvel2 _vel^2 and εscore2 _score^2. Substitution into the explicit sampling theorem, followed by the Wasserstein conversion, proves Theorem 2. Technical Novelty. Our proof makes two main contributions. First, for fixed Gaussian smoothing, we obtain near-linear rather than quadratic dimension dependence in the number of uniform Euler steps, with D5D_5, σ, and target accuracy held fixed. We bound velocity derivatives and mixed velocity–score terms using displacement moments, and keep the noise factors in the score terms explicit. With exact fields, choosing τ2=σD5/dτ^2=σ D_5/d then reduces the dimension dependence and makes every discretization term shrink with D5D_5, including the pure-score terms. This gives both improved dimension dependence and full coupling-sensitive control of the discretization error. Second, we give an end-to-end analysis of counterfactual generation with a learned source–target coupling, rather than only independent pairing. We analyze a doubly robust loss that combines prediction-based losses with weighted corrections from observed outcomes, rather than the usual flow-matching loss. Because the training loss itself is approximated using finitely many random draws from the prediction oracle and the coupling, we bound the error caused by this Monte Carlo approximation. We also separate this error from the finite-data error and the Monte Carlo error in the target samples used to learn the coupling. Combining these learning bounds with the Euler and smoothing errors gives a guarantee for the final output distribution. C.1 Summary of Notation Table 2 summarizes the notation used throughout the paper. Table 2: Summary of notation used throughout the paper. Notation Meaning Notation Meaning (C,A,Y)(C,A,Y) Observed covariate, treatment, and outcome. n,nV_n,S_n Clipped sparse-ReLU velocity class and score class induced by clipped sparse-ReLU denoisers. a Fixed target treatment value. R,εR, Network input-clipping radius and approximation tolerance; field-specific choices appear in Proposition 1. ℙY∣C=c,A=aP_Y C=c,A=a Conditional outcome law at covariate value c under treatment a. εvel,εscore _vel, _score Integrated estimation errors of the learned velocity and score fields. μ1 _1 Target law ∫ℙY∣C=c,A=adℙC(c) _Y C=c,A=a\,dP_C(c). τ Noise level of the score-corrected SDE sampler. ℙY|A=aP_Y A=a Observed outcome law among samples with treatment a. N,hN,h Number of sampling steps and Euler step size, with h=1/Nh=1/N. tr,auxD_tr,D_aux Training and auxiliary splits, each containing n observations. tkt_k Sampling grid point tk=kht_k=kh. ℙ^Y∣C=c,A=a P_Y C=c,A=a Learned conditional outcome law, or prediction oracle. M Monte Carlo size in the final doubly robust training objective. ℙ^Y|A=aaux P_Y A=a^aux Empirical source law formed from treatment-a outcomes in auxD_aux. MQM_Q Prediction-oracle samples per auxiliary covariate used to construct ℙ^Y1Q P_Y_1^Q. ℙ^Y1Q P_Y_1^Q Finite-sample fitted-target approximation based on the prediction oracle. αa(c),α^(c) _a(c), α(c) True and estimated inverse propensities for treatment a. Q^n(dy0∣y1) Q_n(dy_0 y_1) Estimated coupling kernel between the empirical source and fitted target. πmin _ Lower bound on ℙ(A=a∣C=c)P(A=a C=c). Q(dy0∣y1),Γμ1,Q(dy_0 y_1), _ _1,Q Population-valid kernel and joint endpoint law Γμ1,Q=Qμ1 _ _1,Q=Q _1. εα _α Estimation error of the inverse propensity model. (Y0,Y1)(Y_0,Y_1) Source–target pair under the specified coupling. εpred,train _pred,train Empirical prediction-oracle error on auxiliary covariates. D5D_5 Coupling displacement moment (‖Y1−Y0‖5)1/5(E\|Y_1-Y_0\|^5)^1/5. εpred,gen _pred,gen Prediction-oracle generalization error. StS_t Linear interpolation Y0+t(Y1−Y0)Y_0+t(Y_1-Y_0). εOT _OT Entropic regularization parameter for EOT. XtX_t Smoothed interpolation St+σGS_t+σ G, G∼(0,Id)G (0,I_d). ℰaE_a Event that auxD_aux contains at least one observation with Ai=aA_i=a. σ Gaussian smoothing level. supp(P)supp(P) Topological support: x such that P(B(x,r))>0P(B(x,r))>0 for every r>0r>0. vt,stv_t,s_t Population velocity and score fields for the smoothed interpolation. d Outcome dimension. qt,Q,vQ⋆,sQ⋆q_t,Q,v_Q ,s_Q Density, exact velocity, and exact score induced by Γμ1,Q _ _1,Q. W1,W1,ρW_1,W_1,ρ Wasserstein distances on the outcome and coupled endpoint spaces. v^M,s^M v_M, s_M Learned doubly robust velocity and score fields. KLKL Kullback–Leibler divergence. Conditioning Convention. In the finite-sample analysis, we first condition on auxD_aux and the prediction-oracle target reservoir. The fitted nuisance models, Q^n Q_n, Q, Γμ1,Q _ _1,Q, qt,Qq_t,Q, vQ⋆v_Q , sQ⋆s_Q , and the corresponding D5D_5 are then interpreted conditionally. Unconditional high-probability statements include the corresponding auxiliary-sample events in their failure probabilities. Appendix D Algorithms and Auxiliary-Model Details D.1 Prediction-Oracle Accuracy We give sufficient conditions for Assumption 4. The first controls the prediction-oracle error on the auxiliary covariates used to construct the coupling. Assumption 5 (Empirical Generator Accuracy on Auxiliary Covariates). The learned conditional outcome model satisfies [1n∑i=n+12nW1(ℙY∣C=Ci,A=a,ℙ^Y∣C=Ci,A=a)2]1/2 [ 1n _i=n+1^2nW_1 (P_Y C=C^i,A=a, P_Y C=C^i,A=a )^2 ]^1/2 ≤εpred,train. ≤ _pred,train. This assumption measures how well the learned conditional outcome model fits the conditional laws at the empirical auxiliary covariates. It is not a population-C quantity. Assumption 6 (Generator Generalization Gap). Let Ψn _n denote the class for prediction oracle, and let ℙ^ψY∣C=c,A=a P^ψ_Y C=c,A=a be the conditional law generated by ψ. Assume supψ∈Ψn|C[W1(ℙY∣C=c,A=a,ℙ^Y∣C=c,A=aψ)2] _ψ∈ _n |E_C [W_1 (P_Y C=c,A=a, P^ψ_Y C=c,A=a )^2 ] . −1n∑i=n+12nW1(ℙY∣C=Ci,A=a,ℙ^Y∣C=Ci,A=aψ)2|≤εpred,gen2. 85.35826pt .- 1n _i=n+1^2nW_1 (P_Y C=C^i,A=a, P^ψ_Y C=C^i,A=a )^2 |≤ _pred,gen^2. Together, these two assumptions imply the population prediction-oracle accuracy used in the main text. Proposition 2 (Population Generator Accuracy From Empirical Accuracy). Under Assumptions 5 and 6, [CW1(ℙY∣C=c,A=a,ℙ^Y∣C=c,A=a)2]1/2 [E_CW_1(P_Y C=c,A=a, P_Y C=c,A=a)^2 ]^1/2 ≤[(εpred,train)2+εpred,gen2]1/2. ≤[( _pred,train)^2+ _pred,gen^2]^1/2. Hence Assumption 4 holds. Proof. Since Assumption 6 holds uniformly over ψ∈Ψnψ∈ _n, it also applies to the learned prediction oracle. Therefore, CW1(ℙY∣C=c,A=a,ℙ^Y∣C=c,A=a)2 _CW_1(P_Y C=c,A=a, P_Y C=c,A=a)^2 ≤1n∑i=n+12nW1(ℙY∣C=Ci,A=a,ℙ^Y∣C=Ci,A=a)2+εpred,gen2 ≤ 1n _i=n+1^2nW_1(P_Y C=C_i,A=a, P_Y C=C_i,A=a)^2+ _pred,gen^2 ≤(εpred,train)2+εpred,gen2. ≤( _pred,train)^2+ _pred,gen^2. Taking square roots proves the claim. ∎ The prediction oracle ℙ^Y∣C=c,A=a P_Y C=c,A=a can be obtained by training an auxiliary conditional flow model on auxD_aux. For each observation (Yi,Ci,Ai)(Y_i,C_i,A_i), conditional on (Ci,Ai)(C_i,A_i), YiY_i is a draw from ℙY∣C=Ci,A=AiP_Y C=C_i,A=A_i. Let P0P_0 be a source distribution, for example (0,Id)N(0,I_d). The population flow-matching loss is ℒaux(u) _aux(u) =C,A,YZ∼P0,t∼Unif[0,1][‖u(Xt,t,C,A)−(Y−Z)‖2],Xt=(1−t)Z+tY. =E_C,A,YE_Z P_0,\,t [0,1][\|u(X_t,t,C,A)-(Y-Z)\|^2], X_t=(1-t)Z+tY. The auxiliary observations (Ci,Ai,Yi)(C_i,A_i,Y_i), i=n+1,…,2ni=n+1,…,2n, provide samples for training this model. After training, fixing A=aA=a and conditioning on C=cC=c gives the prediction oracle ℙ^Y∣C=c,A=a P_Y C=c,A=a. D.2 Score-Field Training Objective We now explain how the score field in the score-corrected SDE is learned. For each t∈[0,1]t∈[0,1], let qtq_t denote the density of XtX_t and define st(x)=∇xlogqt(x)s_t(x)= _x q_t(x). The corresponding population score-matching loss is ℒscore,true(ϕ) _score,true(φ) =∫01[‖sϕ(Xt,t)−st(Xt)‖2]t. = _0^1E [ \|s_φ(X_t,t)-s_t(X_t) \|^2 ]dt. This loss cannot be evaluated directly because sts_t is unknown. By the Gaussian score identity proved in Lemma 3, st(Xt)=[−G/σ∣Xt]s_t(X_t)=E[-G/σ X_t]. Therefore, −G/σ=(St−Xt)/σ2-G/σ=(S_t-X_t)/σ^2 can be used as a regression target for the score. We define the denoising score-matching loss by ℒscore(ϕ)=∫01(Y0,Y1)∼γG∼(0,Id)[‖sϕ(Xt,t)+Gσ‖2]t=∫01[‖sϕ(Xt,t)−St−Xtσ2‖2]t. _score(φ)= _0^1E_(Y_0,Y_1) γE_G (0,I_d) [ \|s_φ(X_t,t)+ Gσ \|^2 ]dt= _0^1E [ \|s_φ(X_t,t)- S_t-X_tσ^2 \|^2 ]dt. The equivalence follows from the conditional Pythagorean identity. For each fixed t, set U=−G/σU=-G/σ. Since st(Xt)=[U∣Xt]s_t(X_t)=E[U X_t], every square-integrable candidate sϕ(t,⋅)s_φ(t,·) satisfies ‖sϕ(Xt,t)−U‖2 \|s_φ(X_t,t)-U\|^2 =‖sϕ(Xt,t)−st(Xt)‖2+‖U−st(Xt)‖2. =E\|s_φ(X_t,t)-s_t(X_t)\|^2+E\|U-s_t(X_t)\|^2. The second term is independent of ϕφ. Integrating over t proves that the denoising and true-score losses have the same minimizers. Since Xt−St=σGX_t-S_t=σ G, the implementable objective may equivalently be written as ℒscore(ϕ) _score(φ) =∫01[‖sϕ(Xt,t)−St−Xtσ2‖2]t. = _0^1E [ \|s_φ(X_t,t)- S_t-X_tσ^2 \|^2 ]dt. D.3 Algorithms Algorithm 1 gives the finite-step score-corrected sampler, while Algorithm 2 gives the doubly robust procedure used to train its velocity and score fields. Algorithm 1 Score-Corrected SDE Sampler 1: Learned velocity field v v, learned score field s s, empirical source distribution ℙ^Y|A=aaux P_Y A=a^aux, smoothing level σ>0σ>0, SDE noise level τ>0τ>0, number of time steps N. 2: Set h=1/Nh=1/N and tk=kht_k=kh, k=0,…,Nk=0,…,N. 3: Draw Y0init∼ℙ^Y|A=aauxY_0^init P_Y A=a^aux and Ginit∼(0,Id)G^init N(0,I_d). 4: Initialize Z^0=Y0init+σGinit Z_0=Y_0^init+σ G^init. 5: for k=0,…,N−1k=0,…,N-1 do 6: Draw Gk∼(0,Id)G_k N(0,I_d). 7: Update Z^k+1=Z^k+h(v^(Z^k,tk)+τ22s^(Z^k,tk))+τhGk Z_k+1= Z_k+h( v( Z_k,t_k)+ τ^22 s( Z_k,t_k))+τ h\,G_k. 8: end for 9: return Z^N Z_N. Algorithm 2 Minibatch Doubly Robust Flow Training for a Fixed Treatment a 1: Training data (Ci,Ai,Yi)i=1n\(C_i,A_i,Y_i)\_i=1^n for the final flow model, and training data (Ci,Ai,Yi)i=n+12n\(C_i,A_i,Y_i)\_i=n+1^2n for the following models: propensity estimator ℙ^(A=a∣C=c) P(A=a C=c), conditional outcome sampler ℙ^Y∣C=c,A=a P_Y C=c,A=a, estimated coupling kernel Q^n(dy0∣y1) Q_n(dy_0 y_1). 2: Vector-field class nV_n, score-field class nS_n, noise level σ>0σ>0, Monte Carlo size M. 3: Initialize v∈nv _n and s∈ns _n. 4: for i∈1,…,ni∈\1,...,n\ do 5: for k=1,…,Mk=1,…,M do 6: Draw Y^i,k∼ℙ^Y∣C=Ci,A=a Y_i,k P_Y C=C_i,A=a. 7: Draw Y^0,i,k∼Q^n(⋅∣Y^i,k) Y_0,i,k Q_n(· Y_i,k), ti,k∼Unift0,…,tN−1t_i,k \t_0,...,t_N-1\, and Gi,k∼(0,Id)G_i,k N(0,I_d). 8: end for 9: if Ai=aA_i=a then 10: Draw Y^0,iobs∼Q^n(⋅∣Yi) Y_0,i^obs Q_n(· Y_i), tiobs∼Unift0,…,tN−1t_i^obs \t_0,...,t_N-1\, and Giobs∼(0,Id)G_i^obs N(0,I_d). 11: end if 12: end for 13: for each training iteration do 14: Draw a minibatch ℬ⊆1,…,nB \1,…,n\. 15: Compute the minibatch doubly robust flow loss L^ℬ,DR(v) L_B,DR(v) =1|ℬ|∑i∈ℬ[Ai=aℙ^(A=a∣C=Ci)ℓ^iobs(v)−1M∑k=1Mℓ^i,kplug(v)+1M∑k=1Mℓ^i,kplug(v)], = 1|B| _i [ 1\A_i=a\ P(A=a C=C_i) \ _i^obs(v)- 1M _k=1^M _i,k^plug(v) \+ 1M _k=1^M _i,k^plug(v) ], where ℓ^iobs(v) _i^obs(v) =‖v(Y^0,iobs+tiobs(Yi−Y^0,iobs)+σGiobs,tiobs)−(Yi−Y^0,iobs)‖2, = \|v ( Y_0,i^obs+t_i^obs(Y_i- Y_0,i^obs)+σ G_i^obs,t_i^obs )-(Y_i- Y_0,i^obs) \|^2, ℓ^i,kplug(v) _i,k^plug(v) =‖v(Y^0,i,k+ti,k(Y^i,k−Y^0,i,k)+σGi,k,ti,k)−(Y^i,k−Y^0,i,k)‖2. = \|v ( Y_0,i,k+t_i,k( Y_i,k- Y_0,i,k)+σ G_i,k,t_i,k )-( Y_i,k- Y_0,i,k) \|^2. 16: Update v by one optimizer step using L^ℬ,DR(v) L_B,DR(v). 17: Compute the minibatch doubly robust score loss L^ℬ,DRscore(s) L_B,DR^score(s) =1|ℬ|∑i∈ℬ[Ai=aℙ^(A=a∣C=Ci)r^iobs(s)−1M∑k=1Mr^i,kplug(s)+1M∑k=1Mr^i,kplug(s)], = 1|B| _i [ 1\A_i=a\ P(A=a C=C_i) \ r_i^obs(s)- 1M _k=1^M r_i,k^plug(s) \+ 1M _k=1^M r_i,k^plug(s) ], where r^iobs(s) r_i^obs(s) =‖s(Y^0,iobs+tiobs(Yi−Y^0,iobs)+σGiobs,tiobs)+Giobsσ‖2, = \|s ( Y_0,i^obs+t_i^obs (Y_i- Y_0,i^obs )+σ G_i^obs,t_i^obs )+ G_i^obsσ \|^2, r^i,kplug(s) r_i,k^plug(s) =‖s(Y^0,i,k+ti,k(Y^i,k−Y^0,i,k)+σGi,k,ti,k)+Gi,kσ‖2. = \|s ( Y_0,i,k+t_i,k ( Y_i,k- Y_0,i,k )+σ G_i,k,t_i,k )+ G_i,kσ \|^2. 18: Update s by one optimizer step using L^ℬ,DRscore(s) L_B,DR^score(s). 19: end for 20: return v and s. Appendix E Proof of Example 1 Proof of Example 1. In this proof, write Z^k Z_k for the Euler ODE iterates, so Z^N=X^NODE Z_N= X_N^ODE in the example. Starting from Z^0=X0 Z_0 d=X_0, the recursion is Z^k+1=Z^k+hvkh(Z^k),k=0,…,N−1. Z_k+1= Z_k+hv_kh( Z_k), k=0,…,N-1. Here N is the number of Euler steps, and h=1/Nh=1/N is the step size. Write D=Y1−Y0D=Y_1-Y_0. We have vt(x)=[D∣Xt=x]v_t(x)=E[D X_t=x]. Let φV _V denote the density of (0,V)N(0,V). Conditional on B=0B=0 we have X0∼(0,1+σ2)X_0 (0,1+σ^2) and conditional on B=1B=1 we have X0∼(0,4+σ2)X_0 (0,4+σ^2). Bayes’ rule implies ℙ(B=1∣X0=x)=pφ4+σ2(x)(1−p)φ1+σ2(x)+pφ4+σ2(x). (B=1 X_0=x)= p _4+σ^2(x)(1-p) _1+σ^2(x)+p _4+σ^2(x). Since D=0D=0 on B=0\B=0\, v0(x) v_0(x) =ℙ(B=1∣X0=x)[D∣X0=x,B=1]+ℙ(B=0∣X0=x)[D∣X0=x,B=0] =P(B=1 X_0=x)E[D X_0=x,B=1]+P(B=0 X_0=x)E[D X_0=x,B=0] =ℙ(B=1∣X0=x)[D∣X0=x,B=1]. =P(B=1 X_0=x)E[D X_0=x,B=1]. We have that [U∣2U+σG=x]=∫−∞uPu|2u+σG=x(u)u. [U 2U+σ G=x]= _-∞^∞uP_u 2u+σ G=x(u)du. To compute Pu|2u+σG=x(u)P_u 2u+σ G=x(u), we use the formula pX|Y=y(x)=pX,Y(x,y)pY(y)p_X Y=y(x)= p_X,Y(x,y)p_Y(y). We get pU|2U+σG=x(u)=pU,2U+σG(u,x)p2U+σG(x)∝pU,2U+σG(u,x)=1σpU,G(u,x−2uσ), p_U 2U+σ G=x(u)= p_U,2U+σ G(u,x)p_2U+σ G(x) p_U,2U+σ G(u,x)= 1σp_U,G(u, x-2uσ), where we used change of variable formula. Then we have pU|2U+σG=x(u) p_U 2U+σ G=x(u) ∝1σexp(−u2/2−(x−2u)2/(2σ2))∝exp(−(u−2xσ2+4)22σ2/(σ2+4)). 1σ (-u^2/2-(x-2u)^2/(2σ^2)) (- (u- 2xσ^2+4)^22σ^2/(σ^2+4)). Hence [U∣2U+σG=x]=24+σ2xE[U 2U+σ G=x]= 24+σ^2x. Using the definition of λ, [D∣X0=x,B=1]=−2λ24+σ2x=−xh, [D X_0=x,B=1]=-2λ 24+σ^2x=- xh, so we have v0(x)=−ℙ(B=1∣X0=x)hx.v_0(x)=- P(B=1 X_0=x)hx. We next examine the uniform spatial regularity of the population velocity. By the expression above for ℙ(B=1∣X0=x)P(B=1 X_0=x), ℙ(B=1∣X0=x) (B=1 X_0=x) =pφ4+σ2(x)(1−p)φ1+σ2(x)+pφ4+σ2(x)=(1+1−pφ1+σ2(x)φ4+σ2(x))−1 = p _4+σ^2(x)(1-p) _1+σ^2(x)+p _4+σ^2(x)= (1+ 1-pp _1+σ^2(x) _4+σ^2(x) )^-1 =(1+1−p4+σ21+σ2exp−x22(1+σ2)+x22(4+σ2))−1 = (1+ 1-pp 4+σ^21+σ^2 \- x^22(1+σ^2)+ x^22(4+σ^2) \ )^-1 =(1+1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2))−1. = (1+ 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \ )^-1. Therefore, v0(x)=−xh(1+1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2))−1. v_0(x)=- xh (1+ 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \ )^-1. Differentiating with respect to x gives ∂xv0(x)= _xv_0(x)= −1h(1+1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2))−1 - 1h (1+ 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \ )^-1 −3x2h(1+σ2)(4+σ2)1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2)(1+1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2))2. - 3x^2h(1+σ^2)(4+σ^2) 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \ (1+ 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \ )^2. For all sufficiently small p, we have 1−p4+σ21+σ2>1. 1-pp 4+σ^21+σ^2>1. Choose x>0x>0 such that x2=2(1+σ2)(4+σ2)3log(1−p4+σ21+σ2)x^2= 2(1+σ^2)(4+σ^2)3 ( 1-pp 4+σ^21+σ^2 ), i.e., 1−p4+σ21+σ2exp−3x22(1+σ2)(4+σ2)=1 1-pp 4+σ^21+σ^2 \- 3x^22(1+σ^2)(4+σ^2) \=1. Substituting this value of x into the expression for ∂xv0(x) _xv_0(x) gives ∂xv0(x)=−12h−3x24h(1+σ2)(4+σ2)=−12h[1+log(1−p4+σ21+σ2)]. _xv_0(x)=- 12h- 3x^24h(1+σ^2)(4+σ^2)=- 12h [1+ ( 1-pp 4+σ^21+σ^2 ) ]. Consequently, supt∈[0,1]supx∈ℝ|∂xvt(x)|≥supx∈ℝ|∂xv0(x)|≥12h[1+log(1−p4+σ21+σ2)]. _t∈[0,1] _x | _xv_t(x)|≥ _x | _xv_0(x)|≥ 12h [1+ ( 1-pp 4+σ^21+σ^2 ) ]. By construction, D=0D=0 on B=0\B=0\ and D=−2λUD=-2λ U on B=1\B=1\. Hence D5=(|D|5)1/5=2λ(|U|5)1/5p1/5D_5= (E|D|^5 )^1/5=2λ (E|U|^5 )^1/5p^1/5. So we conclude that when p→0p→ 0, supt∈[0,1]supx∈ℝ|∂xvt(x)|→+∞ _t∈[0,1] _x | _xv_t(x)|→+∞ but D5→0D_5→ 0. For the integrated derivative claim, keep h,σh,σ fixed, so that λ is fixed. Since |D|4=16λ4p|U|4=48λ4pE|D|^4=16λ^4p\,E|U|^4=48λ^4p, Lemma 6 with d=1d=1 gives ∫01|∂xvt(Xt)|2t≤3σ2(48λ4p)1/2=12λ2σ2p⟶0. _0^1E| _xv_t(X_t)|^2\,dt≤ 3σ^2(48λ^4p)^1/2= 12λ^2σ^2 p 0. Next, we show the sampling KL error would blow up. The first Euler update is the deterministic map Z^1=Tp(Z^0),Tp(x)=x+hv0(x)=x(1−ℙ(B=1∣X0=x)). Z_1=T_p( Z_0), T_p(x)=x+hv_0(x)=x(1-P(B=1 X_0=x)). Moreover, 1−ℙ(B=1∣X0=x) 1-P(B=1 X_0=x) ≤1−pφ1+σ2(x)φ4+σ2(x) ≤ 1-pp _1+σ^2(x) _4+σ^2(x) =1−p4+σ21+σ2exp−x22(11+σ2−14+σ2). = 1-pp 4+σ^21+σ^2 \- x^22 ( 11+σ^2- 14+σ^2 ) \. Since TpT_p is continuous, its image is bounded. Thus Z^1 Z_1 is supported on a compact interval. For vt(x)v_t(x), we change the condition X0=2U+σG=xX_0=2U+σ G=x to Xt=2U(1−λt)+σG=xX_t=2U(1-λ t)+σ G=x. [D∣Xt=x,B=1]=−4λ(1−λt)4(1−λt)2+σ2x, [D X_t=x,B=1]=- 4λ(1-λ t)4(1-λ t)^2+σ^2x, so vt(x)=−4λ(1−λt)4(1−λt)2+σ2pφ4(1−λt)2+σ2(x)(1−p)φ1+σ2(x)+pφ4(1−λt)2+σ2(x)x. v_t(x)=- 4λ(1-λ t)4(1-λ t)^2+σ^2 p _4(1-λ t)^2+σ^2(x)(1-p) _1+σ^2(x)+p _4(1-λ t)^2+σ^2(x)x. Note that |4λ(1−λt)4(1−λt)2+σ2|≤|4λ(1−λt)σ2|| 4λ(1-λ t)4(1-λ t)^2+σ^2|≤| 4λ(1-λ t)σ^2| and pφ4(1−λt)2+σ2(x)(1−p)φ1+σ2(x)+pφ4(1−λt)2+σ2(x)≤1 p _4(1-λ t)^2+σ^2(x)(1-p) _1+σ^2(x)+p _4(1-λ t)^2+σ^2(x)≤ 1 are uniformly bounded. Consequently, x↦x+hvkh(x) x x+hv_kh(x) is continuous for every k=0,…,N−1k=0,…,N-1. Since a continuous image of a compact set is compact, Z^N Z_N is supported on some compact set. On the other hand, the density q1(x)q_1(x) of X1X_1 is strictly positive for every x∈ℝx . Therefore, ℒ(X1)L(X_1) is not absolutely continuous with respect to ℒ(Z^N)L( Z_N), which proves KL(ℒ(X1)∥ℒ(Z^N))=+∞. (L(X_1)\,\|\,L( Z_N) )=+∞. The result on integrated regularity follows from Lemma 6. ∎ Appendix F Proof of Example 2 Proof of Example 2. We first compute the coupling displacement. Conditional on Y0=xY_0=x, the displacement Y1−Y0Y_1-Y_0 is equal to δ, −δ-δ, or 00, with respective probabilities pm(x)p_m(x), pm(x)p_m(x), and 1−2pm(x)1-2p_m(x). Therefore [|Y1−Y0|5∣Y0=x]=2pm(x)δ5=δ5(12+14sin(Kmx)).E [|Y_1-Y_0|^5 Y_0=x ]=2p_m(x)δ^5=δ^5 ( 12+ 14 (K_mx) ). Since Y0Y_0 is symmetric about zero and x↦sin(Kmx)x (K_mx) is odd, [sin(KmY0)]=0E[ (K_mY_0)]=0. Hence |Y1−Y0|5=δ5/2E|Y_1-Y_0|^5=δ^5/2, and therefore D5=δ21/5=ε.D_5= δ2^1/5= . In particular, D5D_5 is independent of m even though Km→∞K_m→∞. We next verify that the score is well defined despite the absence of smoothing. Let ϕ(x)=12πe−x2/2φ(x)= 1 2πe^-x^2/2 denote the standard Gaussian density. Conditional on the three possible values of Y1−Y0Y_1-Y_0, the interpolation satisfies Yt=Y0+tδ,Y1−Y0=δ,Y0−tδ,Y1−Y0=−δ,Y0,Y1−Y0=0.Y_t= casesY_0+tδ,&Y_1-Y_0=δ,\\ Y_0-tδ,&Y_1-Y_0=-δ,\\ Y_0,&Y_1-Y_0=0. cases It follows that qt(m)(y)=ϕ(y−tδ)pm(y−tδ)+ϕ(y+tδ)pm(y+tδ)+ϕ(y)(1−2pm(y)). q_t^(m)(y)=φ(y-tδ)p_m(y-tδ)+φ(y+tδ)p_m(y+tδ)+φ(y) (1-2p_m(y) ). Since pm(y)≤3/8p_m(y)≤ 3/8, 1−2pm(y)≥141-2p_m(y)≥ 14, and hence qt(m)(y)≥14ϕ(y)>0q_t^(m)(y)≥ 14φ(y)>0. Moreover, qt(m)q_t^(m) is smooth because it is a finite sum of products of smooth functions. Thus st(m)=∂xlogqt(m)s_t^(m)= _x q_t^(m) and ∂xst(m) _xs_t^(m) are well defined for every t∈[0,1]t∈[0,1]. Since ∂xst(m)(x)=∂x(qt(m))′(x)qt(m)(x)=(qt(m))′(x)qt(m)(x)−((qt(m))′(x))2(qt(m)(x))2=(qt(m))′(x)qt(m)(x)−st(m)(x)2 _xs_t^(m)(x)= _x (q_t^(m) ) (x)q_t^(m)(x)= (q_t^(m) ) (x)q_t^(m)(x)- ( (q_t^(m) ) (x) )^2 (q_t^(m)(x) )^2= (q_t^(m) ) (x)q_t^(m)(x)-s_t^(m)(x)^2, we have [∂xst(m)(Yt)] [ _xs_t^(m)(Y_t)] =∫ℝ∂xst(m)(y)qt(m)(y)y = _R _xs_t^(m)(y)q_t^(m)(y)\,dy =∫ℝ(qt(m))′(y)y−[st(m)(Yt)2]=−[st(m)(Yt)2], = _R (q_t^(m) ) (y)\,dy-E[s_t^(m)(Y_t)^2]=-E[s_t^(m)(Y_t)^2], because (qt(m))′(y)→0 (q_t^(m) ) (y)→ 0 as |y|→∞|y|→∞. Jensen’s inequality therefore yields [|∂xst(m)(Yt)|2]≥|[∂xst(m)(Yt)]|2=([st(m)(Yt)2])2.E[| _xs_t^(m)(Y_t)|^2]≥ |E[ _xs_t^(m)(Y_t)] |^2= (E[s_t^(m)(Y_t)^2] )^2. It therefore remains to lower bound [st(m)(Yt)2]E[s_t^(m)(Y_t)^2]. Rather than differentiating the explicit expression for qt(m)q_t^(m) and attempting to control the resulting cross terms pointwise, we use a variational consequence of the score identity. For any smooth bounded function f, [f′(Yt)] [f (Y_t)] =∫ℝf′(y)qt(m)(y)y = _Rf (y)q_t^(m)(y)\,dy =−∫ℝf(y)(qt(m))′(y)dy =- _Rf(y) (q_t^(m) ) (y)\,dy =−[f(Yt)st(m)(Yt)], =-E[f(Y_t)s_t^(m)(Y_t)], where the boundary term vanishes because qt(m)q_t^(m) has Gaussian tails. Hence, by Cauchy–Schwarz, |[f′(Yt)]|2 |E[f (Y_t)]|^2 ≤[f(Yt)2][st(m)(Yt)2], [f(Y_t)^2]E[s_t^(m)(Y_t)^2], and therefore [st(m)(Yt)2]≥|[f′(Yt)]|2[f(Yt)2]=(∫qt(m)(y)f′(y)y)2∫qt(m)(y)f(y)2y. [s_t^(m)(Y_t)^2]≥ |E[f (Y_t)]|^2E[f(Y_t)^2]= ( q_t^(m)(y)f (y)\,dy )^2 q_t^(m)(y)f(y)^2\,dy. We want to choose f so that the lower bound doesn’t decrease, as m increases. Take f(y)=cos(Kmy)f(y)= (K_my). Its derivative is f′(y)=−Kmsin(Kmy)f (y)=-K_m (K_my), so the preceding inequality gives [st(m)(Yt)2]≥Km2|[sin(KmYt)]|2[cos2(KmYt)]≥Km2|[sin(KmYt)]|2, [s_t^(m)(Y_t)^2]≥ K_m^2|E[ (K_mY_t)]|^2E[ ^2(K_mY_t)]≥ K_m^2|E[ (K_mY_t)]|^2, where the last inequality uses cos2(KmYt)≤1 ^2(K_mY_t)≤ 1. By construction, [sin(KmYt)∣Y0=x]= [ (K_mY_t) Y_0=x]= pm(x)sin(Kmx+tKmδ)+pm(x)sin(Kmx−tKmδ) p_m(x) (K_mx+tK_mδ)+p_m(x) (K_mx-tK_mδ) +(1−2pm(x))sin(Kmx). + (1-2p_m(x) ) (K_mx). Using sin(a+b)+sin(a−b)=2sin(a)cos(b) (a+b)+ (a-b)=2 (a) (b) we obtain [sin(KmYt)∣Y0=x] [ (K_mY_t) Y_0=x] =2pm(x)sin(Kmx)cos(tKmδ)+(1−2pm(x))sin(Kmx) =2p_m(x) (K_mx) (tK_mδ)+ (1-2p_m(x) ) (K_mx) =sin(Kmx)[1+2pm(x)(cos(tKmδ)−1)]. = (K_mx) [1+2p_m(x) ( (tK_mδ)-1 ) ]. Taking expectation with respect to Y0Y_0 and using [sin(KmY0)]=0E[ (K_mY_0)]=0 gives [sin(KmYt)] [ (K_mY_t)] =(cos(tKmδ)−1)[2pm(Y0)sin(KmY0)]. = ( (tK_mδ)-1 )E [2p_m(Y_0) (K_mY_0) ]. Recall by definition 2pm(Y0)=12+14sin(KmY0)2p_m(Y_0)= 12+ 14 (K_mY_0), so we have [2pm(Y0)sin(KmY0)]=12[sin(KmY0)]+14[sin2(KmY0)]=14[sin2(KmY0)]. [2p_m(Y_0) (K_mY_0) ]= 12E[ (K_mY_0)]+ 14E[ ^2(K_mY_0)]= 14E[ ^2(K_mY_0)]. Using sin2(u)=1−cos(2u)2 ^2(u)= 1- (2u)2 and [cos(aY0)]=∫ℝϕ(x)cos(ax)dx=e−a2/2E[ (aY_0)]= _Rφ(x) (ax)dx=e^-a^2/2, we obtain [sin2(KmY0)]=1−e−2Km22.E[ ^2(K_mY_0)]= 1-e^-2K_m^22. Consequently, [sin(KmYt)]=−(1−cos(tKmδ))(1−e−2Km2)8=−(1−cos((2m+1)πt))(1−e−2Km2)8, [ (K_mY_t)]=- (1- (tK_mδ) ) (1-e^-2K_m^2 )8=- (1- ((2m+1)π t) ) (1-e^-2K_m^2 )8, where the second equality uses Kmδ=(2m+1)πK_mδ=(2m+1)π. Substituting this expression back yields [st(m)(Yt)2] [s_t^(m)(Y_t)^2] ≥Km2|[sin(KmYt)]|2 ≥ K_m^2|E[ (K_mY_t)]|^2 =Km2(1−e−2Km2)264(1−cos((2m+1)πt))2. = K_m^2(1-e^-2K_m^2)^264 (1- ((2m+1)π t) )^2. It follows that [|∂xst(m)(Yt)|2] [| _xs_t^(m)(Y_t)|^2] ≥(Km2(1−e−2Km2)264(1−cos((2m+1)πt))2)2 ≥ ( K_m^2(1-e^-2K_m^2)^264 (1- ((2m+1)π t) )^2 )^2 =Km4(1−e−2Km2)44096(1−cos((2m+1)πt))4. = K_m^4(1-e^-2K_m^2)^44096 (1- ((2m+1)π t) )^4. Integrating over t∈[0,1]t∈[0,1] therefore gives ∫01[|∂xst(m)(Yt)|2]t≥Km4(1−e−2Km2)44096∫01(1−cos((2m+1)πt))4t. _0^1E[| _xs_t^(m)(Y_t)|^2]\,dt≥ K_m^4(1-e^-2K_m^2)^44096 _0^1 (1- ((2m+1)π t) )^4dt. It remains only to check that the oscillatory factor does not disappear after integration in time. Expanding, (1−cosu)4=1−4cosu+6cos2u−4cos3u+cos4u. (1- u)^4=1-4 u+6 ^2u-4 ^3u+ ^4u. With the change of variables u=(2m+1)πtu=(2m+1)π t, ∫01(1−cos((2m+1)πt))4t=1(2m+1)π∫0(2m+1)π(1−cosu)4u. _0^1 (1- ((2m+1)π t) )^4dt= 1(2m+1)π _0^(2m+1)π(1- u)^4du. Since 2m+12m+1 is an integer, ∫0(2m+1)πcosuu _0^(2m+1)π u\,du =0, =0, ∫0(2m+1)πcos3uu _0^(2m+1)π ^3u\,du =0, =0, ∫0(2m+1)πcos2uu _0^(2m+1)π ^2u\,du =(2m+1)π2, = (2m+1)π2, ∫0(2m+1)πcos4uu _0^(2m+1)π ^4u\,du =3(2m+1)π8. = 3(2m+1)π8. Hence ∫01(1−cos((2m+1)πt))4t=1+6⋅12+38=358. _0^1 (1- ((2m+1)π t) )^4dt=1+6· 12+ 38= 358. Consequently, ∫01[|∂xst(m)(Yt)|2]t≥3532768Km4(1−e−2Km2)4. _0^1E[| _xs_t^(m)(Y_t)|^2]\,dt≥ 3532768K_m^4(1-e^-2K_m^2)^4. Recall that D5=εD_5= for every m, whereas Km=(2m+1)πδ⟶∞.K_m= (2m+1)πδ ∞. Therefore, ∫01[|∂xst(m)(Yt)|2]t⟶+∞ _0^1E[| _xs_t^(m)(Y_t)|^2]\,dt +∞ while D5D_5 remains fixed at the arbitrarily prescribed value ε . Thus, without smoothing, a small coupling displacement does not imply small integrated regularity. ∎ Appendix G Proof of Theorem 1 Standard Sampling Conditions. For the sampling analysis, we require that the exact score-corrected drift btb_t is jointly continuous on [0,1]×ℝd[0,1]×R^d, and that the exact SDE initialized from q0q_0, dZt=bt(Zt)dt+τdBt, dZ_t=b_t(Z_t)\,dt+τ\,dB_t, admits a weak solution on [0,1][0,1] satisfying ℒ(Zt)=qtL(Z_t)=q_t for every t∈[0,1]t∈[0,1]. We also require each grid-frozen learned drift b^tk b_t_k to be continuous. The exact-process requirements are verified in Lemma 1. Continuity of the learned drifts follows from that of the neural networks. Lemma 1 (Sampling well-posedness). Let σ,τ>0σ,τ>0 and D5<∞D_5<∞. Then bt(x)=vt(x)+τ22st(x)b_t(x)=v_t(x)+ τ^22s_t(x) is jointly continuous on [0,1]×ℝd[0,1]×R^d, and the exact SDE dZt=bt(Zt)dt+τdBt,Z0∼q0, dZ_t=b_t(Z_t)\,dt+τ\,dB_t, Z_0 q_0, admits a weak solution on [0,1][0,1] satisfying ℒ(Zt)=qtL(Z_t)=q_t for every t∈[0,1]t∈[0,1]. Proof. Let φσ _σ be the density of (0,σ2Id)N(0,σ^2I_d) and recall that St=Y0+tDS_t=Y_0+tD, where D=Y1−Y0D=Y_1-Y_0. Then qt(x)=[φσ(x−St)],qt(x)vt(x)=[Dφσ(x−St)]. q_t(x)=E[ _σ(x-S_t)], q_t(x)v_t(x)=E[D _σ(x-S_t)]. Fix (tn,xn)→(t,x)(t_n,x_n)→(t,x). Since Stn→StS_t_n→ S_t almost surely, φσ(xn−Stn) _σ(x_n-S_t_n) →φσ(x−St), → _σ(x-S_t), ∇xφσ(xn−Stn) _x _σ(x_n-S_t_n) →∇xφσ(x−St) → _x _σ(x-S_t) almost surely. Both φσ _σ and ∇xφσ _x _σ are bounded, so dominated convergence gives joint continuity of qt(x)q_t(x) and ∇xqt(x) _xq_t(x). Moreover, Dφσ(xn−Stn)→Dφσ(x−St) D _σ(x_n-S_t_n)→ D _σ(x-S_t) almost surely and ‖Dφσ(xn−Stn)‖≤‖φσ‖∞‖D‖. \|D _σ(x_n-S_t_n)\|≤\| _σ\|_∞\|D\|. Since D5<∞D_5<∞ implies ‖D‖<∞E\|D\|<∞, dominated convergence also gives joint continuity of qt(x)vt(x)q_t(x)v_t(x). Because φσ>0 _σ>0, we have qt(x)>0q_t(x)>0 for every (t,x)(t,x). Hence vt(x)=[Dφσ(x−St)]qt(x),st(x)=∇xqt(x)qt(x) v_t(x)= E[D _σ(x-S_t)]q_t(x), s_t(x)= _xq_t(x)q_t(x) are jointly continuous, and therefore so is btb_t. We next verify the Fokker–Planck equation satisfied by (qt)t∈[0,1](q_t)_t∈[0,1]. For every f∈Cc∞(ℝd)f∈ C_c^∞(R^d), dt∫ℝdf(x)qt(x)x ddt _R^df(x)q_t(x)\,dx =dt[f(Xt)] = ddtE[f(X_t)] =[∇xf(Xt)⊤D] =E[ _xf(X_t) D] =∫ℝd∇xf(x)⊤vt(x)qt(x)x, = _R^d _xf(x) v_t(x)q_t(x)\,dx, where differentiation under the expectation is justified by ‖∇xf(Xt)⊤D‖≤‖∇xf‖∞‖D‖\| _xf(X_t) D\|≤\| _xf\|_∞\|D\|. Since qtst=∇xqtq_ts_t= _xq_t, integration by parts gives ∫ℝd∇xf(x)⊤st(x)qt(x)dx=−∫ℝdΔxf(x)qt(x)dx. _R^d _xf(x) s_t(x)q_t(x)\,dx=- _R^d _xf(x)q_t(x)\,dx. Thus dt∫ℝdf(x)qt(x)x=∫ℝd(∇xf(x)⊤bt(x)+τ22Δxf(x))qt(x)x. ddt _R^df(x)q_t(x)\,dx= _R^d ( _xf(x) b_t(x)+ τ^22 _xf(x) )q_t(x)\,dx. It remains to verify the integrability required by the superposition principle. By conditional Jensen’s inequality, ∫ℝd‖vt(x)‖qt(x)x _R^d\|v_t(x)\|q_t(x)\,dx =|[D∣Xt]|≤‖D‖, =E\|E[D X_t]\| \|D\|, while the Gaussian score identity gives ∫ℝd‖st(x)‖qt(x)x _R^d\|s_t(x)\|q_t(x)\,dx =1σ|[G∣Xt]|≤‖G‖σ. = 1σE\|E[G X_t]\|≤ E\|G\|σ. Consequently, ∫01∫ℝd‖bt(x)‖qt(x)xt≤‖D‖+τ22σ|G|<∞. _0^1 _R^d\|b_t(x)\|q_t(x)\,dx\,dt \|D\|+ τ^22σE\|G\|<∞. Moreover, t↦qt(x)dxt q_t(x)\,dx is narrowly continuous: for every bounded continuous f, almost-sure continuity of t↦Xtt X_t and dominated convergence imply that t↦[f(Xt)]t [f(X_t)] is continuous. Therefore, (Trevisan, 2016, Theorem 2.5) yields a probability measure on C([0,1],ℝd)C([0,1];R^d) solving the martingale problem associated with drift btb_t and diffusion matrix τ2Idτ^2I_d, with time-t marginal qtq_t. For the canonical process Z, its martingale part has quadratic covariation τ2tIdτ^2tI_d; hence, by the Lévy characterization, it can be written as τBtτ B_t for a Brownian motion B. Thus Z is a weak solution of the displayed SDE and satisfies ℒ(Zt)=qtL(Z_t)=q_t for every t∈[0,1]t∈[0,1]. ∎ G.1 Path-Space Comparison via Girsanov We now introduce a framework Chen et al. (2023b), Chen et al. (2023a) that controls the KL divergence between the laws of two SDEs. Let T>0T>0 (for our flow matching setting, T=1T=1). Let 0=t0<t1<⋯<tN=T0=t_0<t_1<·s<t_N=T be a uniform time grid with step size h=tk+1−tkh=t_k+1-t_k. For every t∈[tk,tk+1)t∈[t_k,t_k+1), define ηh(t)=tk _h(t)=t_k. Recall that our sampling SDE is written as dZt=bt(Zt)dt+τdBtdZ_t=b_t(Z_t)\,dt+τ\,dB_t. For the continuous-time Euler discretization, we have dZ^t d Z_t =b^ηh(t)(Z^ηh(t))dt+τdBt. = b_ _h(t)( Z_ _h(t))\,dt+τ\,dB_t. Equivalently, for t∈[tk,tk+1)t∈[t_k,t_k+1), dZ^t d Z_t =b^tk(Z^tk)dt+τdBt. = b_t_k( Z_t_k)\,dt+τ\,dB_t. Assume both processes start from the same initial law: Z0∼ρ0Z_0 _0 and Z^0∼ρ0 Z_0 _0. Lemma 2 (Girsanov Bound for the Euler Process via Localization). Let τ>0τ>0, and let b and b^tkk=0N−1\ b_t_k\_k=0^N-1 be continuous drift fields. Fix a weak solution Z of the exact SDE on [0,T][0,T], with Z0∼ρ0Z_0 _0, and let Z Z be the full continuous-time Euler process initialized from ρ0 _0. Write ℙ=ℒ(Z)P=L(Z) and ℙ^=ℒ(Z^) P=L( Z) for their laws on C([0,T],ℝd)C([0,T];R^d). Define Δt(Z)=b^ηh(t)(Zηh(t))−bt(Zt) _t(Z)= b_ _h(t)(Z_ _h(t))-b_t(Z_t), and assume that ∫0T[‖Δt(Z)‖2]t<∞ _0^TE [\| _t(Z)\|^2\, ]dt<∞. Also assume ∫0T‖Δt(Z^)‖2t<∞,ℙ^-a.s. _0^T\| _t( Z)\|^2dt<∞, P-a.s. Then KL(ℒ(ZT)∥ℒ(Z^T))≤12τ2∫0T[‖bt(Zt)−b^ηh(t)(Zηh(t))‖2]dt. (L(Z_T)\|L( Z_T) )≤ 12τ^2 _0^TE [ \|b_t(Z_t)- b_ _h(t)(Z_ _h(t)) \|^2 ]dt. The proof has four steps. First, we define a path-dependent stopping-time construction ζm(x) _m(x), which gives the stopping times ζm(Z) _m(Z) along the exact process and ζm(Z^) _m( Z) along the full Euler process. Second, for ζm(Z) _m(Z), Novikov’s condition holds for each localized approximation, so we can apply Girsanov’s theorem and obtain a KL bound for each of them. Third, the additional integrability condition on Z Z is used to show that ζm(Z^)↑T _m( Z) T. Finally, we show that the localized Euler laws converge to the full Euler law on every truncated interval [0,T−ϵ][0,T-ε]; lower semicontinuity of relative entropy and then ϵ↓0ε 0 complete the argument. Proof. Let (Ω,ℱ,(ℱt)0≤t≤T,)( ,F,(F_t)_0≤ t≤ T,P) be a filtered probability space carrying the exact weak solution Z discussed in Lemma 1 and its driving Brownian motion W, so that dZt=bt(Zt)dt+τdWt. dZ_t=b_t(Z_t)\,dt+τ\,dW_t. Let ℙ=ℒ(Z)P=L_P(Z) be the corresponding path law on C([0,T],ℝd)C([0,T];R^d), and let ℙ P denote the path law of the full continuous-time Euler process. For every integer m≥1m≥ 1, define the stopping time ζm(x) _m(x) =inft∈[0,T]:1τ2∫0t‖Δs(x)‖2s≥m∧T. = \t∈[0,T]: 1τ^2 _0^t\| _s(x)\|^2ds≥ m \ T. Since t↦τ−2∫0t‖Δs(Z)‖2st τ^-2 _0^t\| _s(Z)\|^2ds is adapted and continuous, its first hitting time of [m,∞)[m,∞) is an (ℱt)(F_t)-stopping time. Thus ζm _m is a stopping time. When no confusion arises, we write ζm _m for ζm(Z) _m(Z) in the Girsanov construction, and for ζm(Z^) _m( Z) in the synchronous coupling that will be discussed later. By construction, 1τ2∫0ζm∥Δt(Z)∥2dt=1τ2∫0T∥Δt(Z)t≤ζm∥2dt≤m 1τ^2 _0 _m\| _t(Z)\|^2dt= 1τ^2 _0^T\| _t(Z)1_\t≤ _m\\|^2dt≤ m. Therefore Novikov’s condition holds for the stopped drift perturbation: [exp(12τ2∫0T‖Δt(Z)t≤ζm‖2dt)] _P [ ( 12τ^2 _0^T \| _t(Z)1_\t≤ _m\ \|^2dt ) ] ≤exp(m2)<∞. ≤ ( m2 )<∞. Hence the stopped stochastic exponential MmT:=exp(1τ∫0T(Δt(Z)t≤ζm)⊤dWt−12τ2∫0T‖Δt(Z)t≤ζm‖2dt) M_m^T:= ( 1τ _0^T ( _t(Z)1_\t≤ _m\ ) dW_t- 12τ^2 _0^T \| _t(Z)1_\t≤ _m\ \|^2dt ) is a true martingale with expectation [MmT]=[Mm0]=1E_P[M_m^T]=E_P[M_m^0]=1. Thus it defines a probability measure ^(m) P^(m) on (Ω,ℱ)( ,F) by d^(m)d=MmT d P^(m)dP=M_m^T (note that ∫d^(m)=∫MmT=[MmT]=1 d P^(m)= M_m^TdP=E_P[M_m^T]=1 so ^(m) P^(m) is a probability measure). By Girsanov’s theorem, under ^(m) P^(m), Wt(m)=Wt−1τ∫0tΔs(Z)s≤ζmds W_t^(m)=W_t- 1τ _0^t _s(Z)1_\s≤ _m\\,ds is a Brownian motion. Therefore, dWt dW_t =dWt(m)+1τΔt(Z)t≤ζmdt. =dW_t^(m)+ 1τ _t(Z)1_\t≤ _m\dt. Since, under P, dZt dZ_t =bt(Zt)dt+τdWt, =b_t(Z_t)dt+τ dW_t, we may rewrite the same process under ^(m) P^(m) as dZt dZ_t =bt(Zt)dt+τ(dWt(m)+1τΔt(Z)t≤ζmdt) =b_t(Z_t)dt+τ (dW_t^(m)+ 1τ _t(Z)1_\t≤ _m\dt ) =(bt(Zt)+Δt(Z)t≤ζm)dt+τdWt(m). = (b_t(Z_t)+ _t(Z)1_\t≤ _m\ )dt+τ dW_t^(m). Thus, before the stopping time ζm _m, the drift is bt(Zt)+Δt(Z) b_t(Z_t)+ _t(Z) =b^ηh(t)(Zηh(t)), = b_ _h(t)(Z_ _h(t)), and after the stopping time, the drift returns to bt(Zt)b_t(Z_t). Define ℙ^(m):=ℒ^(m)(Z) P^(m):=L_ P^(m)(Z). Thus ℙ^(m) P^(m) is the path law of the localized Euler approximation. For each m, the path-space KL divergence is well-defined. Since logdd^(m)=−1τ∫0T(Δt(Z)t≤ζm)⊤dWt+12τ2∫0T‖Δt(Z)t≤ζm‖2dt. dPd P^(m)=- 1τ _0^T ( _t(Z)1_\t≤ _m\ ) dW_t+ 12τ^2 _0^T \| _t(Z)1_\t≤ _m\ \|^2dt. Since Z is continuous and adapted, bt(Zt)b_t(Z_t) is predictable, while the grid-frozen term admits a predictable modification differing only at the deterministic grid points. Hence Δt(Z)t≤ζm _t(Z)1_\t≤ _m\ admits a predictable version. Moreover, by the definition of ζm _m, ∫0T∥Δt(Z)t≤ζm∥2dt≤mτ2, _0^T\| _t(Z)1_\t≤ _m\\|^2dt≤ mτ^2, so it is square-integrable and the stochastic integral has mean zero. Taking expectation under P, we obtain KL(∥^(m)) (P \| P^(m) ) =12τ2[∫0ζm‖Δt(Z)‖2t] = 12τ^2E_P [ _0 _m\| _t(Z)\|^2dt ] ≤12τ2[∫0T‖Δt(Z)‖2t]. ≤ 12τ^2E_P [ _0^T\| _t(Z)\|^2dt ]. Consequently, by the data-processing inequality under the map ω↦Z(ω)ω Z(ω), KL(ℙ∥ℙ^(m))≤KL(∥^(m)). (P \| P^(m) ) (P \| P^(m) ). Therefore, KL(ℙ∥ℙ^(m)) (P \| P^(m) ) ≤12τ2[∫0T‖Δt(Z)t≤ζm‖2dt]=12τ2[∫0ζm∥Δt(Z)∥2dt] ≤ 12τ^2E_P [ _0^T \| _t(Z)1_\t≤ _m\ \|^2dt ]= 12τ^2E_P [ _0 _m\| _t(Z)\|^2dt ] ≤12τ2[∫0T‖Δt(Z)‖2t]. ≤ 12τ^2E_P [ _0^T\| _t(Z)\|^2dt ]. It remains to pass from the localized Euler approximation to the full continuous-time Euler approximation. For each fixed m, since MmtM_m^t is a martingale with Mm0=1M_m^0=1, [MmT∣ℱ0]=1,E_P[M_m^T _0]=1, so the change of measure preserves the initial law and Z0∼ρ0Z_0 _0 under ^(m) P^(m). Under ^(m) P^(m), on the same probability space, construct the continuous time Euler process Z Z using the same initial value Z0Z_0 and Brownian motion W(m)W^(m): Z^t Z_t =Z^tk+(t−tk)b^tk(Z^tk)+τ(Wt(m)−Wtk(m)),t∈[tk,tk+1], = Z_t_k+(t-t_k) b_t_k( Z_t_k)+τ (W_t^(m)-W_t_k^(m) ), t∈[t_k,t_k+1], with Z^0=Z0 Z_0=Z_0. Because W(m)W^(m) is Brownian with respect to (ℱt)(F_t) and Z0Z_0 is ℱ0F_0-measurable, W(m)W^(m) is independent of Z0Z_0. The change of measure preserves the law ρ0 _0 of Z0Z_0, and the grid-by-grid Euler recursion is a measurable function of (Z0,W(m))(Z_0,W^(m)). Hence ℒ^(m)(Z^)=ℙ^L_ P^(m)( Z)= P. Let ρm=ζm(Z)∧ζm(Z^) _m= _m(Z) _m( Z). Before ρm _m, both Z and Z Z satisfy the same continuous-time Euler update with the same initial value and Brownian motion, so they agree grid-by-grid on [0,ρm][0, _m]. Hence ∫0t‖Δs(Z)‖2s=∫0t‖Δs(Z^)‖2s _0^t\| _s(Z)\|^2ds= _0^t\| _s( Z)\|^2ds for every t≤ρmt≤ _m. Suppose ζm(Z)<ζm(Z^) _m(Z)< _m( Z). Then ρm=ζm(Z)<T _m= _m(Z)<T, so continuity of the defining integral gives τ−2∫0ρm‖Δs(Z)‖2s=mτ^-2 _0 _m\| _s(Z)\|^2ds=m. The preceding equality implies the same for Z Z, hence ζm(Z^)≤ρm=ζm(Z) _m( Z)≤ _m= _m(Z), a contradiction. The reverse strict inequality is ruled out in the same way. Therefore ζm(Z)=ζm(Z^) _m(Z)= _m( Z). Fix ε∈(0,T) ∈(0,T) and define the path truncation map πε:C([0,T],ℝd)→C([0,T],ℝd) _ :C([0,T];R^d)→ C([0,T];R^d) by (πε(ω))(t)=ω(t∧(T−ε)),0≤t≤T. ( _ (ω) )(t)=ω(t (T- )), 0≤ t≤ T. On the event ζm(Z)>T−ε\ _m(Z)>T- \, we have πε(Z)=πε(Z^). _ (Z)= _ ( Z). Moreover, ^(m)(ζm(Z)≤T−ε) P^(m) ( _m(Z)≤ T- ) =^(m)(ζm(Z^)≤T−ε)=ℙ^(ζm(Z^)≤T−ε)⟶0 as m→∞. = P^(m) ( _m( Z)≤ T- )= P ( _m( Z)≤ T- ) 0 as m→∞. Indeed, ∫0T‖Δt(Z^)‖2t<∞ _0^T\| _t( Z)\|^2dt<∞, ℙ P-almost surely, implies that ζm(Z^)=T _m( Z)=T for all sufficiently large m, almost surely. Therefore, for every bounded continuous F:C([0,T],ℝd)→ℝF:C([0,T];R^d) , |^(m)[F(πε(Z))]−^(m)[F(πε(Z^))]| |E_ P^(m) [F( _ (Z)) ]-E_ P^(m) [F( _ ( Z)) ] | ≤2‖F‖∞^(m)(ζm(Z)≤T−ε)⟶0. ≤ 2\|F\|_∞\, P^(m) ( _m(Z)≤ T- ) 0. Since ℒ^(m)(Z^)=ℙ^L_ P^(m)( Z)= P, we obtain ℒ^(m)(πε(Z))⟶(πε)#ℙ^. _ P^(m)( _ (Z)) w ( _ )_\# P. By lower semicontinuity of KL divergence and the data-processing inequality, KL((πε)#ℙ∥(πε)#ℙ^) (( _ )_\#P \|( _ )_\# P ) ≤lim infm→∞KL(ℒ(πε(Z))∥ℒ^(m)(πε(Z))) ≤ _m→∞KL (L_P( _ (Z)) \|L_ P^(m)( _ (Z)) ) ≤lim infm→∞KL(∥^(m)) ≤ _m→∞KL (P \| P^(m) ) ≤12τ2[∫0T‖Δt(Z)‖2t]. ≤ 12τ^2E_P [ _0^T\| _t(Z)\|^2dt ]. Finally, since ℒ(πε(Z))→ℒ(Z)L( _ (Z)) w→L(Z) and ℒ(πε(Z^))→ℒ(Z^)L( _ ( Z)) w→L( Z) as ε→0 → 0, applying lower semicontinuity of KL divergence once more gives KL(ℙ∥ℙ^)≤lim infε↓0KL(ℒ(πε(Z))∥ℒ(πε(Z^)))≤12τ2[∫0T‖Δt(Z)‖2t]. (P \| P )≤ _ 0KL (L( _ (Z)) \|L( _ ( Z)) )≤ 12τ^2E_P [ _0^T\| _t(Z)\|^2dt ]. Again, by the data-processing inequality, KL(ℒ(ZT)∥ℒ(Z^T))≤KL(ℙ∥ℙ^)≤12τ2[∫0T‖Δt(Z)‖2t]. (L(Z_T) \|L( Z_T) ) (P \| P )≤ 12τ^2E_P [ _0^T\| _t(Z)\|^2dt ]. Equivalently, KL(ℒ(ZT)∥ℒ(Z^T)) (L(Z_T) \|L( Z_T) ) ≤12τ2∫0T[‖bt(Zt)−b^ηh(t)(Zηh(t))‖2]t. ≤ 12τ^2 _0^TE_P [ \|b_t(Z_t)- b_ _h(t)(Z_ _h(t)) \|^2 ]dt. ∎ A point concerning the localization limit. The approximation argument in Chen et al. (2023b, Section 5.2) places the localized and full Euler processes on a common probability space and subsequently invokes convergence of the localizing times. The Lipschitz-SDE result used there, cited as Corollary 8.5 in Chen et al. (2023b, Section 5.2), comes from the chapter on stochastic differential equations in Le Gall (2016) and provides a measurable strong-solution representation. Although this result can support a synchronous realization when its hypotheses hold, it does not by itself show that localizing times constructed under the reference law converge to T under the full Euler law. This Euler-side exhaustion is required to obtain agreement of the localized and full Euler paths on every interval [0,T−ε][0,T- ], and hence the weak convergence of the localized laws. We avoid this issue by using the explicit energy stopping times ζm _m defined above. It remains to verify that ∫0T‖Δt(Z^)‖2t<∞,ℙ^-a.s., _0^T \| _t( Z) \|^2\,dt<∞, P-a.s., because this condition implies that ζm(Z^)↑T _m( Z) T almost surely; indeed, ζm(Z^)=T _m( Z)=T for all sufficiently large m. The following proposition establishes this property under our continuity assumptions. Proposition 3 (Almost-Sure Finiteness of the Pathwise Drift Mismatch). Let 0=t0<t1<⋯<tN=T0=t_0<t_1<·s<t_N=T be a finite time grid, and define ηh(t)=tk _h(t)=t_k for t∈[tk,tk+1)t∈[t_k,t_k+1). Let Z^=(Z^t)0≤t≤T Z=( Z_t)_0≤ t≤ T be a continuous-path process under ℙ P. Let b:[0,T]×ℝd→ℝdb:[0,T]×R^d ^d be continuous. For every k∈0,…,N−1k∈\0,…,N-1\, let b^tk:ℝd→ℝd b_t_k:R^d ^d be continuous. Define Δt(Z^) _t( Z) =b^ηh(t)(Z^ηh(t))−bt(Z^t),0≤t≤T. = b_ _h(t)( Z_ _h(t))-b_t( Z_t), 0≤ t≤ T. Then ∫0T‖Δt(Z^)‖2t _0^T \| _t( Z) \|^2dt <∞,ℙ^-a.s. <∞, P-a.s. Proof. It suffices to show, there exists some Ω0 _0 with ℙ^(Ω0)=1 P( _0)=1 s.t. supt∈[0,T]‖Δt(Z^(ω))‖<∞,∀ω∈Ω0 _t∈[0,T]\| _t( Z(ω))\|<∞,∀ω∈ _0. Since Z Z has continuous paths under ℙ P, there exists Ω0 _0 such that ℙ^(Ω0)=1 P( _0)=1 and for every ω∈Ω0ω∈ _0, the map t↦Z^t(ω)t Z_t(ω) is continuous on [0,T][0,T]. Fix ω∈Ω0ω∈ _0. Since [0,T][0,T] is compact and t↦Z^t(ω)t Z_t(ω) is continuous, the image set Kω:=Z^t(ω):0≤t≤TK_ω:= \ Z_t(ω):0≤ t≤ T \ is a compact subset of ℝdR^d. Because b and b^tk b_t_k are continuous, and because [0,T]×Kω[0,T]× K_ω is compact, we have Mb(ω) M_b(ω) :=sup0≤t≤Tsupx∈Kω‖bt(x)‖<∞, := _0≤ t≤ T _x∈ K_ω \|b_t(x) \|<∞, Mb^,k(ω) M_ b,k(ω) :=supx∈Kω‖b^tk(x)‖<∞, := _x∈ K_ω \| b_t_k(x) \|<∞, Mb^(ω) M_ b(ω) :=max0≤k≤N−1Mb^,k(ω)<∞. := _0≤ k≤ N-1M_ b,k(ω)<∞. Therefore ‖bt(Z^t(ω))‖≤Mb(ω),∀t∈[0,T] \|b_t( Z_t(ω)) \|≤ M_b(ω),∀ t∈[0,T] and ‖b^ηh(t)(Z^ηh(t)(ω))‖≤Mb^(ω) \| b_ _h(t) ( Z_ _h(t)(ω) ) \|≤ M_ b(ω). Hence, for every t∈[0,T]t∈[0,T], ‖Δt(Z^(ω))‖ \| _t( Z(ω)) \| =‖bt(Z^t(ω))−b^ηh(t)(Z^ηh(t)(ω))‖≤‖bt(Z^t(ω))‖+‖b^ηh(t)(Z^ηh(t)(ω))‖ = \|b_t( Z_t(ω))- b_ _h(t) ( Z_ _h(t)(ω) ) \|≤ \|b_t( Z_t(ω)) \|+ \| b_ _h(t) ( Z_ _h(t)(ω) ) \| ≤Mb(ω)+Mb^(ω). ≤ M_b(ω)+M_ b(ω). Therefore, ∫0T‖Δt(Z^(ω))‖2t _0^T \| _t( Z(ω)) \|^2dt ≤∫0T(Mb(ω)+Mb^(ω))2t=T(Mb(ω)+Mb^(ω))2<∞. ≤ _0^T (M_b(ω)+M_ b(ω) )^2dt=T (M_b(ω)+M_ b(ω) )^2<∞. Since this holds for every ω∈Ω0ω∈ _0, and since ℙ^(Ω0)=1 P( _0)=1, we conclude that ∫0T‖Δt(Z^)‖2t _0^T \| _t( Z) \|^2dt <∞,ℙ^-a.s. <∞, P-a.s. This proves the claim. ∎ G.2 Posterior Derivative Identities This subsection collects the derivative identities and integrated bounds needed for the combined Itô analysis of the score increment st(Zt)−stk(Ztk). s_t(Z_t)-s_t_k(Z_t_k). The key point is that the term ∇x(∇x⋅vt) _x( _x· v_t) requires control of the full Hessian tensor Dx2vtD_x^2v_t, not merely the componentwise Laplacian Δxvt _xv_t. Lemma 3 (Score Identity Under Gaussian Smoothing). Let Y0Y_0 and Y1Y_1 be random variables taking values in ℝdR^d, and fix t∈[0,T]t∈[0,T]. Let D=Y1−Y0D=Y_1-Y_0. Define St=Y0+tDS_t=Y_0+tD. Let G∼(0,Id)G N(0,I_d) be independent of (Y0,D)(Y_0,D), and define Xt=St+σGX_t=S_t+σ G with σ>0σ>0. Let qtq_t be the density of XtX_t, and define the score st:ℝd→ℝds_t:R^d ^d by st(x)=∇xlogqt(x)s_t(x)= _x q_t(x). Then st(x) s_t(x) =−1σ[G∣Xt=x]. =- 1σE [G X_t=x ]. Proof. Let Y0,Y1 P_Y_0,Y_1 be the law of (Y0,Y1)(Y_0,Y_1). For fixed t∈[0,T]t∈[0,T], define stpath(y0,y1) s_t^path(y_0,y_1) =(1−t)y0+ty1. =(1-t)y_0+ty_1. The superscript “path” is only used in this proof to avoid confusing the interpolation point stpath(y0,y1)s_t^path(y_0,y_1) with the score st(x)s_t(x). Since Xt=St+σGX_t=S_t+σ G, conditional on (Y0,Y1)=(y0,y1)(Y_0,Y_1)=(y_0,y_1), the random variable XtX_t is Gaussian with mean stpath(y0,y1)s_t^path(y_0,y_1) and covariance matrix σ2Idσ^2I_d. Therefore the conditional density of Xt=xX_t=x given (Y0,Y1)=(y0,y1)(Y_0,Y_1)=(y_0,y_1) is 1(2πσ2)d/2exp(−‖x−stpath(y0,y1)‖22σ2). 1(2πσ^2)^d/2 (- \|x-s_t^path(y_0,y_1) \|^22σ^2 ). Hence the marginal density of XtX_t is qt(x) q_t(x) =∫ℝd×ℝd1(2πσ2)d/2exp(−‖x−stpath(y0,y1)‖22σ2)dY0,Y1(y0,y1). = _R^d×R^d 1(2πσ^2)^d/2 (- \|x-s_t^path(y_0,y_1) \|^22σ^2 )\,d P_Y_0,Y_1(y_0,y_1). Because the Gaussian density is strictly positive, qt(x)>0q_t(x)>0 for every x∈ℝdx ^d. We now compute ∇xqt(x) _xq_t(x). For each coordinate j∈1,…,dj∈\1,…,d\, ∂xjexp(−‖x−stpath(y0,y1)‖22σ2) _x_j (- \|x-s_t^path(y_0,y_1) \|^22σ^2 ) =−xj−st,jpath(y0,y1)σ2exp(−‖x−stpath(y0,y1)‖22σ2). =- x_j-s_t,j^path(y_0,y_1)σ^2 (- \|x-s_t^path(y_0,y_1) \|^22σ^2 ). Therefore, ∂xjqt(x) _x_jq_t(x) =∫ℝd×ℝd−xj−st,jpath(y0,y1)σ21(2πσ2)d/2exp(−‖x−stpath(y0,y1)‖22σ2)dY0,Y1(y0,y1). = _R^d×R^d- x_j-s_t,j^path(y_0,y_1)σ^2 1(2πσ^2)^d/2 (- \|x-s_t^path(y_0,y_1) \|^22σ^2 )\,d P_Y_0,Y_1(y_0,y_1). Since qt(x)>0q_t(x)>0, ∂xjlogqt(x) _x_j q_t(x) =∂xjqt(x)qt(x) = _x_jq_t(x)q_t(x) =−1σ2∫ℝd×ℝd(xj−st,jpath(y0,y1))1(2πσ2)d/2exp(−‖x−stpath(y0,y1)‖22σ2)dY0,Y1(y0,y1)qt(x). =- 1σ^2 _R^d×R^d (x_j-s_t,j^path(y_0,y_1) ) 1(2πσ^2)^d/2 (- \|x-s_t^path(y_0,y_1) \|^22σ^2 )\,d P_Y_0,Y_1(y_0,y_1)q_t(x). By Bayes’ rule, the conditional law of (Y0,Y1)(Y_0,Y_1) given Xt=xX_t=x is obtained by weighting the prior law Y0,Y1 P_Y_0,Y_1 by the same Gaussian likelihood. Thus ∂xjlogqt(x) _x_j q_t(x) =−1σ2[xj−St,j∣Xt=x]. =- 1σ^2E [x_j-S_t,j X_t=x ]. Recall by definition x=St+σGx=S_t+σ G, so we have xj−St,j=σGjx_j-S_t,j=σ G_j inside the conditional expectation. Therefore, ∂xjlogqt(x)=−1σ2[σGj∣Xt=x]=−1σ[Gj∣Xt=x]. _x_j q_t(x)=- 1σ^2E [σ G_j X_t=x ]=- 1σE [G_j X_t=x ]. Since this holds for every j∈1,…,dj∈\1,…,d\, we obtain ∇xlogqt(x) _x q_t(x) =−1σ[G∣Xt=x]. =- 1σE [G X_t=x ]. By definition st(x)=∇xlogqt(x)s_t(x)= _x q_t(x), so st(x) s_t(x) =−1σ[G∣Xt=x]. =- 1σE [G X_t=x ]. This proves the lemma. ∎ Lemma 4 (Posterior Derivative Identities for the Smoothed Velocity and Score). Let Y0Y_0 and Y1Y_1 be random variables taking values in ℝdR^d, and fix t∈[0,T]t∈[0,T]. Let D=Y1−Y0D=Y_1-Y_0. Define St=Y0+tDS_t=Y_0+tD. Let G∼(0,Id)G N(0,I_d) be independent of (Y0,D)(Y_0,D), and define Xt=St+σGX_t=S_t+σ G with σ>0σ>0. Assume ‖D‖3<∞E\|D\|^3<∞. Let qtq_t be the density of XtX_t. Define the smoothed velocity vt:ℝd→ℝdv_t:R^d ^d by vt(x) v_t(x) =[D∣Xt=x], =E [D X_t=x ], and define the score st:ℝd→ℝds_t:R^d ^d by st(x)=∇xlogqt(x)s_t(x)= _x q_t(x). Under the stated moment condition, all derivatives below exist and the required differentiations under the Gaussian-mixture integral are justified. Then the score satisfies st(x) s_t(x) =−1σ[G∣Xt=x]. =- 1σE [G X_t=x ]. The velocity derivatives satisfy Dxvt(x) D_xv_t(x) =−1σCov(D,G∣Xt=x), =- 1σCov (D,G X_t=x ), ∂tvt(x) _tv_t(x) =1σCov(D,D⊤G∣Xt=x), = 1σCov (D,D G X_t=x ), and for every i,j,ℓ∈1,…,di,j, ∈\1,…,d\, ∂xjxℓvt,i(x) _x_jx_ v_t,i(x) =1σ2[(Di−[Di∣Xt=x])(Gj−[Gj∣Xt=x])(Gℓ−[Gℓ∣Xt=x])∣Xt=x]. = 1σ^2E [ (D_i-E[D_i X_t=x] ) (G_j-E[G_j X_t=x] ) (G_ -E[G_ X_t=x] ) X_t=x ]. In particular, Δxvt(x) _xv_t(x) =1σ2[(D−[D∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. = 1σ^2E [ (D-E[D X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. The score derivatives satisfy Dxst(x) D_xs_t(x) =1σ2(Cov(G,G∣Xt=x)−Id), = 1σ^2 (Cov (G,G X_t=x )-I_d ), and Δxst(x) _xs_t(x) =−1σ3[(G−[G∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. =- 1σ^3E [ (G-E[G X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. Proof. We first record the posterior derivative identity used throughout the proof. Let F:ℝd×ℝd→ℝmF:R^d×R^d ^m be differentiable in its second argument and assume that F(D,G)F(D,G) and ∂gjF(D,G) _g_jF(D,G) are integrable against the relevant Gaussian posterior weights. In the expression F(D,G)F(D,G), the first input is D, and the second input is G. For x∈ℝdx ^d, define mF(x) m_F(x) =[F(D,G)∣Xt=x]. =E [F(D,G) X_t=x ]. Here ∂xjmF(x) _x_jm_F(x) denotes the derivative of the function x↦mF(x)x m_F(x) with respect to the j-th coordinate of x. Also, ∂gjF(D,G) _g_jF(D,G) denotes the derivative of the function g↦F(D,g)g F(D,g) with respect to the j-th coordinate of g, evaluated at g=Gg=G. Then, for every j∈1,…,dj∈\1,…,d\, ∂xjmF(x) _x_jm_F(x) =1σ[∂gjF(D,G)∣Xt=x]−1σCov(F(D,G),Gj∣Xt=x). = 1σE [ _g_jF(D,G) X_t=x ]- 1σCov (F(D,G),G_j X_t=x ). If F is vector-valued, the covariance is interpreted componentwise: Cov(F(D,G),Gj∣Xt=x) (F(D,G),G_j X_t=x ) =[(F(D,G)−[F(D,G)∣Xt=x])(Gj−[Gj∣Xt=x])∣Xt=x]. =E [ (F(D,G)-E[F(D,G) X_t=x] ) (G_j-E[G_j X_t=x] ) X_t=x ]. We prove the identity. Since Xt=St+σGX_t=S_t+σ G, conditioning on Xt=xX_t=x implies G=(x−St)/σG=(x-S_t)/σ. Thus mF(x)m_F(x) can be written as mF(x) m_F(x) =Γ(Y0,Y1)[F(D,x−Stσ)exp(−12‖x−Stσ‖2)]Γ(Y0,Y1)[exp(−12‖x−Stσ‖2)]. = E_ (Y_0,Y_1) [F (D, x-S_tσ ) (- 12 \| x-S_tσ \|^2 ) ]E_ (Y_0,Y_1) [ (- 12 \| x-S_tσ \|^2 ) ]. The normalizing constant of the standard Gaussian density is omitted because it appears in both numerator and denominator and cancels. Define AF(x) A_F(x) =Γ(Y0,Y1)[F(D,x−Stσ)exp(−12‖x−Stσ‖2)], =E_ (Y_0,Y_1) [F (D, x-S_tσ ) (- 12 \| x-S_tσ \|^2 ) ], B(x) B(x) =Γ(Y0,Y1)[exp(−12‖x−Stσ‖2)]. =E_ (Y_0,Y_1) [ (- 12 \| x-S_tσ \|^2 ) ]. Then mF(x)=AF(x)/B(x)m_F(x)=A_F(x)/B(x). We differentiate AF(x)A_F(x) with respect to xjx_j. The derivative of the F-term is ∂xjF(D,x−Stσ) _x_jF (D, x-S_tσ ) =1σ∂gjF(D,G). = 1σ _g_jF(D,G). The derivative of the Gaussian weight is ∂xjexp(−12‖x−Stσ‖2) _x_j (- 12 \| x-S_tσ \|^2 ) =−1σGjexp(−‖G‖22). =- 1σG_j (- \|G\|^22 ). Therefore, ∂xjAF(x) _x_jA_F(x) =1σΓ(Y0,Y1)[∂gjF(D,G)exp(−‖G‖22)]−1σΓ(Y0,Y1)[F(D,G)Gjexp(−‖G‖22)]. = 1σE_ (Y_0,Y_1) [ _g_jF(D,G) (- \|G\|^22 ) ]- 1σE_ (Y_0,Y_1) [F(D,G)G_j (- \|G\|^22 ) ]. Similarly, ∂xjB(x) _x_jB(x) =−1σΓ(Y0,Y1)[Gjexp(−‖G‖22)]. =- 1σE_ (Y_0,Y_1) [G_j (- \|G\|^22 ) ]. By the quotient rule, ∂xjmF(x) _x_jm_F(x) =∂xjAF(x)B(x)−AF(x)B(x)2∂xjB(x). = _x_jA_F(x)B(x)- A_F(x)B(x)^2 _x_jB(x). Substituting the formulas for ∂xjAF(x) _x_jA_F(x) and ∂xjB(x) _x_jB(x), we obtain ∂xjmF(x) _x_jm_F(x) =1σΓ(Y0,Y1)[∂gjF(D,G)exp(−‖G‖22)]B(x)−1σΓ(Y0,Y1)[F(D,G)Gjexp(−‖G‖22)]B(x) = 1σ E_ (Y_0,Y_1) [ _g_jF(D,G) (- \|G\|^22 ) ]B(x)- 1σ E_ (Y_0,Y_1) [F(D,G)G_j (- \|G\|^22 ) ]B(x) +1σAF(x)B(x)Γ(Y0,Y1)[Gjexp(−‖G‖22)]B(x). + 1σ A_F(x)B(x) E_ (Y_0,Y_1) [G_j (- \|G\|^22 ) ]B(x). Each weighted ratio is a conditional expectation given Xt=xX_t=x. Hence ∂xjmF(x) _x_jm_F(x) =1σ[∂gjF(D,G)∣Xt=x]−1σ[F(D,G)Gj∣Xt=x] = 1σE [ _g_jF(D,G) X_t=x ]- 1σE [F(D,G)G_j X_t=x ] +1σ[F(D,G)∣Xt=x][Gj∣Xt=x]. + 1σE [F(D,G) X_t=x ]E [G_j X_t=x ]. The last two terms are minus the conditional covariance between F(D,G)F(D,G) and GjG_j. Therefore, ∂xjmF(x) _x_jm_F(x) =1σ[∂gjF(D,G)∣Xt=x]−1σCov(F(D,G),Gj∣Xt=x). = 1σE [ _g_jF(D,G) X_t=x ]- 1σCov (F(D,G),G_j X_t=x ). We now derive the score identity. Recall we have st(x)=−1σ[G∣Xt=x]s_t(x)=- 1σE[G X_t=x]. Apply the posterior derivative identity with F(D,G)=DiF(D,G)=D_i. Since DiD_i has no explicit dependence on G, we have ∂gjDi=0 _g_jD_i=0. Therefore, ∂xjvt,i(x)=∂xj[Di∣Xt=x]=−1σCov(Di,Gj∣Xt=x). _x_jv_t,i(x)= _x_jE [D_i X_t=x ]=- 1σCov (D_i,G_j X_t=x ). This proves Dxvt(x)=−1σCov(D,G∣Xt=x). D_xv_t(x)=- 1σCov (D,G X_t=x ). Next, we compute the time derivative of vt(x)v_t(x). Fix x∈ℝdx ^d. For (y0,y1)∈ℝd×ℝd(y_0,y_1) ^d×R^d, write st(y0,y1)=y0+t(y1−y0)s_t(y_0,y_1)=y_0+t(y_1-y_0). Let γ denote the coupling law of (Y0,Y1)(Y_0,Y_1). Recall the formula for vt(x)v_t(x) is vt(x) v_t(x) =∫ℝd×ℝd(y1−y0)exp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1)∫ℝd×ℝdexp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1). = _R^d×R^d(y_1-y_0) (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1) _R^d×R^d (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1). For every i∈1,…,di∈\1,…,d\, define the numerator of the i-th coordinate by Ai(t,x) A_i(t,x) =∫ℝd×ℝd(y1−y0)iexp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1), = _R^d×R^d(y_1-y_0)_i (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1), and define the denominator by B(t,x) B(t,x) =∫ℝd×ℝdexp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1). = _R^d×R^d (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1). Then vt,i(x)=Ai(t,x)B(t,x)v_t,i(x)= A_i(t,x)B(t,x). We now differentiate the Gaussian weight with respect to t. Since ∂tst(y0,y1)=y1−y0 _ts_t(y_0,y_1)=y_1-y_0, we have ∂texp(−‖x−st(y0,y1)‖22σ2)=(y1−y0)⊤(x−st(y0,y1))σ2exp(−‖x−st(y0,y1)‖22σ2). _t (- \|x-s_t(y_0,y_1) \|^22σ^2 )= (y_1-y_0) (x-s_t(y_0,y_1) )σ^2 (- \|x-s_t(y_0,y_1) \|^22σ^2 ). Using this derivative inside the integral defining Ai(t,x)A_i(t,x), we get ∂tAi(t,x) _tA_i(t,x) =1σ2∫ℝd×ℝd(y1−y0)i(y1−y0)⊤(x−st(y0,y1))⋅exp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1), = 1σ^2 _R^d×R^d(y_1-y_0)_i(y_1-y_0) (x-s_t(y_0,y_1) )· (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1), ∂tB(t,x) _tB(t,x) =1σ2∫ℝd×ℝd(y1−y0)⊤(x−st(y0,y1))⋅exp(−‖x−st(y0,y1)‖22σ2)γ(dy0,dy1). = 1σ^2 _R^d×R^d(y_1-y_0) (x-s_t(y_0,y_1) )· (- \|x-s_t(y_0,y_1) \|^22σ^2 )\,γ(dy_0,dy_1). By the quotient rule, ∂tvt,i(x) _tv_t,i(x) =∂tAi(t,x)B(t,x)−Ai(t,x)B(t,x)2∂tB(t,x). = _tA_i(t,x)B(t,x)- A_i(t,x)B(t,x)^2 _tB(t,x). Substituting the formulas for ∂tAi(t,x) _tA_i(t,x) and ∂tB(t,x) _tB(t,x), and rewriting the weighted ratios as posterior conditional expectations given Xt=xX_t=x, we obtain ∂tvt,i(x)=1σ2[DiD⊤(x−St)∣Xt=x]−1σ2[Di∣Xt=x][D⊤(x−St)∣Xt=x]. _tv_t,i(x)= 1σ^2E [D_iD (x-S_t) X_t=x ]- 1σ^2E [D_i X_t=x ]E [D (x-S_t) X_t=x ]. The right-hand side is exactly the conditional covariance between DiD_i and D⊤(x−St)D (x-S_t). Hence ∂tvt,i(x) _tv_t,i(x) =1σ2Cov(Di,D⊤(x−St)∣Xt=x). = 1σ^2Cov (D_i,D (x-S_t) X_t=x ). Since this holds for every i∈1,…,di∈\1,…,d\, we get the vector identity ∂tvt(x) _tv_t(x) =1σ2Cov(D,D⊤(x−St)∣Xt=x). = 1σ^2Cov (D,D (x-S_t) X_t=x ). Finally, under the conditioning Xt=xX_t=x, we have x−St=σGx-S_t=σ G. Therefore, ∂tvt(x) _tv_t(x) =1σCov(D,D⊤G∣Xt=x). = 1σCov (D,D G X_t=x ). Now apply the posterior derivative identity with F(D,G)=GiF(D,G)=G_i. Since ∂gjGi _g_jG_i equals 11 if i=ji=j, and equals 00 otherwise, ∂xj[Gi∣Xt=x] _x_jE [G_i X_t=x ] =1σi=j−1σCov(Gi,Gj∣Xt=x). = 1σ1_\i=j\- 1σCov (G_i,G_j X_t=x ). Since st,i(x)=−(1/σ)[Gi∣Xt=x]s_t,i(x)=-(1/σ)E[G_i X_t=x], we get ∂xjst,i(x) _x_js_t,i(x) =1σ2(Cov(Gi,Gj∣Xt=x)−i=j). = 1σ^2 (Cov (G_i,G_j X_t=x )-1_\i=j\ ). Therefore, Dxst(x) D_xs_t(x) =1σ2(Cov(G,G∣Xt=x)−Id). = 1σ^2 (Cov (G,G X_t=x )-I_d ). Finally, we compute the Laplacian of the score. Differentiating the last display once more with respect to xjx_j, and using the same centered-product calculation as above, gives ∂xjxjst,i(x) _x_jx_js_t,i(x) =−1σ3[(Gi−[Gi∣Xt=x])(Gj−[Gj∣Xt=x])2∣Xt=x]. =- 1σ^3E [ (G_i-E[G_i X_t=x] ) (G_j-E[G_j X_t=x] )^2 X_t=x ]. Summing over j=1,…,dj=1,…,d gives Δxst,i(x) _xs_t,i(x) =−1σ3[(Gi−[Gi∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. =- 1σ^3E [ (G_i-E[G_i X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. Since this holds for every i∈1,…,di∈\1,…,d\, Δxst(x) _xs_t(x) =−1σ3[(G−[G∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. =- 1σ^3E [ (G-E[G X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. The preceding argument proves all identities in the statement except the full Hessian formula for vtv_t. We now establish that remaining identity. Fix i,j,ℓ∈1,…,di,j, ∈\1,…,d\. From the Jacobian identity, ∂xjvt,i(x) _x_jv_t,i(x) =−1σCov(Di,Gj∣Xt=x). =- 1σCov (D_i,G_j X_t=x ). Therefore, ∂xjxℓvt,i(x) _x_jx_ v_t,i(x) =−1σ∂xℓCov(Di,Gj∣Xt=x). =- 1σ _x_ Cov (D_i,G_j X_t=x ). By definition, Cov(Di,Gj∣Xt=x) (D_i,G_j X_t=x ) =[DiGj∣Xt=x]−[Di∣Xt=x][Gj∣Xt=x]. =E [D_iG_j X_t=x ]-E [D_i X_t=x ]E [G_j X_t=x ]. Here the conditional expectations are taken with respect to the posterior law of (Y0,Y1)(Y_0,Y_1) given Xt=xX_t=x. Explicitly, under this posterior law, Di D_i =Y1,i−Y0,i, =Y_1,i-Y_0,i, Gj G_j =xj−((1−t)Y0,j+tY1,j)σ. = x_j- ((1-t)Y_0,j+tY_1,j )σ. We differentiate each of the three conditional expectations appearing above. First, apply the posterior derivative identity with F(D,G) F(D,G) =DiGj. =D_iG_j. Since DiD_i does not depend on G, while GjG_j depends on the j-th coordinate of G, we have ∂gℓ(DiGj) _g_ (D_iG_j ) =Dij=ℓ. =D_i1_\j= \. Therefore, ∂xℓ[DiGj∣Xt=x] _x_ E [D_iG_j X_t=x ] =1σ[Dij=ℓ∣Xt=x]−1σCov(DiGj,Gℓ∣Xt=x). = 1σE [D_i1_\j= \ X_t=x ]- 1σCov (D_iG_j,G_ X_t=x ). Second, apply the posterior derivative identity with F(D,G)=DiF(D,G)=D_i, we obtain ∂xℓ[Di∣Xt=x]=−1σCov(Di,Gℓ∣Xt=x). _x_ E [D_i X_t=x ]=- 1σCov (D_i,G_ X_t=x ). Third, apply the posterior derivative identity with F(D,G)=GjF(D,G)=G_j, we get ∂xℓ[Gj∣Xt=x]=1σj=ℓ−1σCov(Gj,Gℓ∣Xt=x). _x_ E [G_j X_t=x ]= 1σ1_\j= \- 1σCov (G_j,G_ X_t=x ). Now differentiate the covariance formula: ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =∂xℓ[DiGj∣Xt=x]−(∂xℓ[Di∣Xt=x])[Gj∣Xt=x] = _x_ E [D_iG_j X_t=x ]- ( _x_ E [D_i X_t=x ] )E [G_j X_t=x ] −[Di∣Xt=x](∂xℓ[Gj∣Xt=x]). -E [D_i X_t=x ] ( _x_ E [G_j X_t=x ] ). Substituting the three derivative identities above gives ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =1σ[Dij=ℓ∣Xt=x]−1σCov(DiGj,Gℓ∣Xt=x) = 1σE [D_i1_\j= \ X_t=x ]- 1σCov (D_iG_j,G_ X_t=x ) +1σCov(Di,Gℓ∣Xt=x)[Gj∣Xt=x] + 1σCov (D_i,G_ X_t=x )E [G_j X_t=x ] −1σ[Di∣Xt=x]j=ℓ+1σ[Di∣Xt=x]Cov(Gj,Gℓ∣Xt=x). - 1σE [D_i X_t=x ]1_\j= \+ 1σE [D_i X_t=x ]Cov (G_j,G_ X_t=x ). The two terms containing j=ℓ1_\j= \ cancel: 1σ([Di∣Xt=x]−[Di∣Xt=x])j=ℓ 1σ (E [D_i X_t=x ]-E [D_i X_t=x ] )1_\j= \ =0. =0. Therefore, ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =−1σCov(DiGj,Gℓ∣Xt=x)+1σCov(Di,Gℓ∣Xt=x)[Gj∣Xt=x] =- 1σCov (D_iG_j,G_ X_t=x )+ 1σCov (D_i,G_ X_t=x )E [G_j X_t=x ] +1σ[Di∣Xt=x]Cov(Gj,Gℓ∣Xt=x). + 1σE [D_i X_t=x ]Cov (G_j,G_ X_t=x ). We now rewrite the right-hand side as a centered third moment. Let D¯i(x) D_i(x) =[Di∣Xt=x], =E [D_i X_t=x ], G¯j(x) G_j(x) =[Gj∣Xt=x], =E [G_j X_t=x ], G¯ℓ(x) G_ (x) =[Gℓ∣Xt=x]. =E [G_ X_t=x ]. First, Cov(DiGj,Gℓ∣Xt=x) (D_iG_j,G_ X_t=x ) =[DiGjGℓ∣Xt=x]−[DiGj∣Xt=x]G¯ℓ(x). =E [D_iG_jG_ X_t=x ]-E [D_iG_j X_t=x ] G_ (x). Second, Cov(Di,Gℓ∣Xt=x) (D_i,G_ X_t=x ) =[DiGℓ∣Xt=x]−D¯i(x)G¯ℓ(x). =E [D_iG_ X_t=x ]- D_i(x) G_ (x). Third, Cov(Gj,Gℓ∣Xt=x) (G_j,G_ X_t=x ) =[GjGℓ∣Xt=x]−G¯j(x)G¯ℓ(x). =E [G_jG_ X_t=x ]- G_j(x) G_ (x). Substituting these three formulas gives ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =−1σ[DiGjGℓ∣Xt=x]+1σ[DiGj∣Xt=x]G¯ℓ(x) =- 1σE [D_iG_jG_ X_t=x ]+ 1σE [D_iG_j X_t=x ] G_ (x) +1σ[DiGℓ∣Xt=x]G¯j(x)−1σD¯i(x)G¯ℓ(x)G¯j(x) + 1σE [D_iG_ X_t=x ] G_j(x)- 1σ D_i(x) G_ (x) G_j(x) +1σD¯i(x)[GjGℓ∣Xt=x]−1σD¯i(x)G¯j(x)G¯ℓ(x). + 1σ D_i(x)E [G_jG_ X_t=x ]- 1σ D_i(x) G_j(x) G_ (x). Combining the two identical last-product terms, this becomes ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =−1σ[[DiGjGℓ∣Xt=x]−[DiGj∣Xt=x]G¯ℓ(x) =- 1σ [E [D_iG_jG_ X_t=x ]-E [D_iG_j X_t=x ] G_ (x) . −[DiGℓ∣Xt=x]G¯j(x)−D¯i(x)[GjGℓ∣Xt=x] .-E [D_iG_ X_t=x ] G_j(x)- D_i(x)E [G_jG_ X_t=x ] . +2D¯i(x)G¯j(x)G¯ℓ(x)]. .+2 D_i(x) G_j(x) G_ (x) ]. The bracket is exactly [(Di−D¯i(x))(Gj−G¯j(x))(Gℓ−G¯ℓ(x))∣Xt=x]. [ (D_i- D_i(x) ) (G_j- G_j(x) ) (G_ - G_ (x) ) X_t=x ]. Indeed, expanding this centered product gives [(Di−D¯i(x))(Gj−G¯j(x))(Gℓ−G¯ℓ(x))∣Xt=x] [ (D_i- D_i(x) ) (G_j- G_j(x) ) (G_ - G_ (x) ) X_t=x ] =[DiGjGℓ∣Xt=x]−[DiGj∣Xt=x]G¯ℓ(x) =E [D_iG_jG_ X_t=x ]-E [D_iG_j X_t=x ] G_ (x) −[DiGℓ∣Xt=x]G¯j(x)−D¯i(x)[GjGℓ∣Xt=x] -E [D_iG_ X_t=x ] G_j(x)- D_i(x)E [G_jG_ X_t=x ] +2D¯i(x)G¯j(x)G¯ℓ(x). +2 D_i(x) G_j(x) G_ (x). Therefore, ∂xℓCov(Di,Gj∣Xt=x) _x_ Cov (D_i,G_j X_t=x ) =−1σ[(Di−[Di∣Xt=x])(Gj−[Gj∣Xt=x])(Gℓ−[Gℓ∣Xt=x])∣Xt=x]. =- 1σE [ (D_i-E[D_i X_t=x] ) (G_j-E[G_j X_t=x] ) (G_ -E[G_ X_t=x] ) X_t=x ]. Returning to the Hessian of vtv_t, we obtain ∂xjxℓvt,i(x) _x_jx_ v_t,i(x) =−1σ∂xℓCov(Di,Gj∣Xt=x) =- 1σ _x_ Cov (D_i,G_j X_t=x ) =1σ2[(Di−[Di∣Xt=x])(Gj−[Gj∣Xt=x])(Gℓ−[Gℓ∣Xt=x])∣Xt=x]. = 1σ^2E [ (D_i-E[D_i X_t=x] ) (G_j-E[G_j X_t=x] ) (G_ -E[G_ X_t=x] ) X_t=x ]. Finally, summing over j=ℓj= gives the componentwise Laplacian: Δxvt,i(x) _xv_t,i(x) =∑j=1d∂xjxjvt,i(x) = _j=1^d _x_jx_jv_t,i(x) =1σ2[(Di−[Di∣Xt=x])∑j=1d(Gj−[Gj∣Xt=x])2∣Xt=x] = 1σ^2E [ (D_i-E[D_i X_t=x] ) _j=1^d (G_j-E[G_j X_t=x] )^2 X_t=x ] =1σ2[(Di−[Di∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. = 1σ^2E [ (D_i-E[D_i X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. Since this holds for every i∈1,…,di∈\1,…,d\, we get Δxvt(x) _xv_t(x) =1σ2[(D−[D∣Xt=x])‖G−[G∣Xt=x]‖2∣Xt=x]. = 1σ^2E [ (D-E[D X_t=x] ) \|G-E[G X_t=x] \|^2 X_t=x ]. ∎ G.3 Integrated Derivative Bounds Lemma 5 (Integrated Bounds Required for the Combined Score Analysis). Let Y0Y_0, Y1Y_1, D, StS_t, G, XtX_t, qtq_t, vtv_t, and sts_t be as in Lemma 4. Assume [‖Y1−Y0‖4] [\|Y_1-Y_0\|^4 ] <∞, <∞, where this expectation is taken under the joint law of (Y0,Y1)(Y_0,Y_1). Then, for every t∈[0,T]t∈[0,T], the following bounds hold. The full Hessian of vtv_t satisfies [‖Dx2vt(Xt)‖F2] [ \|D_x^2v_t(X_t) \|_F^2 ] ≤16d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ 16 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. Consequently, [‖∇x(∇x⋅vt)(Xt)‖2] [ \| _x( _x· v_t)(X_t) \|^2 ] ≤16dd(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ 16d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. The mixed velocity-score term satisfies [‖(Dxvt(Xt))⊤st(Xt)‖2] [ \|(D_xv_t(X_t)) s_t(X_t) \|^2 ] ≤d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. The same bound holds for [‖(Dxvt(Xt))st(Xt)‖2]E [ \|(D_xv_t(X_t))s_t(X_t) \|^2 ]. The pure score terms satisfy [‖Dxst(Xt)‖F2] [ \|D_xs_t(X_t) \|_F^2 ] ≤2d(d+2)+2dσ4, ≤ 2d(d+2)+2dσ^4, [‖Dxst(Xt)‖F4] [ \|D_xs_t(X_t) \|_F^4 ] ≤8d(d+2)(d+4)(d+6)+8d2σ8, ≤ 8d(d+2)(d+4)(d+6)+8d^2σ^8, [‖st(Xt)‖4] [ \|s_t(X_t) \|^4 ] ≤d(d+2)σ4, ≤ d(d+2)σ^4, [‖Dxst(Xt)st(Xt)‖2] [ \|D_xs_t(X_t)s_t(X_t) \|^2 ] ≤(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6, ≤ (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6, [‖Δxst(Xt)‖2] [ \| _xs_t(X_t) \|^2 ] ≤64d(d+2)(d+4)σ6. ≤ 64d(d+2)(d+4)σ^6. All expectations in these bounds are taken under the joint law of (Y0,Y1,G)(Y_0,Y_1,G). Proof. We first prove the full Hessian bound for vtv_t. By Lemma 4, for every i,j,ℓ∈1,…,di,j, ∈\1,…,d\, ∂xjxℓvt,i(Xt) _x_jx_ v_t,i(X_t) =1σ2[(Di−[Di∣Xt])(Gj−[Gj∣Xt])(Gℓ−[Gℓ∣Xt])∣Xt]. = 1σ^2E [ (D_i-E[D_i X_t] ) (G_j-E[G_j X_t] ) (G_ -E[G_ X_t] ) X_t ]. By definition of the Frobenius norm for the full Hessian tensor, ‖Dx2vt(Xt)‖F2 \|D_x^2v_t(X_t) \|_F^2 =∑i=1d∑j=1d∑ℓ=1d|∂xjxℓvt,i(Xt)|2. = _i=1^d _j=1^d _ =1^d | _x_jx_ v_t,i(X_t) |^2. Substituting the componentwise formula for ∂xjxℓvt,i(Xt) _x_jx_ v_t,i(X_t), we get ‖Dx2vt(Xt)‖F2 \|D_x^2v_t(X_t) \|_F^2 =1σ4∑i=1d∑j=1d∑ℓ=1d|[(Di−[Di∣Xt])(Gj−[Gj∣Xt])(Gℓ−[Gℓ∣Xt])∣Xt]|2. = 1σ^4 _i=1^d _j=1^d _ =1^d |E [ (D_i-E[D_i X_t] ) (G_j-E[G_j X_t] ) (G_ -E[G_ X_t] ) X_t ] |^2. Now apply conditional Cauchy–Schwarz to each fixed triple (i,j,ℓ)(i,j, ). We have |[(Di−[Di∣Xt])(Gj−[Gj∣Xt])(Gℓ−[Gℓ∣Xt])∣Xt]|2 |E [ (D_i-E[D_i X_t] ) (G_j-E[G_j X_t] ) (G_ -E[G_ X_t] ) X_t ] |^2 ≤[|Di−[Di∣Xt]|2∣Xt][|Gj−[Gj∣Xt]|2|Gℓ−[Gℓ∣Xt]|2∣Xt]. [ |D_i-E[D_i X_t] |^2 X_t ]E [ |G_j-E[G_j X_t] |^2 |G_ -E[G_ X_t] |^2 X_t ]. Therefore, ‖Dx2vt(Xt)‖F2 \|D_x^2v_t(X_t) \|_F^2 ≤1σ4∑i=1d∑j=1d∑ℓ=1d[|Di−[Di∣Xt]|2∣Xt][|Gj−[Gj∣Xt]|2|Gℓ−[Gℓ∣Xt]|2∣Xt]. ≤ 1σ^4 _i=1^d _j=1^d _ =1^dE [ |D_i-E[D_i X_t] |^2 X_t ]E [ |G_j-E[G_j X_t] |^2 |G_ -E[G_ X_t] |^2 X_t ]. Since the first conditional expectation does not depend on j or ℓ , and the second conditional expectation does not depend on i, we can rearrange the sums: ‖Dx2vt(Xt)‖F2 \|D_x^2v_t(X_t) \|_F^2 ≤1σ4(∑i=1d[|Di−[Di∣Xt]|2∣Xt]) ≤ 1σ^4 ( _i=1^dE [ |D_i-E[D_i X_t] |^2 X_t ] ) ⋅(∑j=1d∑ℓ=1d[|Gj−[Gj∣Xt]|2|Gℓ−[Gℓ∣Xt]|2∣Xt]). · ( _j=1^d _ =1^dE [ |G_j-E[G_j X_t] |^2 |G_ -E[G_ X_t] |^2 X_t ] ). The first sum is ∑i=1d[|Di−[Di∣Xt]|2∣Xt] _i=1^dE [ |D_i-E[D_i X_t] |^2 X_t ] =[‖D−[D∣Xt]‖2∣Xt]. =E [ \|D-E[D X_t] \|^2 X_t ]. The second double sum is ∑j=1d∑ℓ=1d[|Gj−[Gj∣Xt]|2|Gℓ−[Gℓ∣Xt]|2∣Xt] _j=1^d _ =1^dE [ |G_j-E[G_j X_t] |^2 |G_ -E[G_ X_t] |^2 X_t ] =[∑j=1d∑ℓ=1d|Gj−[Gj∣Xt]|2|Gℓ−[Gℓ∣Xt]|2∣Xt] =E [ _j=1^d _ =1^d |G_j-E[G_j X_t] |^2 |G_ -E[G_ X_t] |^2 X_t ] =[(∑j=1d|Gj−[Gj∣Xt]|2)(∑ℓ=1d|Gℓ−[Gℓ∣Xt]|2)∣Xt] =E [ ( _j=1^d |G_j-E[G_j X_t] |^2 ) ( _ =1^d |G_ -E[G_ X_t] |^2 ) X_t ] =[‖G−[G∣Xt]‖4∣Xt]. =E [ \|G-E[G X_t] \|^4 X_t ]. Combining the two identities gives ‖Dx2vt(Xt)‖F2 \|D_x^2v_t(X_t) \|_F^2 ≤1σ4[‖D−[D∣Xt]‖2∣Xt][‖G−[G∣Xt]‖4∣Xt]. ≤ 1σ^4E [ \|D-E[D X_t] \|^2 X_t ]E [ \|G-E[G X_t] \|^4 X_t ]. Taking expectation and applying Cauchy–Schwarz gives [‖Dx2vt(Xt)‖F2]≤1σ4([([‖D−[D∣Xt]‖2∣Xt])2])1/2([([‖G−[G∣Xt]‖4∣Xt])2])1/2. [ \|D_x^2v_t(X_t) \|_F^2 ]≤ 1σ^4 (E [ (E [ \|D-E[D X_t] \|^2 X_t ] )^2 ] )^1/2 (E [ (E [ \|G-E[G X_t] \|^4 X_t ] )^2 ] )^1/2. For the first factor, conditional variance contraction and conditional Jensen give [‖D−[D∣Xt]‖2∣Xt] \! [\|D-E[D X_t]\|^2 X_t ] ≤[‖D‖2∣Xt], [\|D\|^2 X_t], ([([‖D−[D∣Xt]‖2∣Xt])2])1/2 (E [ (E [ \|D-E[D X_t] \|^2 X_t ] )^2 ] )^1/2 ≤([‖D‖4])1/2. ≤ (E [\|D\|^4 ] )^1/2. For the second factor, Jensen’s inequality gives ([([‖G−[G∣Xt]‖4∣Xt])2])1/2 (E [ (E [ \|G-E[G X_t] \|^4 X_t ] )^2 ] )^1/2 ≤([‖G−[G∣Xt]‖8])1/2. ≤ (E [ \|G-E[G X_t] \|^8 ] )^1/2. Using ‖G−[G∣Xt]‖8 \|G-E[G X_t] \|^8 ≤27(‖G‖8+‖[G∣Xt]‖8), ≤ 2^7 (\|G\|^8+ \|E[G X_t] \|^8 ), and Jensen’s inequality [‖[G∣Xt]‖8]≤[‖G‖8]E[\|E[G X_t]\|^8] [\|G\|^8], we get [‖G−[G∣Xt]‖8]≤28[‖G‖8]. [ \|G-E[G X_t] \|^8 ]≤ 2^8E [\|G\|^8 ]. Since G∼(0,Id)G N(0,I_d), we know [‖G‖8]=d(d+2)(d+4)(d+6)E[\|G\|^8]=d(d+2)(d+4)(d+6). Therefore, ([‖G−[G∣Xt]‖8])1/2 (E [ \|G-E[G X_t] \|^8 ] )^1/2 ≤16d(d+2)(d+4)(d+6). ≤ 16 d(d+2)(d+4)(d+6). Combining the estimates gives [‖Dx2vt(Xt)‖F2] [ \|D_x^2v_t(X_t) \|_F^2 ] ≤16d(d+2)(d+4)(d+6)σ4([‖D‖4])1/2. ≤ 16 d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2. Since D=Y1−Y0D=Y_1-Y_0, this is the first displayed bound. Next, we explain how the bound for the full Hessian Dx2vtD_x^2v_t implies a bound for ∇x(∇x⋅vt) _x( _x· v_t) (gradient of divergence). The full Hessian Dx2vt(x)D_x^2v_t(x) contains all second derivatives ∂xjxℓvt,i(x),i,j,ℓ∈1,…,d. _x_jx_ v_t,i(x), i,j, ∈\1,…,d\. Its Frobenius norm is defined by ‖Dx2vt(x)‖F2 \|D_x^2v_t(x) \|_F^2 =∑i=1d∑j=1d∑ℓ=1d|∂xjxℓvt,i(x)|2. = _i=1^d _j=1^d _ =1^d | _x_jx_ v_t,i(x) |^2. By contrast, ∇x(∇x⋅vt)(x) _x( _x· v_t)(x) is a vector in ℝdR^d. First, ∇x⋅vt(x) _x· v_t(x) =∑i=1d∂xivt,i(x). = _i=1^d _x_iv_t,i(x). Therefore, for every ℓ∈1,…,d ∈\1,…,d\, the ℓ -th coordinate of ∇x(∇x⋅vt)(x) _x( _x· v_t)(x) is [∇x(∇x⋅vt)(x)]ℓ [ _x( _x· v_t)(x) ]_ =∂xℓ(∑i=1d∂xivt,i(x)) = _x_ ( _i=1^d _x_iv_t,i(x) ) =∑i=1d∂xℓxivt,i(x). = _i=1^d _x_ x_iv_t,i(x). For each fixed ℓ , Cauchy–Schwarz gives |∑i=1d∂xℓxivt,i(x)|2 | _i=1^d _x_ x_iv_t,i(x) |^2 ≤d∑i=1d|∂xℓxivt,i(x)|2. ≤ d _i=1^d | _x_ x_iv_t,i(x) |^2. Now sum this inequality over ℓ=1,…,d =1,…,d. We obtain ‖∇x(∇x⋅vt)(x)‖2 \| _x( _x· v_t)(x) \|^2 =∑ℓ=1d|[∇x(∇x⋅vt)(x)]ℓ|2=∑ℓ=1d|∑i=1d∂xℓxivt,i(x)|2≤d∑ℓ=1d∑i=1d|∂xℓxivt,i(x)|2. = _ =1^d | [ _x( _x· v_t)(x) ]_ |^2= _ =1^d | _i=1^d _x_ x_iv_t,i(x) |^2≤ d _ =1^d _i=1^d | _x_ x_iv_t,i(x) |^2. The last double sum is only part of the full Hessian norm, because the full Hessian norm sums over all triples (i,j,ℓ)(i,j, ). Hence ∑ℓ=1d∑i=1d|∂xℓxivt,i(x)|2 _ =1^d _i=1^d | _x_ x_iv_t,i(x) |^2 ≤∑i=1d∑j=1d∑ℓ=1d|∂xjxℓvt,i(x)|2=‖Dx2vt(x)‖F2. ≤ _i=1^d _j=1^d _ =1^d | _x_jx_ v_t,i(x) |^2= \|D_x^2v_t(x) \|_F^2. Therefore, ‖∇x(∇x⋅vt)(x)‖2 \| _x( _x· v_t)(x) \|^2 ≤d‖Dx2vt(x)‖F2. ≤ d \|D_x^2v_t(x) \|_F^2. Evaluating this inequality at x=Xtx=X_t and taking expectation gives [‖∇x(∇x⋅vt)(Xt)‖2] [ \| _x( _x· v_t)(X_t) \|^2 ] ≤d[‖Dx2vt(Xt)‖F2]. ≤ dE [ \|D_x^2v_t(X_t) \|_F^2 ]. Using the full Hessian estimate established above in this proof, [‖Dx2vt(Xt)‖F2] [ \|D_x^2v_t(X_t) \|_F^2 ] ≤16d(d+2)(d+4)(d+6)σ4([‖D‖4])1/2, ≤ 16 d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2, we conclude that [‖∇x(∇x⋅vt)(Xt)‖2] [ \| _x( _x· v_t)(X_t) \|^2 ] ≤16dd(d+2)(d+4)(d+6)σ4([‖D‖4])1/2. ≤ 16d d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2. We now prove the mixed velocity-score bound directly. From the posterior derivative identity for vtv_t, Dxvt(Xt) D_xv_t(X_t) =−1σCov(D,G∣Xt). =- 1σCov (D,G X_t ). From the Gaussian score identity, st(Xt) s_t(X_t) =−1σ[G∣Xt]. =- 1σE [G X_t ]. Therefore, (Dxvt(Xt))⊤st(Xt) (D_xv_t(X_t)) s_t(X_t) =1σ2Cov(D,G∣Xt)⊤[G∣Xt]. = 1σ^2Cov (D,G X_t ) E [G X_t ]. Hence ‖(Dxvt(Xt))⊤st(Xt)‖2 \|(D_xv_t(X_t)) s_t(X_t) \|^2 =1σ4‖Cov(D,G∣Xt)⊤[G∣Xt]‖2. = 1σ^4 \|Cov (D,G X_t ) E [G X_t ] \|^2. We now bound the conditional quantity. Let D¯(Xt)=[D∣Xt] D(X_t)=E [D X_t ] and G¯(Xt)=[G∣Xt] G(X_t)=E [G X_t ]. By the definition of conditional covariance, Cov(D,G∣Xt)⊤G¯(Xt) (D,G X_t ) G(X_t) =[(G−G¯(Xt))(D−D¯(Xt))⊤∣Xt]G¯(Xt) =E [ (G- G(X_t) ) (D- D(X_t) ) X_t ] G(X_t) =[(G−G¯(Xt))(D−D¯(Xt))⊤G¯(Xt)∣Xt]. =E [ (G- G(X_t) ) (D- D(X_t) ) G(X_t) X_t ]. (6) Since G¯(Xt) G(X_t) is fixed after conditioning on XtX_t, conditional Cauchy–Schwarz gives ‖Cov(D,G∣Xt)⊤G¯(Xt)‖2 \|Cov (D,G X_t ) G(X_t) \|^2 ≤[‖G−G¯(Xt)‖2∣Xt][|(D−D¯(Xt))⊤G¯(Xt)|2∣Xt]. [ \|G- G(X_t) \|^2 X_t ]E [ | (D- D(X_t) ) G(X_t) |^2 X_t ]. For the second factor, |(D−D¯(Xt))⊤G¯(Xt)|2 | (D- D(X_t) ) G(X_t) |^2 ≤‖D−D¯(Xt)‖2‖G¯(Xt)‖2. ≤ \|D- D(X_t) \|^2 \| G(X_t) \|^2. Therefore, ‖Cov(D,G∣Xt)⊤G¯(Xt)‖2 \|Cov (D,G X_t ) G(X_t) \|^2 ≤[‖G−G¯(Xt)‖2∣Xt][‖D−D¯(Xt)‖2∣Xt]‖G¯(Xt)‖2. [ \|G- G(X_t) \|^2 X_t ]E [ \|D- D(X_t) \|^2 X_t ] \| G(X_t) \|^2. Using the conditional variance inequality, [‖D−D¯(Xt)‖2∣Xt] [ \|D- D(X_t) \|^2 X_t ] ≤[‖D‖2∣Xt], [\|D\|^2 X_t ], [‖G−G¯(Xt)‖2∣Xt] [ \|G- G(X_t) \|^2 X_t ] ≤[‖G‖2∣Xt], [\|G\|^2 X_t ], and Jensen’s inequality, ‖G¯(Xt)‖2=‖[G∣Xt]‖2≤[‖G‖2∣Xt] \| G(X_t) \|^2= \|E [G X_t ] \|^2 [\|G\|^2 X_t ], we get ‖Cov(D,G∣Xt)⊤G¯(Xt)‖2≤[‖D‖2∣Xt]([‖G‖2∣Xt])2. \|Cov (D,G X_t ) G(X_t) \|^2 [\|D\|^2 X_t ] (E [\|G\|^2 X_t ] )^2. Taking expectation gives [‖Cov(D,G∣Xt)⊤[G∣Xt]‖2]≤[[‖D‖2∣Xt]([‖G‖2∣Xt])2]. [ \|Cov (D,G X_t ) E[G X_t] \|^2 ] [E [\|D\|^2 X_t ] (E [\|G\|^2 X_t ] )^2 ]. By Cauchy–Schwarz, [[‖D‖2∣Xt]([‖G‖2∣Xt])2] [E [\|D\|^2 X_t ] (E [\|G\|^2 X_t ] )^2 ] ≤([([‖D‖2∣Xt])2])1/2([([‖G‖2∣Xt])4])1/2. ≤ (E [ (E [\|D\|^2 X_t ] )^2 ] )^1/2 (E [ (E [\|G\|^2 X_t ] )^4 ] )^1/2. By Jensen’s inequality, [([‖D‖2∣Xt])2] [ (E [\|D\|^2 X_t ] )^2 ] ≤[‖D‖4], [\|D\|^4 ], [([‖G‖2∣Xt])4] [ (E [\|G\|^2 X_t ] )^4 ] ≤[‖G‖8]. [\|G\|^8 ]. Since G∼(0,Id)G N(0,I_d), [‖G‖8] [\|G\|^8 ] =d(d+2)(d+4)(d+6). =d(d+2)(d+4)(d+6). Therefore, [‖Cov(D,G∣Xt)⊤[G∣Xt]‖2] [ \|Cov (D,G X_t ) E[G X_t] \|^2 ] ≤d(d+2)(d+4)(d+6)([‖D‖4])1/2. ≤ d(d+2)(d+4)(d+6) (E [\|D\|^4 ] )^1/2. Finally, recalling the factor 1/σ41/σ^4, we conclude that [‖(Dxvt(Xt))⊤st(Xt)‖2] [ \|(D_xv_t(X_t)) s_t(X_t) \|^2 ] ≤d(d+2)(d+4)(d+6)σ4([‖D‖4])1/2. ≤ d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2. The non-transposed orientation requires a separate argument. Write Ct=Cov(D,G∣Xt)C_t=Cov(D,G X_t) and G¯t=[G∣Xt] G_t=E[G X_t]. Conditional Cauchy–Schwarz gives ‖Ct‖F2≤[‖D−[D∣Xt]‖2∣Xt][‖G−G¯t‖2∣Xt].\|C_t\|_F^2 [\|D-E[D X_t]\|^2 X_t]\,E[\|G- G_t\|^2 X_t]. Consequently, ‖CtG¯t‖2≤[‖D‖2∣Xt]([‖G‖2∣Xt])2,\|C_t G_t\|^2 [\|D\|^2 X_t]\, (E[\|G\|^2 X_t] )^2, where we used conditional variance contraction and ‖G¯t‖2≤[‖G‖2∣Xt]\| G_t\|^2 [\|G\|^2 X_t]. The same outer Cauchy–Schwarz and Jensen calculation as above therefore proves ‖(Dxvt(Xt))st(Xt)‖2≤d(d+2)(d+4)(d+6)σ4(‖D‖4)1/2.E\|(D_xv_t(X_t))s_t(X_t)\|^2≤ d(d+2)(d+4)(d+6)σ^4 (E\|D\|^4 )^1/2. We next prove the pure score bounds. By Lemma 4, Dxst(Xt) D_xs_t(X_t) =1σ2(Cov(G,G∣Xt)−Id). = 1σ^2 (Cov (G,G X_t )-I_d ). Using ‖A−B‖F2≤2‖A‖F2+2‖B‖F2\|A-B\|_F^2≤ 2\|A\|_F^2+2\|B\|_F^2, ‖Dxst(Xt)‖F2 \|D_xs_t(X_t) \|_F^2 ≤2σ4‖Cov(G,G∣Xt)‖F2+2σ4‖Id‖F2. ≤ 2σ^4 \|Cov (G,G X_t ) \|_F^2+ 2σ^4 \|I_d \|_F^2. Since ‖Id‖F2=d\|I_d\|_F^2=d, and since the conditional covariance matrix is positive semidefinite, ‖Cov(G,G∣Xt)‖F2 \|Cov (G,G X_t ) \|_F^2 ≤(trCov(G,G∣Xt))2=([‖G−[G∣Xt]‖2∣Xt])2 ≤ (trCov (G,G X_t ) )^2= (E [ \|G-E[G X_t] \|^2 X_t ] )^2 =([‖G‖2∣Xt]−‖[G∣Xt]‖2)2 = (E [\|G\|^2 X_t ]- \|E[G X_t] \|^2 )^2 ≤([‖G‖2∣Xt])2. ≤ (E [\|G\|^2 X_t ] )^2. Taking expectation and using Jensen’s inequality gives [‖Cov(G,G∣Xt)‖F2] [ \|Cov (G,G X_t ) \|_F^2 ] ≤[‖G‖4] [\|G\|^4 ] =d(d+2). =d(d+2). Therefore, [‖Dxst(Xt)‖F2] [ \|D_xs_t(X_t) \|_F^2 ] ≤2d(d+2)+2dσ4. ≤ 2d(d+2)+2dσ^4. For the fourth moment of Dxst(Xt)D_xs_t(X_t), use ‖A−B‖F4≤8‖A‖F4+8‖B‖F4\|A-B\|_F^4≤ 8\|A\|_F^4+8\|B\|_F^4: ‖Dxst(Xt)‖F4 \|D_xs_t(X_t) \|_F^4 ≤8σ8‖Cov(G,G∣Xt)‖F4+8σ8‖Id‖F4. ≤ 8σ^8 \|Cov (G,G X_t ) \|_F^4+ 8σ^8 \|I_d \|_F^4. Since ‖Id‖F4=d2\|I_d\|_F^4=d^2, and since the conditional covariance matrix is positive semidefinite, ‖Cov(G,G∣Xt)‖F4≤(trCov(G,G∣Xt))4≤([‖G‖2∣Xt])4. \|Cov (G,G X_t ) \|_F^4≤ (trCov (G,G X_t ) )^4≤ (E [\|G\|^2 X_t ] )^4. Taking expectation and applying conditional Jensen only at this stage gives [‖Cov(G,G∣Xt)‖F4] [ \|Cov(G,G X_t) \|_F^4 ] ≤[([‖G‖2∣Xt])4] [ (E[\|G\|^2 X_t] )^4 ] ≤‖G‖8=d(d+2)(d+4)(d+6). \|G\|^8=d(d+2)(d+4)(d+6). Therefore, [‖Dxst(Xt)‖F4] [ \|D_xs_t(X_t) \|_F^4 ] ≤8d(d+2)(d+4)(d+6)+8d2σ8. ≤ 8d(d+2)(d+4)(d+6)+8d^2σ^8. The fourth moment of the score follows directly from st(Xt)=−σ−1[G∣Xt]s_t(X_t)=-σ^-1E[G X_t]: [‖st(Xt)‖4] [ \|s_t(X_t) \|^4 ] =1σ4‖[G∣Xt]‖4 = 1σ^4E \|E[G X_t] \|^4 ≤1σ4[[‖G‖4∣Xt]]=d(d+2)σ4. ≤ 1σ^4E\! [E[\|G\|^4 X_t] ]= d(d+2)σ^4. Hence, by Cauchy–Schwarz, [‖Dxst(Xt)st(Xt)‖2] [ \|D_xs_t(X_t)s_t(X_t) \|^2 ] ≤([‖Dxst(Xt)‖F4])1/2([‖st(Xt)‖4])1/2 ≤ (E [ \|D_xs_t(X_t) \|_F^4 ] )^1/2 (E [ \|s_t(X_t) \|^4 ] )^1/2 ≤(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6. ≤ (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6. Finally, Lemma 4 gives Δxst(Xt) _xs_t(X_t) =−1σ3[(G−[G∣Xt])‖G−[G∣Xt]‖2∣Xt]. =- 1σ^3E [ (G-E[G X_t] ) \|G-E[G X_t] \|^2 X_t ]. By Jensen’s inequality, ‖Δxst(Xt)‖2 \| _xs_t(X_t) \|^2 ≤1σ6[‖G−[G∣Xt]‖6∣Xt]. ≤ 1σ^6E [ \|G-E[G X_t] \|^6 X_t ]. Taking expectation gives [‖Δxst(Xt)‖2] [ \| _xs_t(X_t) \|^2 ] ≤1σ6[‖G−[G∣Xt]‖6]. ≤ 1σ^6E [ \|G-E[G X_t] \|^6 ]. Using ‖u−w‖6≤25‖u‖6+25‖w‖6\|u-w\|^6≤ 2^5\|u\|^6+2^5\|w\|^6, and using Jensen’s inequality, [‖G−[G∣Xt]‖6] [ \|G-E[G X_t] \|^6 ] ≤26[‖G‖6]. ≤ 2^6E [\|G\|^6 ]. Since G∼(0,Id)G N(0,I_d), [‖G‖6] [\|G\|^6 ] =d(d+2)(d+4). =d(d+2)(d+4). Therefore, [‖Δxst(Xt)‖2] [ \| _xs_t(X_t) \|^2 ] ≤64d(d+2)(d+4)σ6. ≤ 64d(d+2)(d+4)σ^6. This completes the proof. ∎ Lemma 6 (Integrated Derivative Bounds in Terms of Displacement Moments). Let Y0Y_0, Y1Y_1, D, StS_t, G, XtX_t, qtq_t, and vtv_t be as in Lemma 4. Assume that [‖Y1−Y0‖5] [\|Y_1-Y_0\|^5 ] <∞, <∞, where the expectation is taken under the joint law of (Y0,Y1)(Y_0,Y_1). Then, for every t∈[0,T]t∈[0,T], [‖Dxvt(Xt)‖F2] [ \|D_xv_t(X_t) \|_F^2 ] ≤d(d+2)σ2([‖Y1−Y0‖4])1/2, ≤ d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2, [‖∂tvt(Xt)‖2] [ \| _tv_t(X_t) \|^2 ] ≤d(d+2)σ2[‖Y1−Y0‖4], ≤ d(d+2)σ^2E [\|Y_1-Y_0\|^4 ], [‖Δxvt(Xt)‖2] [ \| _xv_t(X_t) \|^2 ] ≤16d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ 16 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. Moreover, [‖Dxvt(Xt)bt(Xt)‖2] [ \|D_xv_t(X_t)b_t(X_t) \|^2 ] ≤2σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 ≤ 2σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +τ42σ4d(d+2)(d+4)(d+6)([‖Y1−Y0‖4])1/2. + τ^42σ^4 d(d+2)(d+4)(d+6) (E [\|Y_1-Y_0\|^4 ] )^1/2. All expectations in these bounds are taken under the joint law of (Y0,Y1,G)(Y_0,Y_1,G). Proof. We first bound the Jacobian. By Lemma 4, Dxvt(Xt) D_xv_t(X_t) =−1σCov(D,G∣Xt). =- 1σCov (D,G X_t ). By Cauchy-Schwarz, ‖Cov(D,G∣Xt)‖F2≤[‖D‖2∣Xt][‖G‖2∣Xt] \|Cov (D,G X_t ) \|_F^2 [\|D\|^2 X_t ]E [\|G\|^2 X_t ]. Taking expectation over XtX_t and apply Cauchy-Schwarz, we get [‖Cov(D,G∣Xt)‖F2] [ \|Cov (D,G X_t ) \|_F^2 ] ≤[[‖D‖2∣Xt][‖G‖2∣Xt]] [E [\|D\|^2 X_t ]E [\|G\|^2 X_t ] ] ≤([([‖D‖2∣Xt])2])1/2([([‖G‖2∣Xt])2])1/2 ≤ (E [ (E [\|D\|^2 X_t ] )^2 ] )^1/2 (E [ (E [\|G\|^2 X_t ] )^2 ] )^1/2 ≤([‖D‖4])1/2([‖G‖4])1/2. ≤ (E [\|D\|^4 ] )^1/2 (E [\|G\|^4 ] )^1/2. Since G∼(0,Id)G N(0,I_d), [‖G‖4]=d(d+2)E [\|G\|^4 ]=d(d+2). Therefore, [‖Dxvt(Xt)‖F2]=1σ2[‖Cov(D,G∣Xt)‖F2]≤d(d+2)σ2([‖D‖4])1/2. [ \|D_xv_t(X_t) \|_F^2 ]= 1σ^2E [ \|Cov (D,G X_t ) \|_F^2 ]≤ d(d+2)σ^2 (E [\|D\|^4 ] )^1/2. For time derivative, by Lemma 4, ∂tvt(Xt) _tv_t(X_t) =1σCov(D,D⊤G∣Xt). = 1σCov (D,D G X_t ). Similarly, apply Cauchy-Schwarz, we have ‖Cov(D,D⊤G∣Xt)‖2≤[‖D‖2∣Xt][|D⊤G|2∣Xt]. \|Cov (D,D G X_t ) \|^2 [\|D\|^2 X_t ]E [|D G|^2 X_t ]. Take expectation and apply Cauchy-Schwarz, we have ‖Cov(D,D⊤G∣Xt)‖2 \|Cov (D,D G X_t ) \|^2 ≤[([‖D‖2∣Xt])2]12[([‖D‖2‖G‖2∣Xt])2]12 [(E [\|D\|^2 X_t ])^2 ] 12E [(E [\|D\|^2\|G\|^2 X_t ])^2 ] 12 ≤[‖D‖4]12[‖D‖4‖G‖4]12=[‖D‖4][‖G‖4]12, [\|D\|^4 ] 12E [\|D\|^4\|G\|^4 ] 12=E [\|D\|^4 ]E [\|G\|^4 ] 12, where note that D and G are independent. Therefore, [‖∂tvt(Xt)‖2] [ \| _tv_t(X_t) \|^2 ] ≤d(d+2)σ2[‖D‖4]. ≤ d(d+2)σ^2E [\|D\|^4 ]. Now consider the Laplacian. By Lemma 4, Δxvt(Xt) _xv_t(X_t) =1σ2[(D−[D∣Xt])‖G−[G∣Xt]‖2∣Xt]. = 1σ^2E [ (D-E[D X_t] ) \|G-E[G X_t] \|^2 X_t ]. Conditional Cauchy–Schwarz gives ‖Δxvt(Xt)‖2≤1σ4[‖D−[D∣Xt]‖2∣Xt][‖G−[G∣Xt]‖4∣Xt]. \| _xv_t(X_t) \|^2≤ 1σ^4E [ \|D-E[D X_t] \|^2 X_t ]E [ \|G-E[G X_t] \|^4 X_t ]. Following the same proof as we did for Dx2vtD_x^2v_t, we obtain [‖Δxvt(Xt)‖2] [ \| _xv_t(X_t) \|^2 ] ≤16d(d+2)(d+4)(d+6)σ4([‖D‖4])1/2. ≤ 16 d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2. We now bound the term [‖Dxvt(Xt)bt(Xt)‖2]. [ \|D_xv_t(X_t)b_t(X_t) \|^2 ]. Recall that st(Xt)=−(1/σ)[G∣Xt]s_t(X_t)=-(1/σ)E[G X_t] and bt(Xt)=vt(Xt)+τ22st(Xt)b_t(X_t)=v_t(X_t)+ τ^22s_t(X_t). Therefore, Dxvt(Xt)bt(Xt) D_xv_t(X_t)b_t(X_t) =Dxvt(Xt)vt(Xt)+τ22Dxvt(Xt)st(Xt). =D_xv_t(X_t)v_t(X_t)+ τ^22D_xv_t(X_t)s_t(X_t). Using ‖a+b‖2≤2‖a‖2+2‖b‖2\|a+b\|^2≤ 2\|a\|^2+2\|b\|^2, we get ‖Dxvt(Xt)bt(Xt)‖2 \|D_xv_t(X_t)b_t(X_t) \|^2 ≤2‖Dxvt(Xt)vt(Xt)‖2+τ42‖Dxvt(Xt)st(Xt)‖2. ≤ 2 \|D_xv_t(X_t)v_t(X_t) \|^2+ τ^42 \|D_xv_t(X_t)s_t(X_t) \|^2. Thus, [‖Dxvt(Xt)bt(Xt)‖2] [ \|D_xv_t(X_t)b_t(X_t) \|^2 ] ≤2[‖Dxvt(Xt)vt(Xt)‖2]+τ42[‖Dxvt(Xt)st(Xt)‖2]. ≤ 2E [ \|D_xv_t(X_t)v_t(X_t) \|^2 ]+ τ^42E [ \|D_xv_t(X_t)s_t(X_t) \|^2 ]. We first bound the velocity part. From the posterior derivative identity, Dxvt(Xt) D_xv_t(X_t) =−1σCov(D,G∣Xt). =- 1σCov (D,G X_t ). Since vt(Xt)=[D∣Xt]v_t(X_t)=E[D X_t], we have Dxvt(Xt)vt(Xt) D_xv_t(X_t)v_t(X_t) =−1σCov(D,G∣Xt)[D∣Xt]. =- 1σCov (D,G X_t )E [D X_t ]. For readability, define D¯(Xt) D(X_t) =[D∣Xt], =E [D X_t ], G¯(Xt) G(X_t) =[G∣Xt]. =E [G X_t ]. By the definition of conditional covariance, Cov(D,G∣Xt)D¯(Xt) (D,G X_t ) D(X_t) =[(D−D¯(Xt))(G−G¯(Xt))⊤∣Xt]D¯(Xt) =E [ (D- D(X_t) ) (G- G(X_t) ) X_t ] D(X_t) =[(D−D¯(Xt))(G−G¯(Xt))⊤D¯(Xt)∣Xt]. =E [ (D- D(X_t) ) (G- G(X_t) ) D(X_t) X_t ]. Since D¯(Xt) D(X_t) is fixed after conditioning on XtX_t, conditional Cauchy–Schwarz gives ‖Cov(D,G∣Xt)D¯(Xt)‖2 \|Cov (D,G X_t ) D(X_t) \|^2 ≤[‖D−D¯(Xt)‖2∣Xt][|(G−G¯(Xt))⊤D¯(Xt)|2∣Xt]. [ \|D- D(X_t) \|^2 X_t ]E [ | (G- G(X_t) ) D(X_t) |^2 X_t ]. For the second conditional expectation, |(G−G¯(Xt))⊤D¯(Xt)|2 | (G- G(X_t) ) D(X_t) |^2 ≤‖G−G¯(Xt)‖2‖D¯(Xt)‖2. ≤ \|G- G(X_t) \|^2 \| D(X_t) \|^2. Hence ‖Cov(D,G∣Xt)D¯(Xt)‖2 \|Cov (D,G X_t ) D(X_t) \|^2 ≤[‖D−D¯(Xt)‖2∣Xt][‖G−G¯(Xt)‖2∣Xt]‖D¯(Xt)‖2. [ \|D- D(X_t) \|^2 X_t ]E [ \|G- G(X_t) \|^2 X_t ] \| D(X_t) \|^2. Using the conditional variance inequalities, [‖D−D¯(Xt)‖2∣Xt] [ \|D- D(X_t) \|^2 X_t ] ≤[‖D‖2∣Xt], [\|D\|^2 X_t ], [‖G−G¯(Xt)‖2∣Xt] [ \|G- G(X_t) \|^2 X_t ] ≤[‖G‖2∣Xt], [\|G\|^2 X_t ], and Jensen’s inequality, ‖D¯(Xt)‖2 \| D(X_t) \|^2 =‖[D∣Xt]‖2 = \|E [D X_t ] \|^2 ≤[‖D‖2∣Xt], [\|D\|^2 X_t ], we obtain ‖Cov(D,G∣Xt)D¯(Xt)‖2 \|Cov (D,G X_t ) D(X_t) \|^2 ≤([‖D‖2∣Xt])2[‖G‖2∣Xt]. ≤ (E [\|D\|^2 X_t ] )^2E [\|G\|^2 X_t ]. Therefore, [‖Dxvt(Xt)vt(Xt)‖2] [ \|D_xv_t(X_t)v_t(X_t) \|^2 ] ≤1σ2[([‖D‖2∣Xt])2[‖G‖2∣Xt]]. ≤ 1σ^2E [ (E [\|D\|^2 X_t ] )^2E [\|G\|^2 X_t ] ]. Apply Hölder’s inequality with conjugate exponents 5/45/4 and 55. Then [([‖D‖2∣Xt])2[‖G‖2∣Xt]] [ (E [\|D\|^2 X_t ] )^2E [\|G\|^2 X_t ] ] ≤([([‖D‖2∣Xt])5/2])4/5([([‖G‖2∣Xt])5])1/5. ≤ (E [ (E [\|D\|^2 X_t ] )^5/2 ] )^4/5 (E [ (E [\|G\|^2 X_t ] )^5 ] )^1/5. For the first factor, Jensen’s inequality gives [([‖D‖2∣Xt])5/2] [ (E [\|D\|^2 X_t ] )^5/2 ] ≤[‖D‖5]. [\|D\|^5 ]. For the second factor, Jensen’s inequality gives [([‖G‖2∣Xt])5] [ (E [\|G\|^2 X_t ] )^5 ] ≤[‖G‖10]. [\|G\|^10 ]. Since G∼(0,Id)G N(0,I_d), [‖G‖10] [\|G\|^10 ] =d(d+2)(d+4)(d+6)(d+8). =d(d+2)(d+4)(d+6)(d+8). Thus, [‖Dxvt(Xt)vt(Xt)‖2] [ \|D_xv_t(X_t)v_t(X_t) \|^2 ] ≤1σ2([‖D‖5])4/5 ≤ 1σ^2 (E [\|D\|^5 ] )^4/5 ⋅(d(d+2)(d+4)(d+6)(d+8))1/5. · (d(d+2)(d+4)(d+6)(d+8) )^1/5. Also recall [‖Dxvt(Xt)st(Xt)‖2] [ \|D_xv_t(X_t)s_t(X_t) \|^2 ] ≤d(d+2)(d+4)(d+6)σ4([‖D‖4])1/2. ≤ d(d+2)(d+4)(d+6)σ^4 (E [\|D\|^4 ] )^1/2. Combining the velocity part and the score part, we obtain [‖Dxvt(Xt)bt(Xt)‖2] [ \|D_xv_t(X_t)b_t(X_t) \|^2 ] ≤2σ2([‖D‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 ≤ 2σ^2 (E [\|D\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +τ42σ4d(d+2)(d+4)(d+6)([‖D‖4])1/2. + τ^42σ^4 d(d+2)(d+4)(d+6) (E [\|D\|^4 ] )^1/2. Since D=Y1−Y0D=Y_1-Y_0, this can be written as [‖Dxvt(Xt)bt(Xt)‖2] [ \|D_xv_t(X_t)b_t(X_t) \|^2 ] ≤2σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 ≤ 2σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +τ42σ4d(d+2)(d+4)(d+6)([‖Y1−Y0‖4])1/2. + τ^42σ^4 d(d+2)(d+4)(d+6) (E [\|Y_1-Y_0\|^4 ] )^1/2. ∎ G.4 Euler Increment Bounds Lemma 7 (Itô Control of the Euler Velocity Term). Let Y0Y_0, Y1Y_1, D, StS_t, G, XtX_t, qtq_t, vtv_t, and sts_t be as in Lemma 4, assume ‖Y1−Y0‖5<∞E\|Y_1-Y_0\|^5<∞, and set bt=vt+τ22stb_t=v_t+ τ^22s_t. Let ZtZ_t solve dZt dZ_t =bt(Zt)dt+τdBt, =b_t(Z_t)dt+τ dB_t, and assume that Zt∼qtZ_t q_t for every t∈[0,T]t∈[0,T]. Let 0=t0<t1<⋯<tN=T0=t_0<t_1<·s<t_N=T be a uniform grid with step size h, so tk+1−tk=ht_k+1-t_k=h and Nh=TNh=T. Then ∑k=0N−1∫tktk+1[‖vt(Zt)−vtk(Ztk)‖2]t _k=0^N-1 _t_k^t_k+1E [ \|v_t(Z_t)-v_t_k(Z_t_k) \|^2 ]dt ≤2Th23[3d(d+2)σ2[∥Y1−Y0∥4] ≤ 2Th^23 [ 3 d(d+2)σ^2E [\|Y_1-Y_0\|^4 ] +6σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 + 6σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +14τ4d(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2] + 14τ^4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 ] +τ2Thd(d+2)σ2([‖Y1−Y0‖4])1/2. + τ^2Th d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2. All expectations in the displayed bound are taken under the coupling law of (Y0,Y1)(Y_0,Y_1), except the expectation on the left-hand side, which is taken under the law of the exact SDE path (Zt)0≤t≤T(Z_t)_0≤ t≤ T. Proof. For t∈[tk,tk+1)t∈[t_k,t_k+1), Itô’s formula applied to vt(Zt)v_t(Z_t) gives vt(Zt)−vtk(Ztk) v_t(Z_t)-v_t_k(Z_t_k) =∫tkt(∂rvr(Zr)+Dxvr(Zr)br(Zr)+τ22Δxvr(Zr))r = _t_k^t ( _rv_r(Z_r)+D_xv_r(Z_r)b_r(Z_r)+ τ^22 _xv_r(Z_r) )dr +τ∫tktDxvr(Zr)dBr. +τ _t_k^tD_xv_r(Z_r)dB_r. Using ‖u+w‖2≤2‖u‖2+2‖w‖2\|u+w\|^2≤ 2\|u\|^2+2\|w\|^2, Jensen’s inequality, and Itô isometry, we get [‖vt(Zt)−vtk(Ztk)‖2] [ \|v_t(Z_t)-v_t_k(Z_t_k) \|^2 ] ≤2(t−tk)∫tkt[‖∂rvr(Zr)+Dxvr(Zr)br(Zr)+τ22Δxvr(Zr)‖2]r ≤ 2(t-t_k) _t_k^tE [ \| _rv_r(Z_r)+D_xv_r(Z_r)b_r(Z_r)+ τ^22 _xv_r(Z_r) \|^2 ]dr +2τ2∫tkt[‖Dxvr(Zr)‖F2]dr. +2τ^2 _t_k^tE [ \|D_xv_r(Z_r) \|_F^2 ]dr. Since Zr∼qrZ_r q_r and Xr∼qrX_r q_r, Lemma 6 applies with XrX_r replaced by ZrZ_r. Using ‖u+w+z‖2≤3‖u‖2+3‖w‖2+3‖z‖2\|u+w+z\|^2≤ 3\|u\|^2+3\|w\|^2+3\|z\|^2, we have [‖∂rvr(Zr)+Dxvr(Zr)br(Zr)+τ22Δxvr(Zr)‖2] [ \| _rv_r(Z_r)+D_xv_r(Z_r)b_r(Z_r)+ τ^22 _xv_r(Z_r) \|^2 ] ≤3[‖∂rvr(Zr)‖2]+3[‖Dxvr(Zr)br(Zr)‖2]+3[‖τ22Δxvr(Zr)‖2] ≤ 3E [ \| _rv_r(Z_r) \|^2 ]+3E [ \|D_xv_r(Z_r)b_r(Z_r) \|^2 ]+3E [ \| τ^22 _xv_r(Z_r) \|^2 ] ≤3d(d+2)σ2[‖Y1−Y0‖4] ≤ 3 d(d+2)σ^2E [\|Y_1-Y_0\|^4 ] +6σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 + 6σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +3τ42σ4d(d+2)(d+4)(d+6)([‖Y1−Y0‖4])1/2 + 3τ^42σ^4 d(d+2)(d+4)(d+6) (E [\|Y_1-Y_0\|^4 ] )^1/2 +12τ4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. + 12τ^4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. Also, [‖Dxvr(Zr)‖F2] [ \|D_xv_r(Z_r) \|_F^2 ] ≤d(d+2)σ2([‖Y1−Y0‖4])1/2. ≤ d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2. Therefore, for t∈[tk,tk+1)t∈[t_k,t_k+1), [‖vt(Zt)−vtk(Ztk)‖2] [ \|v_t(Z_t)-v_t_k(Z_t_k) \|^2 ] ≤2(t−tk)2[3d(d+2)σ2[∥Y1−Y0∥4] ≤ 2(t-t_k)^2 [ 3 d(d+2)σ^2E [\|Y_1-Y_0\|^4 ] +6σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 + 6σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +3τ42σ4d(d+2)(d+4)(d+6)([‖Y1−Y0‖4])1/2 + 3τ^42σ^4 d(d+2)(d+4)(d+6) (E [\|Y_1-Y_0\|^4 ] )^1/2 +12τ4d(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2] + 12τ^4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 ] +2τ2(t−tk)d(d+2)σ2([‖Y1−Y0‖4])1/2. +2τ^2(t-t_k) d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2. Integrating this estimate over t∈[tk,tk+1]t∈[t_k,t_k+1] gives ∫tktk+1[‖vt(Zt)−vtk(Ztk)‖2]t _t_k^t_k+1E [ \|v_t(Z_t)-v_t_k(Z_t_k) \|^2 ]dt ≤2h33[3d(d+2)σ2[∥Y1−Y0∥4] ≤ 2h^33 [ 3 d(d+2)σ^2E [\|Y_1-Y_0\|^4 ] +6σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 + 6σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +14τ4d(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2] + 14τ^4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 ] +τ2h2d(d+2)σ2([‖Y1−Y0‖4])1/2. + τ^2h^2 d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2. Finally, summing over k=0,…,N−1k=0,…,N-1 and using Nh=TNh=T, we obtain ∑k=0N−1∫tktk+1[‖vt(Zt)−vtk(Ztk)‖2]t _k=0^N-1 _t_k^t_k+1E [ \|v_t(Z_t)-v_t_k(Z_t_k) \|^2 ]dt ≤2Th23[3d(d+2)σ2[∥Y1−Y0∥4] ≤ 2Th^23 [ 3 d(d+2)σ^2E [\|Y_1-Y_0\|^4 ] +6σ2([‖Y1−Y0‖5])4/5(d(d+2)(d+4)(d+6)(d+8))1/5 + 6σ^2 (E [\|Y_1-Y_0\|^5 ] )^4/5 (d(d+2)(d+4)(d+6)(d+8) )^1/5 +14τ4d(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2] + 14τ^4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 ] +τ2Thd(d+2)σ2([‖Y1−Y0‖4])1/2. + τ^2Th d(d+2)σ^2 (E [\|Y_1-Y_0\|^4 ] )^1/2. This proves the lemma. ∎ Lemma 8 (Combined Score Increment Regularity). Let Y0Y_0 and Y1Y_1 be random variables taking values in ℝdR^d. Let D=Y1−Y0D=Y_1-Y_0. Assume that [‖Y1−Y0‖4] [\|Y_1-Y_0\|^4 ] <∞, <∞, where the expectation is taken under the joint law of (Y0,Y1)(Y_0,Y_1). For t∈[0,T]t∈[0,T], define St=Y0+tDS_t=Y_0+tD. Let G∼(0,Id)G N(0,I_d) be independent of (Y0,D)(Y_0,D), and define Xt=St+σGX_t=S_t+σ G with σ>0σ>0. Let qtq_t be the density of XtX_t. Define st(x)=∇xlogqt(x)s_t(x)= _x q_t(x) and vt(x) v_t(x) =[Y1−Y0∣Xt=x]. =E [Y_1-Y_0 X_t=x ]. Let bt(x) b_t(x) =vt(x)+τ22st(x), =v_t(x)+ τ^22s_t(x), where τ>0τ>0. Let ZtZ_t solve the exact SDE dZt dZ_t =bt(Zt)dt+τdBt, =b_t(Z_t)dt+τ dB_t, and assume that Zt∼qtZ_t q_t for every t∈[0,T]t∈[0,T]. Let 0=t0<t1<⋯<tN=T0=t_0<t_1<·s<t_N=T be a uniform grid with step size h, so tk+1−tk=ht_k+1-t_k=h and Nh=TNh=T. Then ∑k=0N−1∫tktk+1[‖st(Zt)−stk(Ztk)‖2]t _k=0^N-1 _t_k^t_k+1E [ \|s_t(Z_t)-s_t_k(Z_t_k) \|^2 ]dt ≤τ2Th(2d(d+2)+2d)σ4 ≤ τ^2Th (2d(d+2)+2d )σ^4 +2Th23[64dd(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2 + 2Th^23 [ 64d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 .+ 4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +τ4(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6 .+ τ^4 (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6 . +64τ4d(d+2)(d+4)σ6]. .+ 64τ^4d(d+2)(d+4)σ^6 ]. The expectation on the left-hand side is taken under the law of the exact SDE path (Zt)0≤t≤T(Z_t)_0≤ t≤ T. Proof. We control the combined score increment directly. This is different from decomposing it into score-time and score-space pieces. Fix k∈0,…,N−1k∈\0,…,N-1\ and t∈[tk,tk+1]t∈[t_k,t_k+1]. Applying Itô’s formula to the process sr(Zr)s_r(Z_r) over r∈[tk,t]r∈[t_k,t] gives st(Zt)−stk(Ztk) s_t(Z_t)-s_t_k(Z_t_k) =∫tkt(∂rsr(Zr)+Dxsr(Zr)br(Zr)+τ22Δxsr(Zr))r = _t_k^t ( _rs_r(Z_r)+D_xs_r(Z_r)b_r(Z_r)+ τ^22 _xs_r(Z_r) )dr +τ∫tktDxsr(Zr)dBr. +τ _t_k^tD_xs_r(Z_r)dB_r. Since br(x)=vr(x)+τ22sr(x)b_r(x)=v_r(x)+ τ^22s_r(x), the drift term inside the integral is ∂rsr(Zr)+Dxsr(Zr)br(Zr)+τ22Δxsr(Zr) _rs_r(Z_r)+D_xs_r(Z_r)b_r(Z_r)+ τ^22 _xs_r(Z_r) =∂rsr(Zr)+Dxsr(Zr)vr(Zr)+τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr). = _rs_r(Z_r)+D_xs_r(Z_r)v_r(Z_r)+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r). Now use the continuity equation for the Gaussian-smoothed interpolation. Since qrq_r is the density of Xr=Sr+σGX_r=S_r+σ G, and vr(x)=[D∣Xr=x]v_r(x)=E[D X_r=x], the curve (qr)r∈[0,T](q_r)_r∈[0,T] satisfies ∂rqr(x)+∇x⋅(qr(x)vr(x)) _rq_r(x)+ _x· (q_r(x)v_r(x) ) =0. =0. Dividing by qr(x)q_r(x) gives ∂rlogqr(x) _r q_r(x) =−∇x⋅vr(x)−vr(x)⊤sr(x). =- _x· v_r(x)-v_r(x) s_r(x). Taking ∇x _x on both sides gives ∂r∇xlogqr(x)=∂rsr(x) _r _x q_r(x)= _rs_r(x) =−∇x(∇x⋅vr(x))−(Dxvr(x))⊤sr(x)−Dxsr(x)vr(x). =- _x ( _x· v_r(x) )-(D_xv_r(x)) s_r(x)-D_xs_r(x)v_r(x). Substituting x=Zrx=Z_r gives ∂rsr(Zr) _rs_r(Z_r) =−∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr)−Dxsr(Zr)vr(Zr). =- _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r)-D_xs_r(Z_r)v_r(Z_r). Therefore, the term Dxsr(Zr)vr(Zr)D_xs_r(Z_r)v_r(Z_r) cancels and we have ∂rsr(Zr)+Dxsr(Zr)vr(Zr)+τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr) _rs_r(Z_r)+D_xs_r(Z_r)v_r(Z_r)+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r) = = −∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr)+τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr). - _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r)+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r). Hence st(Zt)−stk(Ztk) s_t(Z_t)-s_t_k(Z_t_k) =∫tkt[−∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr) = _t_k^t [- _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r) . +τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr)]dr .+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r) ]dr +τ∫tktDxsr(Zr)dBr. +τ _t_k^tD_xs_r(Z_r)dB_r. Using ‖u+w‖2≤2‖u‖2+2‖w‖2\|u+w\|^2≤ 2\|u\|^2+2\|w\|^2, Jensen’s inequality, and Itô isometry, we obtain [‖st(Zt)−stk(Ztk)‖2] [ \|s_t(Z_t)-s_t_k(Z_t_k) \|^2 ] ≤2(t−tk)∫tkt[‖−∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr) ≤ 2(t-t_k) _t_k^tE [ \|- _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r) . . +τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr)‖2]dr . .+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r) \|^2 ]dr +2τ2∫tkt[‖Dxsr(Zr)‖F2]dr. +2τ^2 _t_k^tE [ \|D_xs_r(Z_r) \|_F^2 ]dr. Using ‖u1+u2+u3+u4‖2≤4‖u1‖2+4‖u2‖2+4‖u3‖2+4‖u4‖2\|u_1+u_2+u_3+u_4\|^2≤ 4\|u_1\|^2+4\|u_2\|^2+4\|u_3\|^2+4\|u_4\|^2, we have [‖−∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr)+τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr)‖2] [ \|- _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r)+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r) \|^2 ] ≤4[‖∇x(∇x⋅vr)(Zr)‖2]+4[‖(Dxvr(Zr))⊤sr(Zr)‖2] ≤ 4E [ \| _x ( _x· v_r )(Z_r) \|^2 ]+4E [ \|(D_xv_r(Z_r)) s_r(Z_r) \|^2 ] +τ4[‖Dxsr(Zr)sr(Zr)‖2]+τ4[‖Δxsr(Zr)‖2]. +τ^4E [ \|D_xs_r(Z_r)s_r(Z_r) \|^2 ]+τ^4E [ \| _xs_r(Z_r) \|^2 ]. We now bound each term. First, by the posterior-velocity Hessian bound in Lemma 5, [‖∇x(∇x⋅vr)(Zr)‖2] [ \| _x ( _x· v_r )(Z_r) \|^2 ] ≤16dd(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ 16d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. Also, [‖(Dxvr(Zr))⊤sr(Zr)‖2] [ \|(D_xv_r(Z_r)) s_r(Z_r) \|^2 ] ≤d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2. ≤ d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2. Third, [‖Dxsr(Zr)sr(Zr)‖2] [ \|D_xs_r(Z_r)s_r(Z_r) \|^2 ] ≤(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6. ≤ (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6. Fourth, we use the already proved Gaussian score-geometry bound [‖Δxsr(Zr)‖2] [ \| _xs_r(Z_r) \|^2 ] ≤64d(d+2)(d+4)σ6. ≤ 64d(d+2)(d+4)σ^6. Finally, using the already proved Gaussian score-geometry bound [‖Dxsr(Zr)‖F2] [ \|D_xs_r(Z_r) \|_F^2 ] ≤2d(d+2)+2dσ4. ≤ 2d(d+2)+2dσ^4. Therefore, for every r∈[0,T]r∈[0,T], [‖−∇x(∇x⋅vr)(Zr)−(Dxvr(Zr))⊤sr(Zr)+τ22Dxsr(Zr)sr(Zr)+τ22Δxsr(Zr)‖2] [ \|- _x ( _x· v_r )(Z_r)-(D_xv_r(Z_r)) s_r(Z_r)+ τ^22D_xs_r(Z_r)s_r(Z_r)+ τ^22 _xs_r(Z_r) \|^2 ] ≤64dd(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 ≤ 64d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 +4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 + 4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 +τ4(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6 + τ^4 (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6 +64τ4d(d+2)(d+4)σ6. + 64τ^4d(d+2)(d+4)σ^6. Plugging this and the martingale bound into the Itô estimate gives, for every t∈[tk,tk+1]t∈[t_k,t_k+1], [‖st(Zt)−stk(Ztk)‖2] [ \|s_t(Z_t)-s_t_k(Z_t_k) \|^2 ] ≤2(t−tk)2[64dd(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2 ≤ 2(t-t_k)^2 [ 64d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 .+ 4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +τ4(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6 .+ τ^4 (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6 . +64τ4d(d+2)(d+4)σ6] .+ 64τ^4d(d+2)(d+4)σ^6 ] +2τ2(t−tk)2d(d+2)+2dσ4. +2τ^2(t-t_k) 2d(d+2)+2dσ^4. Integrating over t∈[tk,tk+1]t∈[t_k,t_k+1] gives ∫tktk+1[‖st(Zt)−stk(Ztk)‖2]t _t_k^t_k+1E [ \|s_t(Z_t)-s_t_k(Z_t_k) \|^2 ]dt ≤2h33[64dd(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2 ≤ 2h^33 [ 64d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 .+ 4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +τ4(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6 .+ τ^4 (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6 . +64τ4d(d+2)(d+4)σ6] .+ 64τ^4d(d+2)(d+4)σ^6 ] +τ2h22d(d+2)+2dσ4. +τ^2h^2 2d(d+2)+2dσ^4. Summing over k=0,…,N−1k=0,…,N-1 and using Nh=TNh=T, we get ∑k=0N−1∫tktk+1[‖st(Zt)−stk(Ztk)‖2]t _k=0^N-1 _t_k^t_k+1E [ \|s_t(Z_t)-s_t_k(Z_t_k) \|^2 ]dt ≤2Th23[64dd(d+2)(d+4)(d+6)σ4([∥Y1−Y0∥4])1/2 ≤ 2Th^23 [ 64d d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +4d(d+2)(d+4)(d+6)σ4([‖Y1−Y0‖4])1/2 .+ 4 d(d+2)(d+4)(d+6)σ^4 (E [\|Y_1-Y_0\|^4 ] )^1/2 . +τ4(8d(d+2)(d+4)(d+6)+8d2)d(d+2)σ6 .+ τ^4 (8d(d+2)(d+4)(d+6)+8d^2 )d(d+2)σ^6 . +64τ4d(d+2)(d+4)σ6] .+ 64τ^4d(d+2)(d+4)σ^6 ] +τ2Th(2d(d+2)+2d)σ4. + τ^2Th (2d(d+2)+2d )σ^4. This proves the lemma. ∎ G.5 Completion of the KL Sampling Bound Proof of Theorem 1. Set T=tN=1T=t_N=1, and let Z be the exact weak solution with ℒ(Zt)=qtL(Z_t)=q_t supplied by Lemma 1. Let Z Z denote the continuous-time Euler interpolation, so that Z^tk=Z^k Z_t_k= Z_k. Write Vh V_h =∑k=0N−1∫tktk+1‖vt(Zt)−vtk(Ztk)‖2t, = _k=0^N-1 _t_k^t_k+1E\|v_t(Z_t)-v_t_k(Z_t_k)\|^2\,dt, Sh S_h =∑k=0N−1∫tktk+1‖st(Zt)−stk(Ztk)‖2t. = _k=0^N-1 _t_k^t_k+1E\|s_t(Z_t)-s_t_k(Z_t_k)\|^2\,dt. We prove the theorem in four steps. Step 1: reduction to VhV_h, ShS_h, and the estimation errors. For t∈[tk,tk+1)t∈[t_k,t_k+1), decompose the drift mismatch as bt(Zt)−b^tk(Ztk) b_t(Z_t)- b_t_k(Z_t_k) =vt(Zt)−vtk(Ztk)+vtk(Ztk)−v^tk(Ztk) =\v_t(Z_t)-v_t_k(Z_t_k)\+\v_t_k(Z_t_k)- v_t_k(Z_t_k)\ +τ22st(Zt)−stk(Ztk)+τ22stk(Ztk)−s^tk(Ztk). + τ^22\s_t(Z_t)-s_t_k(Z_t_k)\+ τ^22\s_t_k(Z_t_k)- s_t_k(Z_t_k)\. Using the inequality ‖a1+a2+a3+a4‖2≤4∑j=14‖aj‖2\|a_1+a_2+a_3+a_4\|^2≤ 4 _j=1^4\|a_j\|^2, and Assumption 1 we obtain ∫01‖bt(Zt)−b^ηh(t)(Zηh(t))‖2t≤4Vh+4εvel2+τ4Sh+τ4εscore2. _0^1E\|b_t(Z_t)- b_ _h(t)(Z_ _h(t))\|^2\,dt≤ 4V_h+4 _vel^2+τ^4S_h+τ^4 _score^2. Steps 2 and 3 below show that Vh,Sh<∞V_h,S_h<∞; hence the condition ∫0Tℙ[‖Δt(Z)‖2]t<∞ _0^TE_P [\| _t(Z)\|^2\, ]dt<∞ in Lemma 2 holds. The corresponding condition for Z Z is justified by Proposition 3. Applying that lemma and dividing by 2τ22τ^2 then gives KL(ℒ(ZtN)∥ℒ(Z^N))≤2τ2Vh+2τ2εvel2+τ22Sh+τ22εscore2. \! (L(Z_t_N)\, \|\,L( Z_N) )≤ 2τ^2V_h+ 2τ^2 _vel^2+ τ^22S_h+ τ^22 _score^2. (7) Step 2: explicit velocity-increment bound. Lyapunov’s inequality yields ‖Y1−Y0‖4≤D54,(‖Y1−Y0‖4)1/2≤D52.E\|Y_1-Y_0\|^4≤ D_5^4, (E\|Y_1-Y_0\|^4 )^1/2≤ D_5^2. For d≥1d≥ 1, d(d+2)≤d+1,d(d+2)(d+4)(d+6)(d+8)1/5≤d+8, d(d+2)≤ d+1, \d(d+2)(d+4)(d+6)(d+8)\^1/5≤ d+8, and d(d+2)(d+4)(d+6)≤(d+6)2. d(d+2)(d+4)(d+6)≤(d+6)^2. Applying these inequalities to Lemma 7 with T=1T=1 gives Vh V_h ≤2(9d+51)3h2D54σ2+283h2τ4(d+6)2D52σ4+hτ2(d+1)D52σ2. ≤ 2(9d+51)3 h^2D_5^4σ^2+ 283 h^2τ^4(d+6)^2D_5^2σ^4+ hτ^2(d+1)D_5^2σ^2. (8) Step 3: explicit score-increment bound. The two displacement-dependent terms in Lemma 8 satisfy (64d+4)d(d+2)(d+4)(d+6)≤68(d+6)3.(64d+4) d(d+2)(d+4)(d+6)≤ 68(d+6)^3. For the two pure-score drift terms, letting g8=d(d+2)(d+4)(d+6)g_8=d(d+2)(d+4)(d+6), we have g8≥d2g_8≥ d^2 and hence (8g8+8d2)d(d+2) (8g_8+8d^2)d(d+2) ≤4(d+6)3, ≤ 4(d+6)^3, 64d(d+2)(d+4) 64d(d+2)(d+4) ≤64(d+6)3. ≤ 64(d+6)^3. Finally, 2d(d+2)+2d=2d(d+3)2d(d+2)+2d=2d(d+3). Therefore Sh S_h ≤1363h2(d+6)3D52σ4+2hτ2d(d+3)σ4+1363h2τ4(d+6)3σ6. ≤ 1363 h^2(d+6)^3D_5^2σ^4+ 2hτ^2d(d+3)σ^4+ 1363 h^2τ^4(d+6)^3σ^6. (9) Step 4: collect constants and balance the leading terms. Substituting Equation (8) and Equation (9) into Equation (7) gives KL(ℒ(ZtN)∥ℒ(Z^N)) \! (L(Z_t_N)\, \|\,L( Z_N) ) ≤1τ24(9d+51)3h2D54σ2+2εvel2 ≤ 1τ^2 \ 4(9d+51)3 h^2D_5^4σ^2+2 _vel^2 \ +τ2563h2(d+6)2D52σ4+683h2(d+6)3D52σ4+12εscore2 +τ^2 \ 563 h^2(d+6)^2D_5^2σ^4+ 683 h^2(d+6)^3D_5^2σ^4+ 12 _score^2 \ +hτ4d(d+3)σ4+683h2τ6(d+6)3σ6+2h(d+1)D52σ2. + hτ^4d(d+3)σ^4+ 683 h^2τ^6(d+6)^3σ^6+ 2h(d+1)D_5^2σ^2. Because (d+6)2≤(d+6)3(d+6)^2≤(d+6)^3, the coefficient in braces on the second line is at most BhB_h. The first line is Ah/τ2A_h/τ^2, and the remaining three terms are exactly the last three terms on the right-hand side of Equation (2). This proves that bound. Since Ah,Bh>0A_h,B_h>0, the choice τ=(Ah/Bh)1/4τ=(A_h/B_h)^1/4 satisfies Ahτ2+Bhτ2=2AhBh. A_hτ^2+B_hτ^2=2 A_hB_h. Thus this choice balances the first two terms, as claimed. ∎ G.6 Wasserstein Conversion to the Unsmoothed Target The following two corollaries convert the terminal KL guarantee in Theorem 1 into W1W_1 bounds relative to the unsmoothed target. Corollary 1 applies to the unprojected Euler output, whereas Corollary 2 gives a sharper conversion for the terminally projected output when the target is supported on [−1,1]d[-1,1]^d. Throughout this subsection, set T=1T=1 and condition on the auxiliary data, the EOT target reservoir, and the resulting coupling kernels. By Lemma 10, the population-valid comparison coupling satisfies ∫Q(⋅∣y1)∫ℙY∣C=c,A=adℙC(c)(y1)=ℙ^Y|A=aaux. Q(· y_1) _Y C=c,A=a\,dP_C(c)(y_1)= P_Y A=a^aux. Consequently, the exact interpolation associated with Q has Y0∼ℙ^Y|A=aauxY_0 P_Y A=a^aux. Algorithm 1 is initialized from the same empirical source distribution. Therefore, ℒ(Z0) (Z_0) =ℙ^Y|A=aaux∗N(0,σ2Id)=ℒ(Z^0). = P_Y A=a^aux*N(0,σ^2I_d)=L( Z_0). Thus KL(ℒ(Z0)∥ℒ(Z^0))=0KL (L(Z_0)\, \|\,L( Z_0) )=0. At terminal time, ℒ(ZtN)=ℒ(Z1)=ℒ(Y1+σG)=∫ℙY∣C=c,A=adℙC(c)∗N(0,σ2Id)L(Z_t_N)=L(Z_1)=L(Y_1+σ G)= _Y C=c,A=a\,dP_C(c)*N(0,σ^2I_d). We treat the discrepancy between ℒ(Y1+σG)L(Y_1+σ G) and ∫ℙY∣C=c,A=adℙC(c) _Y C=c,A=a\,dP_C(c) solely as Gaussian-smoothing error. No stopping time T<1T<1 is introduced. Corollary 1 (Wasserstein Error Relative to the Unsmoothed Target). Assume the conditions of Theorem 1 and Assumption 2. Write the learned score as s^t(x)=u^t(x)−x/σ2 s_t(x)=\ u_t(x)-x\/σ^2, and assume the network class bounds supx,k‖v^tk(x)‖≤2d,supx,k‖u^tk(x)‖≤d. _x,k\| v_t_k(x)\|≤ 2 d, _x,k\| u_t_k(x)\|≤ d. Assume also hτ2/(2σ2)≤1hτ^2/(2σ^2)≤ 1. Then W1(∫ℙY∣C=c,A=adℙC(c),ℒ(Z^N)) W_1 ( _Y C=c,A=a\,dP_C(c),L( Z_N) ) ≤σd+21/4d[1+σ2+(1+σ2+3+minτ,2σ)2]1/2⋅KL(ℒ(Y1+σG)∥ℒ(Z^N))1/4. ≤σ d+2^1/4 d [1+σ^2+ ( 1+σ^2+3+ \τ, 2σ\ )^2 ]^1/2·KL (L(Y_1+σ G)\, \|\,L( Z_N) )^1/4. Proof. Let Y1∼∫ℙY∣C=c,A=adℙC(c)Y_1 _Y C=c,A=a\,dP_C(c) and G∼(0,Id)G N(0,I_d) be independent. Coupling Y1Y_1 with Y1+σGY_1+σ G gives W1(∫ℙY∣C=c,A=adℙC(c),ℒ(Y1+σG)) W_1( _Y C=c,A=a\,dP_C(c),L(Y_1+σ G)) ≤σ‖G‖≤σd. ≤ \|G\|≤σ d. Let (X,X~)(X, X) be a maximal coupling of ℒ(Y1+σG)L(Y_1+σ G) and ℒ(Z^N)L( Z_N). Then ℙ(X≠X~)=TV(ℒ(Y1+σG),ℒ(Z^N)).P(X≠ X)=TV\! (L(Y_1+σ G),L( Z_N) ). Therefore W1(ℒ(Y1+σG),ℒ(Z^N)) W_1 (L(Y_1+σ G),L( Z_N) ) ≤∥X−X~∥=[∥X−X~∥X≠X~] \|X- X\|=E [\|X- X\|1\X≠ X\ ] ≤(‖X−X~‖2)1/2TV(ℒ(Y1+σG),ℒ(Z^N))1/2 ≤ (E\|X- X\|^2 )^1/2TV (L(Y_1+σ G),L( Z_N) )^1/2 ≤(2‖X‖2+2‖X~‖2)1/2[12KL(ℒ(Y1+σG)∥ℒ(Z^N))]1/4 ≤ (2E\|X\|^2+2E\| X\|^2 )^1/2 [ 12KL (L(Y_1+σ G)\, \|\,L( Z_N) ) ]^1/4 =21/4(‖X‖2+‖X~‖2)1/2KL(ℒ(Y1+σG)∥ℒ(Z^N))1/4. =2^1/4 (E\|X\|^2+E\| X\|^2 )^1/2KL (L(Y_1+σ G)\, \|\,L( Z_N) )^1/4. Since Y1∈[−1,1]dY_1∈[-1,1]^d, ‖X‖2=‖Y1+σG‖2=‖Y1‖2+σ2‖G‖2≤d(1+σ2). \|X\|^2=E\|Y_1+σ G\|^2=E\|Y_1\|^2+σ^2E\|G\|^2≤ d(1+σ^2). It remains to bound the second moment of Z^N Z_N. By the definition of the learned score class, s^t(x)=u^t(x)−xσ2, s_t(x)= u_t(x)-xσ^2, the Euler recursion can be written as ah:=1−hτ22σ2∈[0,1],ξk:=v^tk(Z^k)+τ22σ2u^tk(Z^k).a_h:=1- hτ^22σ^2∈[0,1], _k:= v_t_k( Z_k)+ τ^22σ^2 u_t_k( Z_k). Z^k+1 Z_k+1 =ahZ^k+hξk+τhGk. =a_h Z_k+h _k+τ h\,G_k. Iterating this recursion gives Z^N Z_N =ahNZ^0+h∑k=0N−1ahN−1−kξk =a_h^N Z_0+h _k=0^N-1a_h^N-1-k _k +τh∑k=0N−1ahN−1−kGk. +τ h _k=0^N-1a_h^N-1-kG_k. Applying Minkowski’s inequality in L2L^2, we obtain ([‖Z^N‖2])1/2≤ (E[\| Z_N\|^2] )^1/2≤\; ahN([‖Z^0‖2])1/2+h∑k=0N−1ahN−1−k(‖ξk‖2)1/2+τh([‖∑k=0N−1ahN−1−kGk‖2])1/2. a_h^N (E[\| Z_0\|^2] )^1/2+h _k=0^N-1a_h^N-1-k (E\| _k\|^2 )^1/2+τ h (E [ \| _k=0^N-1a_h^N-1-kG_k \|^2 ] )^1/2. Since we have ‖v^tk(Z^k)+τ22σ2u^tk(Z^k)‖≤(2+τ22σ2)d,[‖∑k=0N−1(1−hτ22σ2)N−1−kGk‖2]=d∑k=0N−1(1−hτ22σ2)2k \| v_t_k( Z_k)+ τ^22σ^2 u_t_k( Z_k) \|≤ (2+ τ^22σ^2 ) d, [ \| _k=0^N-1 (1- hτ^22σ^2 )^N-1-kG_k \|^2 ]=d _k=0^N-1 (1- hτ^22σ^2 )^2k where GkG_k are independent, [‖Gk‖2]=dE[\|G_k\|^2]=d, and all cross terms vanish. Therefore, ([‖Z^N‖2])1/2≤ (E[\| Z_N\|^2] )^1/2≤ ([‖Z^0‖2])1/2+h(2+τ22σ2)d∑k=0N−1(1−hτ22σ2)k+τ[hd∑k=0N−1(1−hτ22σ2)2k]1/2. (E[\| Z_0\|^2] )^1/2+h (2+ τ^22σ^2 ) d _k=0^N-1 (1- hτ^22σ^2 )^k+τ [hd _k=0^N-1 (1- hτ^22σ^2 )^2k ]^1/2. Since Nh=1Nh=1, h∑k=0N−1(1−hτ22σ2)k≤hN=1h _k=0^N-1 (1- hτ^22σ^2 )^k≤ hN=1, so we have h∑k=0N−1(1−hτ22σ2)k h _k=0^N-1 (1- hτ^22σ^2 )^k =h2σ2hτ2[1−(1−hτ22σ2)N]≤min1,2σ2τ2, =h 2σ^2hτ^2 [1- (1- hτ^22σ^2 )^N ]≤ \1, 2σ^2τ^2 \, and h∑k=0N−1(1−hτ22σ2)2k≤h∑k=0N−1(1−hτ22σ2)k≤min1,2σ2τ2. h _k=0^N-1 (1- hτ^22σ^2 )^2k≤ h _k=0^N-1 (1- hτ^22σ^2 )^k≤ \1, 2σ^2τ^2 \. Furthermore, observe that (2+τ22σ2)min1,2σ2τ2≤3 (2+ τ^22σ^2 ) \1, 2σ^2τ^2 \≤ 3 and τ(min1,2σ2τ2)1/2=minτ,2στ ( \1, 2σ^2τ^2 \ )^1/2= \τ, 2σ\. It follows that ([‖Z^N‖2])1/2≤([‖Z^0‖2])1/2+3d+dminτ,2σ. (E[\| Z_N\|^2] )^1/2≤ (E[\| Z_0\|^2] )^1/2+3 d+ d \τ, 2σ\. Since [‖Z^0‖2]=[‖Y^0‖2]+σ2d≤d(1+σ2)E[\| Z_0\|^2]=E[\| Y_0\|^2]+σ^2d≤ d(1+σ^2), we obtain [‖Z^N‖2]≤d(1+σ2+3+minτ,2σ)2. [\| Z_N\|^2]≤ d ( 1+σ^2+3+ \τ, 2σ\ )^2. It follows that ‖Z^N‖2 \| Z_N\|^2 ≤d(1+σ2+3+minτ,2σ)2. ≤ d ( 1+σ^2+3+ \τ, 2σ\ )^2. Combining the preceding moment bounds yields W1(ℒ(Y1+σG),ℒ(Z^N)) W_1 (L(Y_1+σ G),L( Z_N) ) ≤21/4d[1+σ2+(1+σ2+3+minτ,2σ)2]1/2 ≤ 2^1/4 d [1+σ^2+ ( 1+σ^2+3+ \τ, 2σ\ )^2 ]^1/2 ⋅KL(ℒ(Y1+σG)∥ℒ(Z^N))1/4. ·KL (L(Y_1+σ G)\, \|\,L( Z_N) )^1/4. The triangle inequality through ℒ(Y1+σG)L(Y_1+σ G) proves the result. ∎ Corollary 2 (Wasserstein Error After Output Clipping). Let Y1∼μ1Y_1 _1 be supported on K=[−1,1]dK=[-1,1]^d, let G∼(0,Id)G (0,I_d) be independent of Y1Y_1, and let Z^N Z_N be any ℝdR^d-valued random vector. Define the Euclidean projection Π=clip1:ℝd→K,Z~N=Π(Z^N). =clip_1:R^d→ K, Z_N= ( Z_N). Then W1(μ1,ℒ(Z~N))≤σd+2dKL(ℒ(Y1+σG)∥ℒ(Z^N)). W_1\! ( _1,L( Z_N) )≤σ d+ 2d\,KL\! (L(Y_1+σ G) \|L( Z_N) ). (10) Under the conditions of Theorem 1, its KL bound may be substituted into the right-hand side of Equation (10). Clipping is applied only after the final Euler step. Proof. Write Pσ=ℒ(Y1+σG)P_σ=L(Y_1+σ G) and P^=ℒ(Z^N) P=L( Z_N). Since Π(Y1)=Y1 (Y_1)=Y_1 and Euclidean projection onto K is nonexpansive, W1(μ1,Π#Pσ) W_1( _1, _\#P_σ) ≤‖Π(Y1)−Π(Y1+σG)‖≤σ|G|≤σd. \| (Y_1)- (Y_1+σ G)\|≤ \|G\|≤σ d. Both Π#Pσ _\#P_σ and Π#P _\# P are supported on K, whose diameter is 2d2 d. A maximal coupling, Pinsker’s inequality, and data processing for KL divergence therefore give W1(Π#Pσ,Π#P^) W_1( _\#P_σ, _\# P) ≤2dTV(Π#Pσ,Π#P^) ≤ 2 d\,TV( _\#P_σ, _\# P) ≤2dKL(Π#Pσ∥Π#P^) ≤ 2d\,KL( _\#P_σ\| _\# P) ≤2dKL(Pσ∥P^). ≤ 2d\,KL(P_σ\| P). The triangle inequality proves Equation (10). ∎ This is a guarantee for the clipped output Z~N Z_N; the KL bound in Theorem 1 continues to concern the Euler output Z^N Z_N without clipping. Appendix H Coupling Construction and Replacement We first introduce some notations. For any distribution ζ and any conditional distribution K(dy0∣y1)K(dy_0 y_1), write Γζ,K(dy0,dy1)=K(dy0∣y1)ζ(dy1). _ζ,K(dy_0,dy_1)=K(dy_0 y_1)ζ(dy_1). This is the joint law obtained by drawing Y1∼ζY_1 ζ, and then drawing Y0Y_0 from K(⋅∣Y1)K(· Y_1). On ℝd×ℝdR^d×R^d, define ρ((y0,y1),(y~0,y~1))=‖y0−y~0‖+‖y1−y~1‖,ρ ((y_0,y_1),( y_0, y_1) )=\|y_0- y_0\|+\|y_1- y_1\|, and let W1,ρW_1,ρ denote the 11-Wasserstein distance induced by ρ. We first construct Q^n Q_n using finitely many samples from the prediction oracle. Define the empirical auxiliary covariate distribution by ℙ^C:=n−1∑i=n+12nδCi P_C:=n^-1 _i=n+1^2n _C_i. Conditional on auxD_aux, for each i=n+1,…,2ni=n+1,…,2n, draw Y^i,1Q,…,Y^i,MQQ∼ind.ℙ^Y∣C=c,A=aevaluated at c=Ci. Y_i,1^Q,…, Y_i,M_Q^Q ind. P_Y C=c,A=a\ evaluated at c=C_i. We then define ℙ^Y1Q=1nMQ∑i=n+12n∑r=1MQδY^i,rQ, P_Y_1^Q= 1nM_Q _i=n+1^2n _r=1^M_Q _ Y_i,r^Q, which approximates ∫ℙ^Y∣C=c,A=adℙ^C(c) P_Y C=c,A=a\,d P_C(c). To keep the coupling arguments readable, write μ1 _1 :=∫ℙY∣C=c,A=adℙC(c), := _Y C=c,A=a\,dP_C(c), μ¯Q μ_Q :=∫ℙ^Y∣C=c,A=adℙ^C(c), := P_Y C=c,A=a\,d P_C(c), μQMC _Q^MC :=ℙ^Y1Q. := P_Y_1^Q. Thus μ1 _1 is the population counterfactual target, μ¯Q μ_Q is the fitted target averaged over auxiliary covariates, and μQMC _Q^MC is its finite-MQM_Q empirical approximation. In this section, we focus on the entropic optimal transport kernel between ℙ^Y|A=aaux P_Y A=a^aux and μQMC _Q^MC: Γ^EOT∈argminΓ∈Π(ℙ^Y|A=aaux,μQMC)∫‖y1−y0‖2Γ(dy0,dy1)+εOTKL(Γ∥ℙ^Y|A=aaux⊗μQMC). _EOT∈ *arg\,min_ ∈ ( P_Y A=a^aux, _Q^MC ) \ \|y_1-y_0\|^2\, (dy_0,dy_1)+ _OTKL ( \, \|\, P_Y A=a^aux _Q^MC ) \. When εOT _OT is close to zero, the EOT objective is close to the unregularized optimal transport problem between ℙ^Y|A=aaux P_Y A=a^aux and μQMC _Q^MC. For a generic kernel, one would need to assume regularity of y1↦Q^n(⋅∣y1)y_1 Q_n(· y_1). This regularity is needed because we need to control the coupling replacement error. The proof evaluates Q^n Q_n not only at samples from μQMC _Q^MC, where the EOT marginal identity ∫Q^n(⋅∣y1)μQMC(dy1)=ℙ^Y|A=aaux Q_n(· y_1) _Q^MC(dy_1)= P_Y A=a^aux holds, but also at y1y_1’s from the population target. For the EOT kernel, this needed regularity can be proved directly: Lemma 9 shows that y1↦Q^n(⋅∣y1)y_1 Q_n(· y_1) is Lipschitz in W1W_1, with constant proportional to 1/εOT1/ _OT. We then establish a coupling replacement error bound, see Lemma 10 and Corollary 3. H.1 The Learned EOT Kernel The entropic optimal transport problem defining Γ^EOT _EOT has a unique optimizer. By (Genevay et al., 2016, Proposition 2.1), there exist φ^n _n and ψ^n ψ_n such that dΓ^EOT(y0,y1)=exp(φ^n(y0)+ψ^n(y1)−‖y0−y1‖2εOT)dℙ^Y|A=aaux(y0)dμQMC(y1) d _EOT(y_0,y_1)= ( _n(y_0)+ ψ_n(y_1)-\|y_0-y_1\|^2 _OT )d P_Y A=a^aux(y_0)d _Q^MC(y_1) for almost every (y0,y1)(y_0,y_1). Consequently, for μQMC _Q^MC-almost every y, the conditional distribution of Y0Y_0 given Y1=yY_1=y under Γ^EOT _EOT is Q^n(dy0∣y)=exp(φ^n(y0)−‖y0−y‖2εOT)∫exp(φ^n(z)−‖z−y‖2εOT)ℙ^Y|A=aaux(z)ℙ^Y|A=aaux(dy0). Q_n(dy_0 y)= ( _n(y_0)-\|y_0-y\|^2 _OT ) ( _n(z)-\|z-y\|^2 _OT ) P_Y A=a^aux(dz) P_Y A=a^aux(dy_0). Motivated by the out-of-sample entropic extension discussed before Equation (13) of Pooladian and Niles-Weed (2021), we use the displayed Gibbs formula to define Q^n(⋅∣y) Q_n(· y) for every y∈ℝdy ^d. We remark that this is an extension of the conditional kernel. The denominator is finite and strictly positive, so the right-hand side is a probability distribution for every y. It holds that Γ^EOT(dy0,dy1)=Q^n(dy0∣y1)μQMC(dy1). _EOT(dy_0,dy_1)= Q_n(dy_0 y_1) _Q^MC(dy_1). Moreover, under Assumption 2, we have Q^n([−1,1]d∣y)=1,∀y∈ℝd Q_n([-1,1]^d y)=1,∀ y ^d. The same kernel can be evaluated at target queries drawn from the training-fold fitted distribution or from the population target distribution. The resulting joint distribution remains well defined, but its Y0Y_0-marginal need not equal ℙ^Y|A=aaux P_Y A=a^aux, because the exact source-marginal identity holds only when Q^n Q_n is integrated against μQMC _Q^MC. By construction, we have the marginal identity ∫Q^n(⋅∣y1)μQMC(dy1)=ℙ^Y|A=aaux. Q_n(· y_1) _Q^MC(dy_1)= P_Y A=a^aux. This identity is not an identity under ∫ℙ^Y∣C=c,A=adℙ^C(c) P_Y C=c,A=a\,d P_C(c), nor under the training-fold fitted target distribution 1n∑i=1nℙ^Y∣C=Ci,A=a 1n _i=1^n P_Y C=C_i,A=a. Lemma 9 (Stability of the Finite-Sample Entropic Coupling). Assume Assumption 2. Then, for every y1,y1′∈ℝdy_1,y_1 ^d, W1(Q^n(⋅∣y1),Q^n(⋅∣y1′))≤2dεOT∥y1−y1′∥. W_1 ( Q_n(· y_1), Q_n(· y_1 ) )≤ 2d _OT\|y_1-y_1 \|. Proof. Recall that Q^n(dy0∣y)=exp(φ^n(y0)−‖y0−y‖2εOT)∫exp(φ^n(z)−‖z−y‖2εOT)ℙ^Y|A=aaux(z)ℙ^Y|A=aaux(dy0). Q_n(dy_0 y)= ( _n(y_0)-\|y_0-y\|^2 _OT ) ( _n(z)-\|z-y\|^2 _OT ) P_Y A=a^aux(dz) P_Y A=a^aux(dy_0). Fix y1,y1′∈ℝdy_1,y_1 ^d and a 11-Lipschitz function f:ℝd→ℝf:R^d . For t∈[0,1]t∈[0,1], substituting y=(1−t)y1+ty1′y=(1-t)y_1+ty_1 into the preceding display gives ∫f(y0)Q^n(dy0∣(1−t)y1+ty1′)=∫f(y0)exp(φ^n(y0)−‖y0−((1−t)y1+ty1′)‖2εOT)ℙ^Y|A=aaux(dy0)∫exp(φ^n(z)−‖z−((1−t)y1+ty1′)‖2εOT)ℙ^Y|A=aaux(z). f(y_0) Q_n (dy_0 (1-t)y_1+ty_1 )= f(y_0) ( _n(y_0)-\|y_0-((1-t)y_1+ty_1 )\|^2 _OT ) P_Y A=a^aux(dy_0) ( _n(z)-\|z-((1-t)y_1+ty_1 )\|^2 _OT ) P_Y A=a^aux(dz). For every y0∈[−1,1]dy_0∈[-1,1]^d, dt[−‖y0−((1−t)y1+ty1′)‖2εOT]=2εOT⟨y0−((1−t)y1+ty1′),y1′−y1⟩. ddt [- \|y_0-((1-t)y_1+ty_1 )\|^2 _OT ]= 2 _OT y_0-((1-t)y_1+ty_1 ),y_1 -y_1 . Applying the quotient rule to the preceding ratio yields dt∫f(y0)Q^n(dy0∣(1−t)y1+ty1′) ddt f(y_0) Q_n (dy_0 (1-t)y_1+ty_1 ) =2εOT∫f(y0)⟨y0−((1−t)y1+ty1′),y1′−y1⟩Q^n(dy0∣(1−t)y1+ty1′) = 2 _OT f(y_0) y_0-((1-t)y_1+ty_1 ),y_1 -y_1 Q_n (dy_0 (1-t)y_1+ty_1 ) −2εOT∫f(y0)Q^n(dy0∣(1−t)y1+ty1′)∫⟨z−((1−t)y1+ty1′),y1′−y1⟩Q^n(dz∣(1−t)y1+ty1′) - 2 _OT f(y_0) Q_n (dy_0 (1-t)y_1+ty_1 ) z-((1-t)y_1+ty_1 ),y_1 -y_1 Q_n (dz (1-t)y_1+ty_1 ) =2εOTCovY0∼Q^n(⋅∣(1−t)y1+ty1′)(f(Y0),⟨Y0,y1′−y1⟩). = 2 _OTCov_Y_0 Q_n(· (1-t)y_1+ty_1 ) (f(Y_0), Y_0,y_1 -y_1 ). Assumption 2 implies that Q^n(⋅∣(1−t)y1+ty1′) Q_n(· (1-t)y_1+ty_1 ) is supported on [−1,1]d[-1,1]^d. Since f is 11-Lipschitz, supy0,z∈[−1,1]d|f(y0)−f(z)|≤2d,andsupy0,z∈[−1,1]d|⟨y0−z,y1′−y1⟩|≤2d‖y1′−y1‖. _y_0,z∈[-1,1]^d|f(y_0)-f(z)|≤ 2 d, _y_0,z∈[-1,1]^d | y_0-z,y_1 -y_1 |≤ 2 d\,\|y_1 -y_1\|. Therefore, Popoviciu’s variance inequality gives VarY0∼Q^n(⋅∣(1−t)y1+ty1′)(f(Y0)) _Y_0 Q_n(· (1-t)y_1+ty_1 ) (f(Y_0) ) ≤d, ≤ d, VarY0∼Q^n(⋅∣(1−t)y1+ty1′)(⟨Y0,y1′−y1⟩) _Y_0 Q_n(· (1-t)y_1+ty_1 ) ( Y_0,y_1 -y_1 ) ≤d‖y1′−y1‖2. ≤ d\|y_1 -y_1\|^2. By Cauchy–Schwarz, |CovY0∼Q^n(⋅∣(1−t)y1+ty1′)(f(Y0),⟨Y0,y1′−y1⟩)|≤d∥y1′−y1∥. |Cov_Y_0 Q_n(· (1-t)y_1+ty_1 ) (f(Y_0), Y_0,y_1 -y_1 ) |≤ d\|y_1 -y_1\|. Hence, for every t∈[0,1]t∈[0,1], |dt∫f(y0)Q^n(dy0∣(1−t)y1+ty1′)|≤2dεOT‖y1′−y1‖. | ddt f(y_0) Q_n (dy_0 (1-t)y_1+ty_1 ) |≤ 2d _OT\|y_1 -y_1\|. Integrating over t∈[0,1]t∈[0,1] yields |∫f(y0)Q^n(dy0∣y1′)−∫f(y0)Q^n(dy0∣y1)|≤2dεOT‖y1′−y1‖. | f(y_0) Q_n(dy_0 y_1 )- f(y_0) Q_n(dy_0 y_1) |≤ 2d _OT\|y_1 -y_1\|. Taking the supremum over all 11-Lipschitz f and applying Kantorovich–Rubinstein duality gives W1(Q^n(⋅∣y1),Q^n(⋅∣y1′))≤2dεOT∥y1−y1′∥. W_1 ( Q_n(· y_1), Q_n(· y_1 ) )≤ 2d _OT\|y_1-y_1 \|. ∎ Proposition 4 (Target-Query Discrepancy Under Empirical C). Assume Assumptions 2 and 5. Then W1(∫ℙY∣C=c,A=adℙC(c),1n∑i=n+12nℙ^Y∣C=Ci,A=a) W_1 ( _Y C=c,A=a\,dP_C(c), 1n _i=n+1^2n P_Y C=C_i,A=a ) ≤W1(∫ℙY∣C=c,A=adℙC(c),1n∑i=n+12nℙY∣C=Ci,A=a)+εpred,train. ≤ W_1 ( _Y C=c,A=a\,dP_C(c), 1n _i=n+1^2nP_Y C=C_i,A=a )+ _pred,train. Consequently, on the event of Lemma 12, W1(∫ℙY∣C=c,A=adℙC(c),1n∑i=n+12nℙ^Y∣C=Ci,A=a)≲n−1/(d+2)+εpred,train. W_1 ( _Y C=c,A=a\,dP_C(c), 1n _i=n+1^2n P_Y C=C_i,A=a ) n^-1/(d+2)+ _pred,train. Proof. The triangle inequality gives W1(∫ℙY∣C=c,A=adℙC(c),1n∑i=n+12nℙ^Y∣C=Ci,A=a) W_1 ( _Y C=c,A=a\,dP_C(c), 1n _i=n+1^2n P_Y C=C_i,A=a ) ≤W1(∫ℙY∣C=c,A=adℙC(c),1n∑i=n+12nℙY∣C=Ci,A=a) ≤ W_1 ( _Y C=c,A=a\,dP_C(c), 1n _i=n+1^2nP_Y C=C_i,A=a ) +W1(1n∑i=n+12nℙY∣C=Ci,A=a,1n∑i=n+12nℙ^Y∣C=Ci,A=a). +W_1 ( 1n _i=n+1^2nP_Y C=C_i,A=a, 1n _i=n+1^2n P_Y C=C_i,A=a ). For each i=n+1,…,2ni=n+1,…,2n, take an optimal coupling of ℙY∣C=Ci,A=aP_Y C=C_i,A=a and ℙ^Y∣C=Ci,A=a P_Y C=C_i,A=a, and mix these couplings uniformly over i. This gives W1(1n∑i=n+12nℙY∣C=Ci,A=a,1n∑i=n+12nℙ^Y∣C=Ci,A=a) W_1 ( 1n _i=n+1^2nP_Y C=C_i,A=a, 1n _i=n+1^2n P_Y C=C_i,A=a ) ≤1n∑i=n+12nW1(ℙY∣C=Ci,A=a,ℙ^Y∣C=Ci,A=a) ≤ 1n _i=n+1^2nW_1 (P_Y C=C_i,A=a, P_Y C=C_i,A=a ) ≤[1n∑i=n+12nW1(ℙY∣C=Ci,A=a,ℙ^Y∣C=Ci,A=a)2]1/2≤εpred,train. ≤ [ 1n _i=n+1^2nW_1 (P_Y C=C_i,A=a, P_Y C=C_i,A=a )^2 ]^1/2≤ _pred,train. Combining this bound with Lemma 12 proves the claim. ∎ Conditional on auxD_aux, trD_tr, and Y^i,rQ\ Y_i,r^Q\, the offline final-flow Monte Carlo array is drawn independently of the coupling reservoir: Y^i,1,…,Y^i,M∼ind.ℙ^Y∣C=c,A=aevaluated at c=Ci,i=1,…,n. Y_i,1,…, Y_i,M ind. P_Y C=c,A=a\ evaluated at c=C_i, i=1,…,n. (11) For each i and k, draw independently Y^0,i,k∼Q^n(⋅∣Y^i,k),Ki,k∼Unif0,…,N−1,Gi,k∼(0,Id), Y_0,i,k Q_n(· Y_i,k), K_i,k \0,…,N-1\, G_i,k N(0,I_d), (12) and set ti,k=tKi,kt_i,k=t_K_i,k. For every i with Ai=aA_i=a, draw independently Y^0,iobs∼Q^n(⋅∣Yi),Kiobs∼Unif0,…,N−1,Giobs∼(0,Id), Y_0,i^obs Q_n(· Y_i), K_i^obs \0,…,N-1\, G_i^obs N(0,I_d), (13) and set tiobs=tKiobst_i^obs=t_K_i^obs. These variables are drawn once before minimizing the final empirical objective and are held fixed during optimization. The M-sample empirical-process error for this fixed objective is separate from the MQM_Q-sample coupling-reservoir error above. Assumption 7 (Nonempty Treatment-a Auxiliary Sample). The auxiliary split contains at least one observation with treatment A=aA=a: ∑i=n+12nAi=a≥1. _i=n+1^2n1\A_i=a\≥ 1. Under the overlap condition ℙ(A=a∣C=c)≥πminP(A=a C=c)≥ _ , the event in this assumption has probability at least 1−(1−πmin)n≥1−e−nπmin.1-(1- _ )^n≥ 1-e^-n _ . Thus the assumption may equivalently be replaced by this additional exponentially small failure probability. Lemma 10 (Existence of an Ideal Coupling Q With the Correct Population Marginals). Under Assumptions 2 and 7, let λ⋆λ be an optimal coupling of μ1 _1 and μQMC _Q^MC, and write λ⋆(dy^1∣y1)λ (d y_1 y_1) for the conditional law of its second coordinate given the first. Define Q(dy0∣y1):=∫Q^n(dy0∣y^1)λ⋆(dy^1∣y1).Q(dy_0 y_1):= Q_n(dy_0 y_1)\,λ (d y_1 y_1). Then Q satisfies the marginal identity ∫Q(⋅∣y1)μ1(dy1) Q(· y_1)\, _1(dy_1) =ℙ^Y|A=aaux = P_Y A=a^aux and the coupling-replacement bound W1,ρ(Γμ1,Q,Γμ1,Q^n) W_1,ρ\! ( _ _1,Q, _ _1, Q_n ) ≤2dεOTW1(μ1,μQMC). ≤ 2d _OTW_1\! ( _1, _Q^MC ). Proof. Under the construction in the statement, draw (Y1,Y^1)∼λ⋆(Y_1, Y_1) λ , followed by Y0∼Q^n(⋅∣Y^1).Y_0 Q_n(· Y_1). The EOT marginal identity implies Y0∼∫Q^n(⋅∣y)μQMC(dy)=ℙ^Y|A=aaux.Y_0 Q_n(· y) _Q^MC(dy)= P_Y A=a^aux. Thus Q has the displayed marginal identity. Conditional on (Y1,Y^1)(Y_1, Y_1), couple Y~0∼Q^n(⋅∣Y1) Y_0 Q_n(· Y_1) and Y0∼Q^n(⋅∣Y^1)Y_0 Q_n(· Y_1) optimally. Then ((Y0,Y1),(Y~0,Y1)) ((Y_0,Y_1),( Y_0,Y_1) ) couples Γμ1,Q _ _1,Q and Γμ1,Q^n _ _1, Q_n, so W1,ρ(Γμ1,Q,Γμ1,Q^n) W_1,ρ( _ _1,Q, _ _1, Q_n) ≤‖Y0−Y~0‖ \|Y_0- Y_0\| =W1(Q^n(⋅∣Y^1),Q^n(⋅∣Y1)) =E\,W_1\! ( Q_n(· Y_1), Q_n(· Y_1) ) ≤2dεOT‖Y^1−Y1‖ ≤ 2d _OTE\| Y_1-Y_1\| =2dεOTW1(μ1,μQMC), = 2d _OTW_1\! ( _1, _Q^MC ), where the penultimate line uses Lemma 9. ∎ Remark 2. Under Assumption 2, the coupling constructed in Lemma 10 is supported on [−1,1]d×[−1,1]d[-1,1]^d×[-1,1]^d. Hence its conditional kernel Q(dy0∣y1)Q(dy_0 y_1) may be chosen so that Q([−1,1]d∣y1)=1,y1∈[−1,1]d.Q([-1,1]^d y_1)=1, y_1∈[-1,1]^d. Corollary 3 (Coupling-Replacement Error). Under Assumptions 2, 5, and 7, with probability at least 1−2n1- 2n, W1,ρ(Γμ1,Q,Γμ1,Q^n) W_1,ρ\! ( _ _1,Q, _ _1, Q_n ) ≲1εOT(n−1/(d+2)+εpred,train+(nMQ)−1/(d+2)), 1 _OT (n^-1/(d+2)+ _pred,train+(nM_Q)^-1/(d+2) ), where d is treated as fixed. Proof. Apply triangle inequality and Lemma 10, W1,ρ(Γμ1,Q,Γμ1,Q^n) W_1,ρ\! ( _ _1,Q, _ _1, Q_n ) ≤2dεOTW1(μ1,μQMC) ≤ 2d _OTW_1\! ( _1, _Q^MC ) ≤2dεOTW1(μ1,μ¯Q)+W1(μ¯Q,μQMC). ≤ 2d _OT \W_1\! ( _1, μ_Q )+W_1\! ( μ_Q, _Q^MC ) \. By Proposition 4, with probability at least 1−1n1- 1n, W1(μ1,μ¯Q) W_1\! ( _1, μ_Q ) ≲n−1/(d+2)+εpred,train. n^-1/(d+2)+ _pred,train. Taking δQ=1/n _Q=1/n in Lemma 11 gives, with probability at least 1−1n1- 1n, W1(μ¯Q,μQMC) W_1\! ( μ_Q, _Q^MC ) ≲(nMQ)−1/(d+2), (nM_Q)^-1/(d+2), For fixed d, substituting the preceding two bounds gives W1,ρ(Γμ1,Q,Γμ1,Q^n) W_1,ρ\! ( _ _1,Q, _ _1, Q_n ) ≲1εOT(n−1/(d+2)+εpred,train+(nMQ)−1/(d+2)). 1 _OT (n^-1/(d+2)+ _pred,train+(nM_Q)^-1/(d+2) ). A union bound over the two events gives probability at least 1−2n1- 2n. ∎ H.2 Finite-Sample Target Approximation In this subsection, we establish two high-probability W1W_1 approximation bounds used in the coupling analysis. Bounds of this type are standard in empirical Wasserstein theory; see, for example, Weed and Bach (2019). We provide a proof for completeness. Lemma 11 (Finite-MQM_Q Approximation of the Fitted Target Distribution). Assume Assumption 2. Conditional on auxD_aux, for every δQ∈(0,1) _Q∈(0,1), with probability at least 1−δQ1- _Q, W1(μ¯Q,μQMC)≤d[(2+2d/2)(nMQ)−1/(d+2)+2log(1/δQ)nMQ]. W_1\! ( μ_Q, _Q^MC )≤ d [ (2+2^d/2 )(nM_Q)^-1/(d+2)+ 2 (1/ _Q)nM_Q ]. In particular, for fixed d and δQ=1/n _Q=1/n, the right-hand side is Od((nMQ)−1/(d+2))O_d((nM_Q)^-1/(d+2)). Proof. Set m=nMQm=nM_Q and K=⌈m1/(d+2)⌉K= m^1/(d+2) . Partition [−1,1]d[-1,1]^d into KdK^d cubes A1,…,AKdA_1,…,A_K^d of side length 2/K2/K, and let aja_j be the center of AjA_j. If ζ♯ζ denotes the law obtained by moving the mass in each AjA_j to aja_j, then W1(ζ,ζ♯)≤dKW_1(ζ,ζ )≤ dK for every ζ supported on the cube. Define the conditional cell probabilities and their empirical counterparts by pj p_j :=μ¯Q(Aj)=1n∑i=n+12nℙ^Y∣C=Ci,A=a(Aj), := μ_Q(A_j)= 1n _i=n+1^2n P_Y C=C^i,A=a(A_j), p^j p_j :=μQMC(Aj)=1m∑i=n+12n∑r=1MQY^i,rQ∈Aj. := _Q^MC(A_j)= 1m _i=n+1^2n _r=1^M_Q1\ Y_i,r^Q∈ A_j\. The discretized measures have masses pjp_j and p^j p_j at aja_j. Since the cube has diameter 2d2 d, W1(μ¯Q♯,μQMC♯)≤2dTV(μ¯Q♯,μQMC♯)=d∑j=1Kd|pj−p^j|.W_1( μ_Q , _Q^MC )≤ 2 d\,TV( μ_Q , _Q^MC )= d _j=1^K^d|p_j- p_j|. Consequently, with F:=∑j=1Kd|pj−p^j|F:= _j=1^K^d|p_j- p_j|, W1(μ¯Q,μQMC)≤2dK+dF.W_1( μ_Q, _Q^MC)≤ 2 dK+ d\,F. (14) We next control F conditionally on auxD_aux. Replacing one of the m independent draws moves empirical mass 1/m1/m from at most one cell to at most one other cell. Hence the empirical histogram changes by at most 2/m2/m in ℓ1 _1, and therefore F changes by at most 2/m2/m. McDiarmid’s inequality gives ℙ(F−[F∣aux]≥t|aux)≤exp(−mt22).P\! (F-E[F _aux]≥ t\; |\;D_aux )≤ \! (- mt^22 ). Thus, with conditional probability at least 1−δQ1- _Q, F≤[F∣aux]+2log(1/δQ)m.F [F _aux]+ 2 (1/ _Q)m. (15) It remains to bound the conditional mean. Cauchy–Schwarz followed by Jensen gives [F∣aux] [F _aux] ≤Kd[∑j=1Kd[(p^j−pj)2∣aux]]1/2. ≤ K^d [ _j=1^K^dE\! [( p_j-p_j)^2 _aux ] ]^1/2. For pi,j:=ℙ^Y∣C=Ci,A=a(Aj)p_i,j:= P_Y C=C^i,A=a(A_j), conditional independence and unbiasedness imply ∑j=1Kd[(p^j−pj)2∣aux] _j=1^K^dE\! [( p_j-p_j)^2 _aux ] =1m2∑i=n+12n∑r=1MQ∑j=1Kdpi,j(1−pi,j) = 1m^2 _i=n+1^2n _r=1^M_Q _j=1^K^dp_i,j(1-p_i,j) ≤1m2∑i=n+12n∑r=1MQ∑j=1Kdpi,j=1m. ≤ 1m^2 _i=n+1^2n _r=1^M_Q _j=1^K^dp_i,j= 1m. Therefore [F∣aux]≤Kdm.E[F _aux]≤ K^dm. Combining this inequality with Equation (14) and Equation (15) yields W1(μ¯Q,μQMC)≤d[2K+Kdm+2log(1/δQ)m].W_1( μ_Q, _Q^MC)≤ d [ 2K+ K^dm+ 2 (1/ _Q)m ]. Finally, m≥1m≥ 1 implies 1K≤m−1/(d+2)andK≤2m1/(d+2), 1K≤ m^-1/(d+2) K≤ 2m^1/(d+2), so Kd/m≤2d/2m−1/(d+2) K^d/m≤ 2^d/2m^-1/(d+2). Substituting m=nMQm=nM_Q proves the stated bound. ∎ Lemma 12 (Empirical Covariate Standardization Error). Assume Assumption 2 and n≥2n≥ 2. Define μ1,n:=1n∑i=n+12nℙY∣C=Ci,A=a,Kn:=⌈n1/(d+2)⌉. _1,n:= 1n _i=n+1^2nP_Y C=C^i,A=a, K_n:= n^1/(d+2) . Then, with probability at least 1−1/n1-1/n over Cn+1,…,C2nC^n+1,…,C^2n, W1(μ1,μ1,n) W_1( _1, _1,n) ≤4dKn+2d(Kndn)1/2+2d(2logn)1/2. ≤ 4 dK_n+2 d ( K_n^dn )^1/2+2 d ( 2 nn )^1/2. Consequently, for fixed d, this error is Od(n−1/(d+2))O_d(n^-1/(d+2)). Proof. Partition [−1,1]d[-1,1]^d into KndK_n^d cubes A1,…,AKndA_1,…,A_K_n^d of side length 2/Kn2/K_n. Moving every point to the center of its cell changes any law supported on the cube by at most 2d/Kn2 d/K_n in W1W_1. For each cell, define qj(c) q_j(c) :=ℙ(Y∈Aj∣C=c,A=a), :=P(Y∈ A_j C=c,A=a), pj p_j :=C[qj(C)], :=E_C[q_j(C)], p^j p_j :=1n∑i=n+12nqj(Ci). := 1n _i=n+1^2nq_j(C^i). The discretizations of μ1 _1 and μ1,n _1,n put masses pjp_j and p^j p_j on the same cell centers. Therefore, with F:=∑j=1Knd|pj−p^j|,F:= _j=1^K_n^d|p_j- p_j|, the triangle inequality and the diameter bound give W1(μ1,μ1,n)≤4dKn+2dF.W_1( _1, _1,n)≤ 4 dK_n+2 d\,F. (16) Changing one covariate CiC^i changes the probability vector (q1(Ci),…,qKnd(Ci))(q_1(C^i),…,q_K_n^d(C^i)) to another probability vector. Its ℓ1 _1-change is at most 22, so F changes by at most 2/n2/n. McDiarmid’s inequality and δ=1/nδ=1/n yield, with probability at least 1−1/n1-1/n, F≤F+2logn.F + 2 nn. (17) For the expectation, Cauchy–Schwarz and Jensen imply F≤Knd[∑j=1KndVar(p^j)]1/2.EF≤ K_n^d [ _j=1^K_n^dVar( p_j) ]^1/2. Because the covariates are i.i.d., ∑j=1KndVar(p^j)=1n∑j=1KndVar(qj(C))≤1nC[∑j=1Kndqj(C)2]≤1n, _j=1^K_n^dVar( p_j)= 1n _j=1^K_n^dVar(q_j(C))≤ 1n\,E_C\! [ _j=1^K_n^dq_j(C)^2 ]≤ 1n, where the last inequality uses 0≤qj(C)≤10≤ q_j(C)≤ 1 and ∑jqj(C)=1 _jq_j(C)=1. Hence F≤Kndn.EF≤ K_n^dn. (18) Combining Equation (16), Equation (17), and Equation (18) proves the explicit bound. Finally, Kn≤2n1/(d+2)K_n≤ 2n^1/(d+2), and for fixed d≥1d≥ 1, logn/n=Od(n−1/(d+2)) n/n=O_d(n^-1/(d+2)). This proves the stated order. ∎ Appendix I Proof of Proposition 1 In this section, we work on ℰaE_a, conditioning on the auxiliary split, coupling reservoir, and fitted nuisance objects. We use the kernels Q,Q^nQ, Q_n and the offline sampling scheme from Equation (11), Equation (12) and Equation (13) in Appendix H. Population expectations below are taken over fresh observations or interpolation variables, with the fitted objects fixed. To align the training analysis with the discrete-time sampling analysis, we evaluate the loss on the same time grid. Let 0=t0<t1<⋯<tN=10=t_0<t_1<·s<t_N=1, with h=1/Nh=1/N, and let K∼Unif0,…,N−1K \0,…,N-1\. Let Θ be a parameter class with metric dΘd_ and finite proper covering numbers (r,Θ,dΘ)N(r, ,d_ ); every cover center belongs to Θ . The class and loss family are fixed under the preceding conditioning. For θ∈Θθ∈ , let Φθ(y0,y1,k,g) _θ(y_0,y_1,k,g) be a measurable nonnegative loss on [−1,1]d×[−1,1]d×0,…,N−1×ℝd[-1,1]^d×[-1,1]^d×\0,…,N-1\×R^d. With K∼Unif0,…,N−1K \0,…,N-1\ and G∼(0,Id)G (0,I_d) mutually independent and independent of the endpoints, define Hθ(y0,y1) H_θ(y_0,y_1) =K,G[Φθ(y0,y1,K,G)], =E_K,G[ _θ(y_0,y_1,K,G)], ℓQ^n,θ(y1) _ Q_n,θ(y_1) =∫Hθ(y0,y1)Q^n(dy0∣y1), = H_θ(y_0,y_1)\, Q_n(dy_0 y_1), La,QΦ(θ) L_a,Q (θ) =∫Hθ(y0,y1)Γμ1,Q(dy0,dy1). = H_θ(y_0,y_1)\, _ _1,Q(dy_0,dy_1). Define La,Q^nΦL_a, Q_n by replacing Q with Q^n Q_n. For O=(C,A,Y)O=(C,A,Y) and Oi=(Ci,Ai,Yi)O_i=(C_i,A_i,Y_i), set mθ(c) m_θ(c) =Y∼ℙY∣C=c,A=a[ℓQ^n,θ(Y)], =E_Y _Y C=c,A=a[ _ Q_n,θ(Y)], m^θ(c) m_θ(c) =Y^∼ℙ^Y∣C=c,A=a[ℓQ^n,θ(Y^)], =E_ Y P_Y C=c,A=a[ _ Q_n,θ( Y)], ℓ^DR,θ(O) _DR,θ(O) =A=aα^(C)ℓQ^n,θ(Y)−m^θ(C)+m^θ(C). =1\A=a\ α(C)\ _ Q_n,θ(Y)- m_θ(C)\+ m_θ(C). For the offline draws, write Φi,θobs _i,θ^obs =Φθ(Y^0,iobs,Yi,Kiobs,Giobs), = _θ( Y_0,i^obs,Y_i,K_i^obs,G_i^obs), Φ¯i,θaux _i,θ^aux =1M∑m=1MΦθ(Y^0,i,m,Y^i,m,Ki,m,Gi,m), = 1M _m=1^M _θ( Y_0,i,m, Y_i,m,K_i,m,G_i,m), with Φi,θobs=0 _i,θ^obs=0 for Ai≠aA_i≠ a. The implemented empirical loss is L^actualΦ(θ) L_actual (θ) =1n∑i=1n[Ai=aα^(Ci)Φi,θobs−Φ¯i,θaux+Φ¯i,θaux]. = 1n _i=1^n [1\A_i=a\ α(C_i)\ _i,θ^obs- _i,θ^aux\+ _i,θ^aux ]. (19) The following lemma shows concretely how a covering number enters the concentration argument: we first approximate every function in the class by one of J cover elements, apply Hoeffding’s inequality and a union bound to these J representatives, and then incur at most an additional 2r2r error when passing back to the full class. Lemma 13 (A finite-cover empirical-process bound). Conditional on a sigma-field ℱF, let W1,…,WmW_1,…,W_m be independent. Let Z be a separable class of measurable functions taking values in a common interval of length Benv>0B_env>0, with a proper uniform r-cover z1,…,zJ\z_1,…,z_J\. Conditional on ℱF, the function class Z and the chosen cover z1,…,zJ\z_1,…,z_J\ are treated as fixed. For 0<δ<10<δ<1, with conditional probability at least 1−δ1-δ, supz∈|1m∑i=1mz(Wi)−[z(Wi)∣ℱ]|≤2r+Benvlog(2J/δ)2m. _z | 1m _i=1^m\z(W_i)-E[z(W_i) ]\ |≤ 2r+B_env (2J/δ)2m. Proof. For each zjz_j, conditional Hoeffding’s inequality gives ℙ(|1m∑i=1mzj(Wi)−[zj(Wi)∣ℱ]|>u∣ℱ)≤2exp(−2mu2Benv2). ( | 1m _i=1^m\z_j(W_i)-E[z_j(W_i) ]\ |>u )≤ 2 (- 2mu^2B_env^2 ). Take u=Benvlog(2J/δ)/(2m)u=B_env (2J/δ)/(2m) and apply a union bound over the cover. Replacing a function by its cover center changes its empirical average and its conditional expected average by at most r each, giving the additional 2r2r. ∎ We now turn minimization of the empirical doubly robust loss into a high-probability bound on the population excess risk, in the following proposition. The error separates into three sources: replacing the ideal coupling Q by the fitted coupling Q^n Q_n, estimation error in the nuisance functions, and finite-sample error from the observed data and offline Monte Carlo draws. The last part is controlled through the covering-number concentration argument above. Proposition 5 (Doubly Robust ERM Bound). Under Assumptions 2, 3, and 4, let the loss satisfy the following bounds for constants BΦ,LY,LΘ>0B_ ,L_Y,L_ >0 and all admissible arguments: 0≤Φθ(y0,y1,k,g) 0≤ _θ(y_0,y_1,k,g) ≤BΦ, ≤ B_ , (20) |Hθ(y0,y1)−Hθ(y0′,y1′)| |H_θ(y_0,y_1)-H_θ(y_0 ,y_1 )| ≤LY(‖y0−y0′‖+‖y1−y1′‖), ≤ L_Y(\|y_0-y_0 \|+\|y_1-y_1 \|), (21) |Φθ(y0,y1,k,g)−Φθ′(y0,y1,k,g)| | _θ(y_0,y_1,k,g)- _θ (y_0,y_1,k,g)| ≤LΘdΘ(θ,θ′). ≤ L_ \,d_ (θ,θ ). (22) Define NDRΦ=(BΦ(1+2πmin−1)LΘ(1+4πmin−1)n,Θ,dΘ),NobsΦ=(BΦLΘn,Θ,dΘ),NauxΦ=(BΦLΘnM,Θ,dΘ). N_DR =N ( B_ (1+2 _ ^-1)L_ (1+4 _ ^-1) n, ,d_ ),N_obs =N ( B_ L_ n, ,d_ ),N_aux =N ( B_ L_ nM, ,d_ ). For 0<δ<10<δ<1, set ℜΦ(δ) R_ (δ) =4BΦ(1+2πmin−1)[1n+2log(6NDRΦ/δ)n] =4B_ (1+2 _ ^-1) [ 1 n+ 2 (6N_DR /δ)n ] +8BΦπmin−1[1n+2log(6NobsΦ/δ)n] +8B_ _ ^-1 [ 1 n+ 2 (6N_obs /δ)n ] +4BΦ(1+2πmin−1)[1nM+2log(6NauxΦ/δ)nM]. +4B_ (1+2 _ ^-1) [ 1 nM+ 2 (6N_aux /δ)nM ]. (23) Write ΔQ=W1,ρ(Γμ1,Q,Γμ1,Q^n) _Q=W_1,ρ( _ _1,Q, _ _1, Q_n). Let θ^M θ_M be an exact minimizer of L^actualΦ L_actual over Θ . With conditional probability at least 1−δ1-δ over the training observations and offline Monte Carlo draws, La,QΦ(θ^M)−infθ∈ΘLa,QΦ(θ) L_a,Q ( θ_M)- _θ∈ L_a,Q (θ) ≤2LYΔQ+2LY(1+2dεOT)εαεpred+ℜΦ(δ). ≤ 2L_Y _Q+2L_Y (1+ 2d _OT ) _α _pred+ R_ (δ). (24) Proof. Empirical optimality gives La,QΦ(θ^M)−infθ∈ΘLa,QΦ(θ)≤2supθ∈Θ|La,QΦ(θ)−L^actualΦ(θ)|. L_a,Q ( θ_M)- _θ∈ L_a,Q (θ)≤ 2 _θ∈ |L_a,Q (θ)- L_actual (θ)|. Inserting the fitted-kernel population loss, the population DR loss, and its empirical average gives La,QΦ(θ^M)−infθ∈ΘLa,QΦ(θ) L_a,Q ( θ_M)- _θ∈ L_a,Q (θ) ≤2supθ∈Θ|La,QΦ(θ)−La,Q^nΦ(θ)| ≤ 2 _θ∈ |L_a,Q (θ)-L_a, Q_n (θ)| +2supθ∈Θ|La,Q^nΦ(θ)−[ℓ^DR,θ(O)]| +2 _θ∈ |L_a, Q_n (θ)-E[ _DR,θ(O)]| +2supθ∈Θ|[ℓ^DR,θ(O)]−1n∑i=1nℓ^DR,θ(Oi)| +2 _θ∈ |E[ _DR,θ(O)]- 1n _i=1^n _DR,θ(O_i) | +2supθ∈Θ|1n∑i=1nℓ^DR,θ(Oi)−L^actualΦ(θ)|. +2 _θ∈ | 1n _i=1^n _DR,θ(O_i)- L_actual (θ) |. (25) Coupling replacement and nuisance error. By Equation (21) and Kantorovich–Rubinstein duality, supθ∈Θ|La,QΦ(θ)−La,Q^nΦ(θ)|≤LYΔQ. _θ∈ |L_a,Q (θ)-L_a, Q_n (θ)|≤ L_Y _Q. Conditioning on C and using ℙ(A=a∣C)=1/αa(C)P(A=a C)=1/ _a(C) gives [ℓ^DR,θ(O)∣C]=α^(C)αa(C)mθ(C)−m^θ(C)+m^θ(C). [ _DR,θ(O) C]= α(C) _a(C)\m_θ(C)- m_θ(C)\+ m_θ(C). Since La,Q^nΦ(θ)=C[mθ(C)]L_a, Q_n (θ)=E_C[m_θ(C)], La,Q^nΦ(θ)−[ℓ^DR,θ(O)]=C[(1−α^(C)αa(C))mθ(C)−m^θ(C)]. L_a, Q_n (θ)-E[ _DR,θ(O)]=E_C [ (1- α(C) _a(C) )\m_θ(C)- m_θ(C)\ ]. (26) Coupling Q^n(⋅∣y) Q_n(· y) and Q^n(⋅∣y′) Q_n(· y ) optimally, Equation (21) and Lemma 9 yield |ℓQ^n,θ(y)−ℓQ^n,θ(y′)| | _ Q_n,θ(y)- _ Q_n,θ(y )| ≤LYW1(Q^n(⋅∣y),Q^n(⋅∣y′))+∥y−y′∥ ≤ L_Y\W_1( Q_n(· y), Q_n(· y ))+\|y-y \|\ ≤LY(1+2dεOT)‖y−y′‖. ≤ L_Y (1+ 2d _OT )\|y-y \|. Kantorovich–Rubinstein duality and Assumption 4 therefore give supθ∈Θ‖mθ−m^θ‖L2(ℙC)≤LY(1+2dεOT)εpred. _θ∈ \|m_θ- m_θ\|_L^2(P_C)≤ L_Y (1+ 2d _OT ) _pred. Cauchy–Schwarz in Equation (26), followed by Assumption 3, bounds the second term of Equation (25) by 2LY(1+2d/εOT)εαεpred2L_Y(1+2d/ _OT) _α _pred. Generalization. The training observations remain independent under the initial conditioning. Integration of Equation (20) and Equation (22) gives |ℓ^DR,θ(O)| | _DR,θ(O)| ≤BΦ(1+2πmin−1), ≤ B_ (1+2 _ ^-1), |ℓ^DR,θ(O)−ℓ^DR,θ′(O)| | _DR,θ(O)- _DR,θ (O)| ≤LΘ(1+4πmin−1)dΘ(θ,θ′). ≤ L_ (1+4 _ ^-1)d_ (θ,θ ). Thus the parameter cover defining NDRΦN_DR induces a proper uniform cover of the DR loss class with radius BΦ(1+2πmin−1)/nB_ (1+2 _ ^-1)/ n. Lemma 13, with m=nm=n, interval length 2BΦ(1+2πmin−1)2B_ (1+2 _ ^-1), and failure probability δ/3δ/3, bounds the third term of Equation (25) by the first term of Equation (23). Monte Carlo approximation. Condition additionally on O1,…,OnO_1,…,O_n. All training observations and fitted objects are now fixed. The observed tuples are independent across i, and the auxiliary tuples are independent across (i,m)(i,m), as specified in Appendix H. Dependence within each tuple, including the conditional draw from Q^n Q_n, is retained. Under this additional conditioning, [Ai=aα^(Ci)Φi,θobs∣O1,…,On] [1\A_i=a\ α(C_i) _i,θ^obs O_1,…,O_n] =Ai=aα^(Ci)ℓQ^n,θ(Yi), =1\A_i=a\ α(C_i) _ Q_n,θ(Y_i), [Φθ(Y^0,i,m,Y^i,m,Ki,m,Gi,m)∣O1,…,On] [ _θ( Y_0,i,m, Y_i,m,K_i,m,G_i,m) O_1,…,O_n] =m^θ(Ci). = m_θ(C_i). Expanding Equation (19) gives 1n∑i=1nℓ^DR,θ(Oi)−L^actualΦ(θ) 1n _i=1^n _DR,θ(O_i)- L_actual (θ) =1n∑i=1nAi=aα^(Ci)ℓQ^n,θ(Yi)−Φi,θobs = 1n _i=1^n1\A_i=a\ α(C_i)\ _ Q_n,θ(Y_i)- _i,θ^obs\ +1n∑i=1n[1−Ai=aα^(Ci)]m^θ(Ci)−Φ¯i,θaux. + 1n _i=1^n[1-1\A_i=a\ α(C_i)]\ m_θ(C_i)- _i,θ^aux\. Apply Lemma 13 to these two averages, including i as a deterministic argument of each integrand. For the observed term, the weighted losses lie in [0,2BΦπmin−1][0,2B_ _ ^-1] and are 2πmin−1LΘ2 _ ^-1L_ -Lipschitz in θ. The cover defining NobsΦN_obs therefore has induced radius 2BΦπmin−1/n2B_ _ ^-1/ n. With n draws and failure probability δ/3δ/3, twice this discrepancy is bounded by the second term of Equation (23). For the auxiliary term, apply the lemma to the nMnM individual weighted losses, not to their n within-observation averages. Their absolute values are at most BΦ(1+2πmin−1)B_ (1+2 _ ^-1), and their parameter Lipschitz constant is (1+2πmin−1)LΘ(1+2 _ ^-1)L_ . The cover defining NauxΦN_aux has induced radius BΦ(1+2πmin−1)/nMB_ (1+2 _ ^-1)/ nM. With interval length 2BΦ(1+2πmin−1)2B_ (1+2 _ ^-1) and failure probability δ/3δ/3, twice this discrepancy is bounded by the third term of Equation (23). The loss classes used above are separable because they are Lipschitz images of the totally bounded class Θ . Each Monte Carlo failure probability is at most δ/3δ/3 conditional on the training observations and the initially fixed objects. Taking conditional expectations removes the additional conditioning. A union bound with the generalization event therefore gives total conditional failure probability at most δ. Substituting all four bounds into Equation (25) proves Equation (24). ∎ We remark that Proposition 5 applies to a generic loss class. Its conditions will be verified separately for the velocity and score fields below. I.1 Velocity-Field Bound We now specialize the generic doubly robust framework above to velocity-field estimation. Define the squared velocity loss Φvvel(y0,y1,k,g)=‖v(y0+tk(y1−y0)+σg,tk)−(y1−y0)‖2. _v^vel(y_0,y_1,k,g)= \|v\! (y_0+t_k(y_1-y_0)+σ g,t_k )-(y_1-y_0) \|^2. Each parameter θ induces a velocity field vθv_θ. To simplify notation in this specialization, we write v=vθv=v_θ, suppress the abstract index θ, and use the velocity field itself to index the loss, writing Φvvel _v^vel in place of the generic Φθ _θ. The generic population risks La,QΦL_a,Q and La,Q^nΦL_a, Q_n then become the velocity losses below; we write them as La,Q(v)L_a,Q(v) and La,Q^n(v)L_a, Q_n(v) for brevity. Then the previous abstract loss becomes ℓQ(v,y1) _Q(v;y_1) =K,Y0∼Q(⋅∣y1),G[‖v(Y0+tK(y1−Y0)+σG,tK)−(y1−Y0)‖2], =E_K,Y_0 Q(· y_1),G\! [ \|v\! (Y_0+t_K(y_1-Y_0)+σ G,t_K )-(y_1-Y_0) \|^2 ], La,Q(v) L_a,Q(v) =Y1∼∫ℙY∣C=c,A=adℙC(c)[ℓQ(v;Y1)], =E_Y_1 _Y C=c,A=a\,dP_C(c)\! [ _Q(v;Y_1) ], ℓQ^n(v,y1) _ Q_n(v;y_1) =K,Y0∼Q^n(⋅∣y1),G[‖v(Y0+tK(y1−Y0)+σG,tK)−(y1−Y0)‖2], =E_K,Y_0 Q_n(· y_1),G\! [ \|v\! (Y_0+t_K(y_1-Y_0)+σ G,t_K )-(y_1-Y_0) \|^2 ], La,Q^n(v) L_a, Q_n(v) =Y1∼∫ℙY∣C=c,A=adℙC(c)[ℓQ^n(v;Y1)]. =E_Y_1 _Y C=c,A=a\,dP_C(c)\! [ _ Q_n(v;Y_1) ]. Since vQ⋆(XtK,tK)=[D∣XtK,K]v_Q (X_t_K,t_K)=E[D X_t_K,K], the residual D−vQ⋆(XtK,tK)D-v_Q (X_t_K,t_K) is orthogonal in L2L^2 to every measurable function of (XtK,K)(X_t_K,K). Expanding the two squared losses gives La,Q(v)−La,Q(vQ⋆) L_a,Q(v)-L_a,Q(v_Q ) =[‖v(XtK,tK)−vQ⋆(XtK,tK)‖2] =E\! [\|v(X_t_K,t_K)-v_Q (X_t_K,t_K)\|^2 ] =1N∑k=0N−1[‖v(Xtk,tk)−vQ⋆(Xtk,tk)‖2]. = 1N _k=0^N-1E\! [\|v(X_t_k,t_k)-v_Q (X_t_k,t_k)\|^2 ]. (27) The right-hand side corresponds to the quantity used in the sampling analysis. I.1.1 Velocity Network Class and Approximation Throughout this subsection, Q denotes the ideal coupling kernel, and we write vQ⋆v_Q for the population velocity field associated with this coupling. We approximate the velocity field by the following scalar neural network. For integers L≥2L≥ 2 and p,J≥1p,J≥ 1, and constants K,κ,γ>0K,κ,γ>0, define the scalar sparse ReLU class scal(q,L,p,J,K,κ,γ) N_scal(q,L,p,J,K,κ,γ) =h:[0,1]q→ℝ:∃p0,…,pL∈ℕ,p0=q,pL=1,max1≤ℓ≤L−1pℓ≤p, = \h:[0,1]^q :∃\,p_0,…,p_L ,\;p_0=q,\;p_L=1,\; _1≤ ≤ L-1p_ ≤ p, ∃Wℓ∈ℝpℓ×pℓ−1,bℓ∈ℝpℓ,ℓ=1,…,L, such that h0(z)=z, ∃\,W_ ^p_ × p_ -1,\;b_ ^p_ ,\; =1,…,L, such that h_0(z)=z, hℓ(z)=ReLU(Wℓhℓ−1(z)+bℓ),ℓ=1,…,L−1, h_ (z)=ReLU (W_ h_ -1(z)+b_ ), =1,…,L-1, h(z)=WLhL−1(z)+bL, h(z)=W_Lh_L-1(z)+b_L, ∑ℓ=1L(‖Wℓ‖0+‖bℓ‖0)≤J,max1≤ℓ≤L(‖Wℓ‖max∨‖bℓ‖max)≤κ,supz∈[0,1]q|h(z)|≤K, _ =1^L (\|W_ \|_0+\|b_ \|_0 )≤ J, _1≤ ≤ L (\|W_ \|_ \|b_ \|_ )≤κ, _z∈[0,1]^q|h(z)|≤ K, |h(z)−h(z′)|≤γ∥z−z′∥∞ for every z,z′∈[0,1]q. |h(z)-h(z )|≤γ\|z-z \|_∞ for every z,z ∈[0,1]^q \. Thus L counts all affine layers, including the output layer. Here ‖Wℓ‖0\|W_ \|_0 and ‖bℓ‖0\|b_ \|_0 denote the number of nonzero entries, and ‖Wℓ‖max\|W_ \|_ and ‖bℓ‖max\|b_ \|_ denote the largest absolute value of their entries. For u,ω∈ℝu,ω , define clipω(u) _ω(u) =−ω+ReLU(u+ω)−ReLU(u−ω). =-ω+ReLU(u+ω)-ReLU(u-ω). For x=(x1,…,xd)x=(x_1,…,x_d), define clipR(x)=(clipR(x1),…,clipR(xd)),SR(x,t)=(clipR(x1)+R2R,…,clipR(xd)+R2R,t). _R(x)= (clip_R(x_1),…,clip_R(x_d) ), S_R(x,t)= ( clip_R(x_1)+R2R,…, clip_R(x_d)+R2R,t ). We first define some constants that will be used to construct a network class. Let R>1R>1, and define Λ=16d(2R+1)σ2 = 16d(2R+1)σ^2 and TR(z)=(−R+2Rz1,…,−R+2Rzd,zd+1),z∈[0,1]d+1. T_R(z)= (-R+2Rz_1,…,-R+2Rz_d,z_d+1 ), z∈[0,1]^d+1. Lemma 14 shows that vQ⋆∘TRv_Q T_R is Λ -Lipschitz jointly in the rescaled spatial and time coordinates. Lemma 14 (Lipschitz Regularity of the True Vector Field). Assume Assumption 2. Then vQ⋆v_Q admits a version such that, for every i=1,…,di=1,…,d, the function gi(z) g_i(z) =vQ,i⋆(TR(z)),z∈[0,1]d+1, =v_Q,i (T_R(z)), z∈[0,1]^d+1, satisfies |gi(z)−gi(z′)| |g_i(z)-g_i(z )| ≤Λ‖z−z′‖∞,z,z′∈[0,1]d+1. ≤ \|z-z \|_∞, z,z ∈[0,1]^d+1. Proof of Lemma 14. All expectations and conditional expectations below are taken under the joint law induced by Q. By Lemma 4, for every i,j∈1,…,di,j∈\1,…,d\, ∂xjvQ,i⋆(x,t) _x_jv_Q,i (x,t) =−1σCov(Di,Gj∣Xt=x), =- 1σCov (D_i,G_j X_t=x ), and ∂tvQ,i⋆(x,t) _tv_Q,i (x,t) =1σCov(Di,D⊤G∣Xt=x). = 1σCov (D_i,D G X_t=x ). Under the conditional law given Xt=xX_t=x, we have G G =x−Stσ. = x-S_tσ. Since x is fixed under this conditional law, ∂xjvQ,i⋆(x,t) _x_jv_Q,i (x,t) =1σ2Cov(Di,St,j∣Xt=x), = 1σ^2Cov (D_i,S_t,j X_t=x ), and ∂tvQ,i⋆(x,t) _tv_Q,i (x,t) =1σ2Cov(Di,D⊤(x−St)∣Xt=x). = 1σ^2Cov (D_i,D (x-S_t) X_t=x ). Since Y0,Y1∈[−1,1]dY_0,Y_1∈[-1,1]^d, we have Di D_i ∈[−2,2], ∈[-2,2], St,j S_t,j ∈[−1,1]. ∈[-1,1]. Hence, conditional on Xt=xX_t=x, |Di−[Di∣Xt=x]| |D_i-E[D_i X_t=x] | ≤4, ≤ 4, |St,j−[St,j∣Xt=x]| |S_t,j-E[S_t,j X_t=x] | ≤2. ≤ 2. Therefore, |∂xjvQ,i⋆(x,t)| | _x_jv_Q,i (x,t) | ≤8σ2. ≤ 8σ^2. For the time derivative, if x∈[−R,R]dx∈[-R,R]^d, then ‖D‖≤2d\|D\|≤ 2 d and ‖x−St‖≤(R+1)d\|x-S_t\|≤(R+1) d. Thus |D⊤(x−St)|≤2d(R+1), |D (x-S_t) |≤ 2d(R+1), and so, conditional on Xt=xX_t=x, |D⊤(x−St)−[D⊤(x−St)∣Xt=x]| |D (x-S_t)-E[D (x-S_t) X_t=x] | ≤4d(R+1). ≤ 4d(R+1). Together with |Di−[Di∣Xt=x]|≤4 |D_i-E[D_i X_t=x] |≤ 4, this gives |∂tvQ,i⋆(x,t)| | _tv_Q,i (x,t) | ≤16d(R+1)σ2. ≤ 16d(R+1)σ^2. For gi(z)=vQ,i⋆(TR(z))g_i(z)=v_Q,i (T_R(z)), the preceding derivative bounds imply |∂zjgi(z)| | _z_jg_i(z) | ≤16Rσ2,j=1,…,d, ≤ 16Rσ^2, j=1,…,d, |∂zd+1gi(z)| | _z_d+1g_i(z) | ≤16d(R+1)σ2. ≤ 16d(R+1)σ^2. Therefore, for every z,z′∈[0,1]d+1z,z ∈[0,1]^d+1, |gi(z)−gi(z′)| |g_i(z)-g_i(z )| ≤[16Rdσ2+16d(R+1)σ2]‖z−z′‖∞=Λ‖z−z′‖∞. ≤ [ 16Rdσ^2+ 16d(R+1)σ^2 ]\|z-z \|_∞= \|z-z \|_∞. ∎ Based on the scalar neural network defined above, we use it to construct a neural network class that we use to approximate our vector field. Let ε>0 >0, recall Λ=16d(2R+1)σ2 = 16d(2R+1)σ^2. Define η=εdΛη= d\, as an auxiliary variable that control the approximation accuracy of each scalar network. Proposition 6 (Approximation Error Under the Noisy Interpolation Law). Assume Assumption 2. Let R>1R>1, q=d+1q=d+1 and ε>0 >0 satisfy 0<η=ε/(dΛ)<10<η= /( d\, )<1 and 32d2exp(−(R−1)22σ2) 32d^2 (- (R-1)^22σ^2 ) ≤ε2. ≤ ^2. Choose the width and parameter bounds so that pLip(q,η)≥qp_Lip(q,η)≥ q and κLip(q,η)≥2 _Lip(q,η)≥ 2. The scalar-network architecture can then be chosen with KLip(q,η) K_Lip(q,η) =q,LLip(q,η)=O(log1η+q),pLip(q,η)=O(q+η−q), = q, L_Lip(q,η)=O ( 1η+q ), p_Lip(q,η)=O (q+η^-q ), JLip(q,η) J_Lip(q,η) =O(η−q[log1η+q]),κLip(q,η)=O(1). =O (η^-q [ 1η+q ] ), _Lip(q,η)=O(1). These are the scalar approximation parameters from (Chen et al., 2020, Lemma 10), with LLipL_Lip denoting the total number of affine layers. Enlarging the width and parameter bounds to enforce pLip≥qp_Lip≥ q and κLip≥2 _Lip≥ 2 does not alter these orders for fixed q. We construct a vector valued neural network coordinate-wisely by viapp(x,t) v_i^app(x,t) =clip2(Λh^i(SR(x,t))+gi(0)),i=1,…,d. =clip_2 ( h_i(S_R(x,t))+g_i(0) ), i=1,…,d. where h^i∈scal(q,LLip(q,η),pLip(q,η),JLip(q,η),KLip(q,η),κLip(q,η),10q) h_i∈ N_scal (q,L_Lip(q,η),p_Lip(q,η),J_Lip(q,η),K_Lip(q,η), _Lip(q,η),10q ) and gi(0)g_i(0) are constants that will be specified in the proof. Then we have 1N∑k=0N−1[‖vapp(Xtk,tk)−vQ⋆(Xtk,tk)‖2] 1N _k=0^N-1E [ \|v^app(X_t_k,t_k)-v_Q (X_t_k,t_k) \|^2 ] ≤2ε2. ≤ 2 ^2. Proof. By Lemma 14, for each i=1,…,di=1,…,d, the function gi(z)=vQ,i⋆(TR(z)),z∈[0,1]d+1g_i(z)=v_Q,i (T_R(z)),z∈[0,1]^d+1 is Λ -Lipschitz with respect to ∥⋅∥∞\|·\|_∞. Define fi(z)=gi(z)−gi(0)Λ. f_i(z)= g_i(z)-g_i(0) . Then fi(0)=0f_i(0)=0 and fif_i is 11-Lipschitz with respect to ∥⋅∥∞\|·\|_∞. Since ‖z−z′‖∞≤‖z−z′‖2\|z-z \|_∞≤\|z-z \|_2, it is also 11-Lipschitz with respect to the Euclidean norm required by (Chen et al., 2020, Lemma 10). Consequently, that lemma provides h^i h_i ∈scal(q,LLip(q,η),pLip(q,η),JLip(q,η),KLip(q,η),κLip(q,η),10q) ∈ N_scal (q,L_Lip(q,η),p_Lip(q,η),J_Lip(q,η),K_Lip(q,η), _Lip(q,η),10q ) such that supz∈[0,1]d+1|h^i(z)−fi(z)| _z∈[0,1]^d+1 | h_i(z)-f_i(z) | ≤η. ≤η. Now recall we constructed viapp(x,t) v_i^app(x,t) =clip2(Λh^i(SR(x,t))+gi(0)),i=1,…,d. =clip_2 ( h_i(S_R(x,t))+g_i(0) ), i=1,…,d. If x∈[−R,R]dx∈[-R,R]^d, then TR(SR(x,t))=(x,t)T_R(S_R(x,t))=(x,t). Since η=ε/(dΛ)η= /( d\, ), |Λh^i(SR(x,t))+gi(0)−vQ,i⋆(x,t)| | h_i(S_R(x,t))+g_i(0)-v_Q,i (x,t) | ≤εd. ≤ d. Because vQ,i⋆(x,t)∈[−2,2]v_Q,i (x,t)∈[-2,2] and clip2clip_2 is the projection onto [−2,2][-2,2], clipping does not increase the error. Therefore our constructed vector field approximation satisfies supx∈[−R,R]d,t∈[0,1]‖vapp(x,t)−vQ⋆(x,t)‖2 _x∈[-R,R]^d,\;t∈[0,1] \|v^app(x,t)-v_Q (x,t) \|^2 ≤ε2. ≤ ^2. It remains to control the event Xt∉[−R,R]dX_t∉[-R,R]^d. Since St∈[−1,1]dS_t∈[-1,1]^d, for every t∈[0,1]t∈[0,1], ℙ(Xt∉[−R,R]d) (X_t∉[-R,R]^d ) ≤2dexp(−(R−1)22σ2). ≤ 2d (- (R-1)^22σ^2 ). On this event, ‖vapp(Xt,t)−vQ⋆(Xt,t)‖2≤16d \|v^app(X_t,t)-v_Q (X_t,t) \|^2≤ 16d. Thus, for every t∈[0,1]t∈[0,1], [‖vapp(Xt,t)−vQ⋆(Xt,t)‖2] [ \|v^app(X_t,t)-v_Q (X_t,t) \|^2 ] ≤ε2+32d2exp(−(R−1)22σ2)≤2ε2. ≤ ^2+32d^2 (- (R-1)^22σ^2 )≤ 2 ^2. The bound holds at every grid time tkt_k; averaging over k=0,…,N−1k=0,…,N-1 proves the claim. ∎ The preceding proposition motivates the following coordinatewise and vector-valued network classes: ℱcoord(R,Λ,L,p,J,K,κ,γ) _coord (R, ,L,p,J,K,κ,γ ) =f:ℝd×[0,1]→ℝ:f(x,t)=clip2(Λh(SR(x,t))+b), = \f:R^d×[0,1] :f(x,t)=clip_2 ( h(S_R(x,t))+b ), h∈scal(d+1,L,p,J,K,κ,γ),|b|≤2 h∈ N_scal (d+1,L,p,J,K,κ,γ ), |b|≤ 2 \ and (R,Λ,L,p,J,K,κ,γ) (R, ,L,p,J,K,κ,γ ) =v:ℝd×[0,1]→ℝd:v(x,t)=(v1(x,t),…,vd(x,t)), = \v:R^d×[0,1] ^d:v(x,t)= (v_1(x,t),…,v_d(x,t) ), vi∈ℱcoord(R,Λ,L,p,J,K,κ,γ),i=1,…,d. v_i _coord (R, ,L,p,J,K,κ,γ ), i=1,…,d \. Concretely, we instantiate the class V as n _n =v:ℝd×[0,1]→ℝd:v(x,t)=(v1(x,t),…,vd(x,t)), = \v:R^d×[0,1] ^d:v(x,t)= (v_1(x,t),…,v_d(x,t) ), vi(x,t)=clip2(Λhi(SR(x,t))+bi),i=1,…,d, v_i(x,t)=clip_2 ( h_i(S_R(x,t))+b_i ), i=1,…,d, hi∈scal(q,LLip(q,η),pLip(q,η),JLip(q,η),KLip(q,η),κLip(q,η),10q),i=1,…,d, h_i∈ N_scal (q,L_Lip(q,η),p_Lip(q,η),J_Lip(q,η),K_Lip(q,η), _Lip(q,η),10q ), i=1,…,d, |bi|≤2,i=1,…,d. |b_i|≤ 2, i=1,…,d \. Then Proposition 6 implies infv∈n1N∑k=0N−1[‖v(Xtk,tk)−vQ⋆(Xtk,tk)‖2] _v _n 1N _k=0^N-1E [ \|v(X_t_k,t_k)-v_Q (X_t_k,t_k) \|^2 ] ≤2ε2. ≤ 2 ^2. We transfer a cover of the scalar subnetworks directly to the clipped vector class. Lemma 15 (Direct Covering-Number Transfer for Clipped Vector Networks). Let ℋ=scal(q,L,p,J,K,κ,γ)H= N_scal(q,L,p,J,K,κ,γ), where L is the total number of affine layers, p≥qp≥ q, and κ>1κ>1. Let =v:vi(x,t)=clipω(Λhi(SR(x,t))+bi),hi∈ℋ,|bi|≤ω,i=1,…,d.V= \v:\ v_i(x,t)=clip_ω ( h_i(S_R(x,t))+b_i ),\ h_i ,\ |b_i|≤ω,\ i=1,…,d \. Then, for every δcov>0 _cov>0, log(δcov,,∥⋅∥∞,[−R,R]d×[0,1]) \! ( _cov,V,\|·\|_∞,[-R,R]^d×[0,1] ) ≤dJlog(1+8L2(p+2)κLpL+1Λdδcov) ≤ dJ \! (1+ 8L^2(p+2)κ^Lp^L+1 d _cov ) +dlog(1+4ωdδcov). +d \! (1+ 4ω d _cov ). (28) Proof. Set ηh=δcov2Λd,ηb=δcov2d. _h= _cov2 d, _b= _cov2 d. After zero-padding hidden layers when necessary, ℋH is a subclass of the sparse-ReLU class in (Chen et al., 2022, Lemma 5.3) with total depth L, maximum layer width p, input bound B=1B=1, sparsity J, and parameter bound κ>1κ>1. The cited result therefore supplies an ηh/2 _h/2-cover, not necessarily with centers in ℋH, whose log-cardinality is at most Jlog(1+8L2(p+2)κLpL+1Λdδcov).J \! (1+ 8L^2(p+2)κ^Lp^L+1 d _cov ). Discard balls that do not intersect ℋH, and select one element of ℋH from each remaining ball. By the triangle inequality, the selected elements form a proper ηh _h-cover with no larger cardinality. The interval [−ω,ω][-ω,ω] has an ηb _b-cover of cardinality at most 1+4ωd/δcov1+4ω d/ _cov (this is by definition of cover). For a coordinate viv_i, choose hi′h_i and bi′b_i from these covers. Because clipωclip_ω is 11-Lipschitz, supx,t|vi(x,t)−vi′(x,t)|≤Λ‖hi−hi′‖∞,[0,1]q+|bi−bi′|≤δcovd. _x,t|v_i(x,t)-v_i (x,t)|≤ \|h_i-h_i \|_∞,[0,1]^q+|b_i-b_i |≤ _cov d. Taking the product cover over the d coordinates gives supx,t‖v(x,t)−v′(x,t)‖≤δcov _x,t\|v(x,t)-v (x,t)\|≤ _cov. Multiplying the scalar and bias cover cardinalities and then taking logarithms proves Equation (28). ∎ For nV_n, take ω=2ω=2, Λ as above, and substitute the scalar approximation parameters LLip(q,η)L_Lip(q,η), pLip(q,η)p_Lip(q,η), JLip(q,η)J_Lip(q,η), and κLip(q,η) _Lip(q,η) into Equation (28). To keep the later bounds readable, define, for r>0r>0, ℭv(r) C_v(r) :=dJLip(q,η)log(1+8LLip(q,η)2(pLip(q,η)+2)κLip(q,η)LLip(q,η)pLip(q,η)LLip(q,η)+1Λdr) :=dJ_Lip(q,η) \! (1+ 8L_Lip(q,η)^2(p_Lip(q,η)+2) _Lip(q,η)^L_Lip(q,η)p_Lip(q,η)^L_Lip(q,η)+1 dr ) +dlog(1+8dr). +d \! (1+ 8 dr ). (29) Thus, for every δcov>0 _cov>0, log(δcov,n,∥⋅∥∞,[−R,R]d×[0,1]) \! ( _cov,V_n,\|·\|_∞,[-R,R]^d×[0,1] ) ≤ℭv(δcov). ≤ C_v( _cov). (30) The function r↦ℭv(r)r C_v(r) is nonincreasing, a fact used below when a loss-class cover is transferred to a cover of nV_n. Lemma 16 (Uniform Bounds for the Vector Network Class). Every v∈nv _n satisfies supx∈ℝd,t∈[0,1]‖v(x,t)‖ _x ^d,\;t∈[0,1]\|v(x,t)\| ≤2d, ≤ 2 d, (31) ‖v(x,t)−v(x′,t)‖ \|v(x,t)-v(x ,t)\| ≤γx,R,ε‖x−x′‖∞,x,x′∈ℝd,t∈[0,1], ≤ _x,R, \|x-x \|_∞, x,x ^d, t∈[0,1], (32) ‖v(x,t)−v(x,s)‖ \|v(x,t)-v(x,s)\| ≤γt,R,ε|t−s|,x∈ℝd,t,s∈[0,1], ≤ _t,R, |t-s|, x ^d, t,s∈[0,1], (33) where γx,R,ε _x,R, =5(d+1)dΛR, = 5(d+1) d\, R, γt,R,ε _t,R, =10(d+1)dΛ. =10(d+1) d\, . Proof. From the coordinatewise clipping, we have |vi(x,t)|≤2,∀i=1,…,d|v_i(x,t)|≤ 2,∀ i=1,…,d. Hence ‖v(x,t)‖≤2d\|v(x,t)\|≤ 2 d. Next, fix x,x′∈ℝdx,x ^d and t∈[0,1]t∈[0,1]. Since clip2clip_2 is 11-Lipschitz and each hih_i is 10q10q-Lipschitz with respect to ∥⋅∥∞\|·\|_∞, |vi(x,t)−vi(x′,t)| |v_i(x,t)-v_i(x ,t)| ≤Λ|hi(SR(x,t))−hi(SR(x′,t))|≤10qΛ‖SR(x,t)−SR(x′,t)‖∞. ≤ |h_i(S_R(x,t))-h_i(S_R(x ,t)) |≤ 10q \|S_R(x,t)-S_R(x ,t)\|_∞. By the definition of SRS_R, ‖SR(x,t)−SR(x′,t)‖∞≤12R‖x−x′‖∞\|S_R(x,t)-S_R(x ,t)\|_∞≤ 12R\|x-x \|_∞. Therefore, |vi(x,t)−vi(x′,t)|≤5(d+1)ΛR‖x−x′‖∞|v_i(x,t)-v_i(x ,t)|≤ 5(d+1) R\|x-x \|_∞. Summing over coordinates gives ‖v(x,t)−v(x′,t)‖ \|v(x,t)-v(x ,t)\| ≤5(d+1)dΛR‖x−x′‖∞. ≤ 5(d+1) d\, R\|x-x \|_∞. For variation in the time coordinate, t,s∈[0,1]t,s∈[0,1], |vi(x,t)−vi(x,s)| |v_i(x,t)-v_i(x,s)| ≤Λ|hi(SR(x,t))−hi(SR(x,s))|≤10qΛ‖SR(x,t)−SR(x,s)‖∞=10(d+1)Λ|t−s|. ≤ |h_i(S_R(x,t))-h_i(S_R(x,s)) |≤ 10q \|S_R(x,t)-S_R(x,s)\|_∞=10(d+1) |t-s|. Summing over coordinates gives ‖v(x,t)−v(x,s)‖≤10(d+1)dΛ|t−s|\|v(x,t)-v(x,s)\|≤ 10(d+1) d\, |t-s|. ∎ I.1.2 Completion of the Velocity Bound Recall that for v∈nv _n, we defined the pointwise velocity loss Φvvel(y0,y1,k,g) _v^vel(y_0,y_1,k,g) =‖v(y0+tk(y1−y0)+σg,tk)−(y1−y0)‖2. = \|v (y_0+t_k(y_1-y_0)+σ g,t_k )-(y_1-y_0) \|^2. (34) Equip nV_n with dv(v,v~)=supx∈[−R,R]d,t∈[0,1]‖v(x,t)−v~(x,t)‖.d_v(v, v)= _x∈[-R,R]^d,\;t∈[0,1]\|v(x,t)- v(x,t)\|. Lemma 17 (Constants for the velocity loss). The loss in Equation (34) satisfies Equation (20)–Equation (22) with Bvel B_vel =16d, =16d, LY,vel L_Y,vel =8d(1+γx,R,ε), =8 d (1+ _x,R, ), LΘ,vel L_ ,vel =8d. =8 d. (35) Moreover, log(r,n,dv) (r,V_n,d_v) ≤ℭv(r). ≤ C_v(r). (36) Proof. Write St=(1−t)y0+ty1,D=y1−y0,S_t=(1-t)y_0+ty_1, D=y_1-y_0, and define St′S_t and D′D from y0′,y1′y_0 ,y_1 . Bounded support and Equation (31) give ‖v(St+σg,t)−D‖≤‖v(St+σg,t)‖+|D|≤4d,\|v(S_t+σ g,t)-D\|≤\|v(S_t+σ g,t)\|+\|D\|≤ 4 d, so the squared loss is at most 16d16d. For two endpoint pairs, ‖St−St′‖∞≤‖y0−y0′‖+‖y1−y1′‖,‖D−D′‖≤‖y0−y0′‖+‖y1−y1′‖.\|S_t-S_t \|_∞≤\|y_0-y_0 \|+\|y_1-y_1 \|, \|D-D \|≤\|y_0-y_0 \|+\|y_1-y_1 \|. Using |‖a‖2−‖b‖2|≤(‖a‖+‖b‖)‖a−b‖ |\|a\|^2-\|b\|^2 |≤(\|a\|+\|b\|)\|a-b\| and Equation (32), we obtain |Φvvel(y0,y1,k,g)−Φvvel(y0′,y1′,k,g)| | _v^vel(y_0,y_1,k,g)- _v^vel(y_0 ,y_1 ,k,g) | ≤8d[γx,R,ε‖Stk−Stk′‖∞+‖D−D′‖] ≤ 8 d [ _x,R, \|S_t_k-S_t_k \|_∞+\|D-D \| ] ≤LY,vel(‖y0−y0′‖+‖y1−y1′‖). ≤ L_Y,vel (\|y_0-y_0 \|+\|y_1-y_1 \| ). Averaging over K,GK,G proves the required endpoint bound for HvvelH_v^vel. For v,v~∈nv, v _n, both residuals have norm at most 4d4 d, and hence |Φvvel(y0,y1,k,g)−Φv~vel(y0,y1,k,g)|≤8d‖v(Stk+σg,tk)−v~(Stk+σg,tk)‖. | _v^vel(y_0,y_1,k,g)- _ v^vel(y_0,y_1,k,g) |≤ 8 d\,\|v(S_t_k+σ g,t_k)- v(S_t_k+σ g,t_k)\|. Every network in nV_n depends on x through clipR(x)clip_R(x), so the last norm is bounded by dv(v,v~)d_v(v, v). This proves the parameter bound. Finally, Equation (36) is exactly Equation (30) under the metric dvd_v. ∎ Proof of the velocity bound in Proposition 1. Choose ε=n−1/(d+5),R=1+σ[2log(32d2ε2)]1/2. =n^-1/(d+5), R=1+σ [2 ( 32d^2 ^2 ) ]^1/2. This choice satisfies the tail condition in Proposition 6. By Equation (27) and that proposition, infv∈nLa,Q(v)−La,Q(vQ⋆) _v _n \L_a,Q(v)-L_a,Q(v_Q ) \ ≤2ε2. ≤ 2 ^2. (37) For Φvvel _v^vel, La,QΦ(v)=La,Q(v)L_a,Q (v)=L_a,Q(v), and L^actualΦ(v) L_actual (v) is the velocity training objective. Lemma 17 verifies every loss hypothesis of Proposition 5, with Θ=n,dΘ=dv,BΦ=Bvel,LY=LY,vel,LΘ=LΘ,vel. =V_n, _ =d_v, B_ =B_vel, L_Y=L_Y,vel, L_ =L_ ,vel. We now evaluate the quantities in that proposition. Since Λ=16d(2R+1)/σ2 =16d(2R+1)/σ^2, η−1 η^-1 =dΛε=16d3/2[3+2σ2log(32d2/ε2)]σ2ε. = d\, = 16d^3/2 [3+2σ 2 (32d^2/ ^2) ]σ^2 . (38) The architecture orders in Proposition 6, substituted into ℭv C_v, imply for the three radii in Proposition 5 logNDRvel+logNobsvel+logNauxvel N_DR^vel+ N_obs^vel+ N_aux^vel =O~(η−(d+1)). = O (η^-(d+1) ). (39) Here O~ O hides logarithms in n,M,σ−1,εOT−1n,M,σ^-1, _OT^-1, and πmin−1 _ ^-1, and constants depending on fixed d. Combining Equation (23), Equation (38), and Equation (39), with δ=n−1δ=n^-1, give ℜvel(n−1) R_vel(n^-1) ≲1+2πmin−1σd+1n−2/(d+5). 1+2 _ ^-1σ^d+1n^-2/(d+5). (40) Indeed, before substituting ε , the leading term is O~((1+2πmin−1)σ−(d+1)ε−(d+1)/2n−1/2), O ((1+2 _ ^-1)σ^-(d+1) ^-(d+1)/2n^-1/2 ), and ε−(d+1)/2n−1/2=n−2/(d+5) ^-(d+1)/2n^-1/2=n^-2/(d+5). The endpoint constant has the explicit order LY,vel L_Y,vel =8d[1+5(d+1)dΛR]=O(σ−2), =8 d [1+ 5(d+1) d\, R ]=O(σ^-2), (41) because R≥1R≥ 1 and Λ/R=16d(2+R−1)/σ2 /R=16d(2+R^-1)/σ^2. On the event in Corollary 3, ΔQ≲εOT−1[n−1/(d+2)+εpred,train+(nMQ)−1/(d+2)]. _Q _OT^-1 [n^-1/(d+2)+ _pred,train+(nM_Q)^-1/(d+2) ]. Substitute this display, Equation (37), Equation (40), and Equation (41) into Equation (24). Since MQ≥1M_Q≥ 1, (nMQ)−1/(d+2)≤n−1/(d+2)(nM_Q)^-1/(d+2)≤ n^-1/(d+2), and since 0<σ≤10<σ≤ 1, the approximation term is absorbed by Equation (40). Thus La,Q(v^M)−La,Q(vQ⋆) L_a,Q( v_M)-L_a,Q(v_Q ) ≲εOT−1σ2[n−1/(d+2)+εpred,train] _OT^-1σ^2 [n^-1/(d+2)+ _pred,train ] +1+εOT−1σ2εαεpred+1+2πmin−1σd+1n−2/(d+5). + 1+ _OT^-1σ^2 _α _pred+ 1+2 _ ^-1σ^d+1n^-2/(d+5). Applying Equation (27) with v=v^Mv= v_M proves Equation (4). The coupling event has probability at least 1−2/n1-2/n, and the conditional ERM bound with δ=1/nδ=1/n has probability at least 1−1/n1-1/n. Finally, ℙ(ℰac)≤e−nπminP(E_a^c)≤ e^-n _ , which gives the stated failure probability. ∎ I.2 Score-Field Bound Recall that St=Y0+t(Y1−Y0)S_t=Y_0+t(Y_1-Y_0) and Xt=St+σGX_t=S_t+σ G. We define the oracle by uQ⋆(x,t)=[St∣Xt=x]u_Q (x,t)=E[S_t X_t=x]. By the Gaussian score identity, sQ⋆(x,t) s_Q (x,t) =uQ⋆(x,t)−xσ2. = u_Q (x,t)-xσ^2. Instead of approximating sQ⋆s_Q directly, we approximate uQ⋆u_Q . For every u:ℝd×[0,1]→ℝdu:R^d×[0,1] ^d, define the induced score field by su(x,t) s_u(x,t) =u(x,t)−xσ2. = u(x,t)-xσ^2. I.2.1 Score Loss and Exact Risk Identity We now specialize the generic doubly robust bound to the score estimator. Here the abstract parameter θ indexes a conditional-mean field uθ∈nu_θ _n, where nU_n is defined later in Equation (52). This in turn induces the score field suθ(x,t)=uθ(x,t)−xσ2.s_u_θ(x,t)= u_θ(x,t)-xσ^2. For brevity, we suppress the abstract index θ and write u for uθu_θ and sus_u for the induced score field. For u∈nu _n, define Φuscore(y0,y1,k,g)=‖su(y0+tk(y1−y0)+σg,tk)+gσ‖2. _u^score(y_0,y_1,k,g)= \|s_u\! (y_0+t_k(y_1-y_0)+σ g,t_k )+ gσ \|^2. Recall population minimizer of the denoising score-matching loss satisfies sQ⋆(x,t)=∇xlogqt,Q(x)=[−Gσ∣Y0+t(Y1−Y0)+σG=x], s_Q (x,t)= _x q_t,Q(x)=E [- Gσ Y_0+t(Y_1-Y_0)+σ G=x ], where Y1∼∫ℙY∣C=c,A=adℙC(c)Y_1 _Y C=c,A=adP_C(c) and Y0∼Q(⋅∣Y1)Y_0 Q(· Y_1). For a score field s:ℝd×[0,1]→ℝds:R^d×[0,1] ^d, let K∼Unif0,…,N−1K \0,…,N-1\, independently of all endpoint and Gaussian variables, we have rQ(s,y1) r_Q(s;y_1) =K,Y0∼Q(⋅∣y1),G∼(0,Id)[‖s(Y0+tK(y1−Y0)+σG,tK)+Gσ‖2], =E_K,Y_0 Q(· y_1),\,G N(0,I_d) [ \|s (Y_0+t_K(y_1-Y_0)+σ G,t_K )+ Gσ \|^2 ], La,Qscore(s) L^score_a,Q(s) =Y1∼∫ℙY∣C=c,A=adℙC[rQ(s;Y1)], =E_Y_1 _Y C=c,A=adP_C [r_Q(s;Y_1) ], rQ^n(s,y1) r_ Q_n(s;y_1) =K,Y0∼Q^n(⋅∣y1),G∼(0,Id)[‖s(Y0+tK(y1−Y0)+σG,tK)+Gσ‖2], =E_K,Y_0 Q_n(· y_1),\,G N(0,I_d) [ \|s (Y_0+t_K(y_1-Y_0)+σ G,t_K )+ Gσ \|^2 ], La,Q^nscore(s) L^score_a, Q_n(s) =Y1∼∫ℙY∣C=c,A=adℙC[rQ^n(s;Y1)]. =E_Y_1 _Y C=c,A=adP_C [r_ Q_n(s;Y_1) ]. Exact Excess-Risk Identity. Because sQ⋆(XtK,tK)=[−G/σ∣XtK,K]s_Q (X_t_K,t_K)=E[-G/σ X_t_K,K], conditional projection gives, for every score field s with finite population loss, La,Qscore(s)−La,Qscore(sQ⋆) L^score_a,Q(s)-L^score_a,Q(s_Q ) =[‖s(XtK,tK)−sQ⋆(XtK,tK)‖2] =E\! [ \|s(X_t_K,t_K)-s_Q (X_t_K,t_K) \|^2 ] =1N∑k=0N−1[‖s(Xtk,tk)−sQ⋆(Xtk,tk)‖2]. = 1N _k=0^N-1E\! [ \|s(X_t_k,t_k)-s_Q (X_t_k,t_k) \|^2 ]. (42) Thus it remains to control the excess risk of the empirical minimizer. I.2.2 Score Network Class and Approximation The denoising score-matching loss is evaluated under the joint sampling procedure in which the same Gaussian draw G is used both to construct Xt=St+σGX_t=S_t+σ G and to form the denoising target −G/σ-G/σ. Consequently, ‖su(St+σG,t)+Gσ‖2=‖u(St+σG,t)−St−σGσ2+Gσ‖2=1σ4‖u(St+σG,t)−St‖2. \|s_u(S_t+σ G,t)+ Gσ \|^2= \| u(S_t+σ G,t)-S_t-σ Gσ^2+ Gσ \|^2= 1σ^4 \|u(S_t+σ G,t)-S_t \|^2. (43) This identity is not a statement about arbitrary independently chosen x∈ℝdx ^d and g∈ℝdg ^d. It applies to the jointly constructed pair x=St+σgx=S_t+σ g, with the same g appearing in the denoising target. Consequently, the population score loss can be written as La,Qscore(su) L^score_a,Q(s_u) =1σ4K[‖u(XtK,tK)−StK‖2]. = 1σ^4E_KE [ \|u(X_t_K,t_K)-S_t_K \|^2 ]. Since uQ⋆(Xt,t)=[St∣Xt]u_Q (X_t,t)=E[S_t X_t], the conditional projection identity gives La,Qscore(su)−La,Qscore(sQ⋆) L^score_a,Q(s_u)-L^score_a,Q(s_Q ) =1σ4K[‖u(XtK,tK)−uQ⋆(XtK,tK)‖2]. = 1σ^4E_KE [ \|u(X_t_K,t_K)-u_Q (X_t_K,t_K) \|^2 ]. (44) Moreover, su(Xt,t)−sQ⋆(Xt,t)=u(Xt,t)−uQ⋆(Xt,t)/σ2s_u(X_t,t)-s_Q (X_t,t)=\u(X_t,t)-u_Q (X_t,t)\/σ^2, so La,Qscore(su)−La,Qscore(sQ⋆) L^score_a,Q(s_u)-L^score_a,Q(s_Q ) =K[‖su(XtK,tK)−sQ⋆(XtK,tK)‖2]. =E_KE [ \|s_u(X_t_K,t_K)-s_Q (X_t_K,t_K) \|^2 ]. (45) Under Assumption 2, St∈[−1,1]dS_t∈[-1,1]^d almost surely. Therefore, uQ⋆(x,t) u_Q (x,t) =[St∣Xt=x]∈[−1,1]d =E[S_t X_t=x]∈[-1,1]^d (46) for every x∈ℝdx ^d and t∈[0,1]t∈[0,1]. Thus uQ⋆u_Q is globally bounded, although the induced score sQ⋆(x,t)=uQ⋆(x,t)−x/σ2s_Q (x,t)=\u_Q (x,t)-x\/σ^2 is not globally bounded. Recall that for R>1R>1, we defined TR(z) T_R(z) =(−R+2Rz1,…,−R+2Rzd,zd+1),z∈[0,1]d+1. = (-R+2Rz_1,…,-R+2Rz_d,z_d+1 ), z∈[0,1]^d+1. Lemma 18 (Lipschitz Regularity of the Conditional Mean Oracle). Assume Assumption 2. For i=1,…,di=1,…,d, define gi(z)=uQ,i⋆(TR(z))g_i(z)=u_Q,i (T_R(z)). Then |gi(z)−gi(z′)| |g_i(z)-g_i(z )| ≤Λu‖z−z′‖∞,z,z′∈[0,1]d+1, ≤ _u\|z-z \|_∞, z,z ∈[0,1]^d+1, where Λu _u =2+8d(2R+1)σ2. =2+ 8d(2R+1)σ^2. (47) Proof. By the score identity, uQ⋆(x,t)=x+σ2sQ⋆(x,t)u_Q (x,t)=x+σ^2s_Q (x,t). By Lemma 4, DxsQ⋆(x,t) D_xs_Q (x,t) =1σ2(Cov(G,G∣Xt=x)−Id). = 1σ^2 (Cov(G,G X_t=x)-I_d ). Therefore, DxuQ⋆(x,t) D_xu_Q (x,t) =Id+σ2DxsQ⋆(x,t)=Cov(G,G∣Xt=x). =I_d+σ^2D_xs_Q (x,t)=Cov(G,G X_t=x). Under the conditioning Xt=xX_t=x, we have G=(x−St)/σG=(x-S_t)/σ, and hence G−[G∣Xt=x] G-E[G X_t=x] =−St−[St∣Xt=x]σ. =- S_t-E[S_t X_t=x]σ. Consequently, for every i,j∈1,…,di,j∈\1,…,d\, ∂xjuQ,i⋆(x,t) _x_ju_Q,i (x,t) =Cov(Gi,Gj∣Xt=x)=1σ2Cov(St,i,St,j∣Xt=x). =Cov(G_i,G_j X_t=x)= 1σ^2Cov(S_t,i,S_t,j X_t=x). Since St∈[−1,1]dS_t∈[-1,1]^d, |St,i−[St,i∣Xt=x]| |S_t,i-E[S_t,i X_t=x] | ≤2,|St,j−[St,j∣Xt=x]|≤2. ≤ 2, |S_t,j-E[S_t,j X_t=x] |≤ 2. It follows that |∂xjuQ,i⋆(x,t)| | _x_ju_Q,i (x,t) | ≤1σ2[|St,i−[St,i∣Xt=x]||St,j−[St,j∣Xt=x]|∣Xt=x]≤4σ2. ≤ 1σ^2E [ |S_t,i-E[S_t,i X_t=x] | |S_t,j-E[S_t,j X_t=x] | X_t=x ]≤ 4σ^2. (48) We next control the time derivative. Write D=Y1−Y0D=Y_1-Y_0 and, for fixed x, define the unnormalized posterior weight wt(Y0,Y1,x)=exp(−‖x−St‖22σ2).w_t(Y_0,Y_1;x)= \! (- \|x-S_t\|^22σ^2 ). Bounded support permits differentiation under the endpoint expectation, and ∂twt(Y0,Y1,x)=D⊤(x−St)σ2wt(Y0,Y1,x). _tw_t(Y_0,Y_1;x)= D (x-S_t)σ^2w_t(Y_0,Y_1;x). Therefore, for any differentiable scalar Ft=Ft(Y0,Y1)F_t=F_t(Y_0,Y_1), the quotient rule applied to [Ft∣Xt=x]=[Ftwt(Y0,Y1,x)][wt(Y0,Y1,x)]E[F_t X_t=x]= E[F_tw_t(Y_0,Y_1;x)]E[w_t(Y_0,Y_1;x)] gives the posterior differentiation identity ∂t[Ft∣Xt=x] _tE[F_t X_t=x] =[∂tFt∣Xt=x]+1σ2Cov(Ft,D⊤(x−St)∣Xt=x). =E[ _tF_t X_t=x]+ 1σ^2Cov\! (F_t,D (x-S_t) X_t=x ). Taking Ft=St,iF_t=S_t,i, for which ∂tFt=Di _tF_t=D_i, yields ∂tuQ,i⋆(x,t) _tu_Q,i (x,t) =[Y1,i−Y0,i∣Xt=x]+1σ2Cov(St,i,(Y1−Y0)⊤(x−St)∣Xt=x). =E[Y_1,i-Y_0,i X_t=x]+ 1σ^2Cov (S_t,i,(Y_1-Y_0) (x-S_t) X_t=x ). The first term satisfies |[Y1,i−Y0,i∣Xt=x]| |E[Y_1,i-Y_0,i X_t=x] | ≤2. ≤ 2. If x∈[−R,R]dx∈[-R,R]^d, then |(Y1−Y0)⊤(x−St)|≤∑j=1d|Y1,j−Y0,j||xj−St,j|≤2d(R+1). |(Y_1-Y_0) (x-S_t) |≤ _j=1^d|Y_1,j-Y_0,j||x_j-S_t,j|≤ 2d(R+1). Therefore, |(Y1−Y0)⊤(x−St)−[(Y1−Y0)⊤(x−St)∣Xt=x]| |(Y_1-Y_0) (x-S_t)-E [(Y_1-Y_0) (x-S_t) X_t=x ] | ≤|(Y1−Y0)⊤(x−St)|+[|(Y1−Y0)⊤(x−St)|∣Xt=x] ≤ |(Y_1-Y_0) (x-S_t) |+E [ |(Y_1-Y_0) (x-S_t) | X_t=x ] ≤4d(R+1). ≤ 4d(R+1). Together with |St,i−[St,i∣Xt=x]|≤2 |S_t,i-E[S_t,i X_t=x] |≤ 2, this gives |∂tuQ,i⋆(x,t)| | _tu_Q,i (x,t) | ≤2+1σ2[|St,i−[St,i∣Xt=x]| ≤ 2+ 1σ^2E [ |S_t,i-E[S_t,i X_t=x] | . ⋅|(Y1−Y0)⊤(x−St)−[(Y1−Y0)⊤(x−St)∣Xt=x]|∣Xt=x] 96.73918pt .· |(Y_1-Y_0) (x-S_t)-E [(Y_1-Y_0) (x-S_t) X_t=x ] | X_t=x ] ≤2+8d(R+1)σ2. ≤ 2+ 8d(R+1)σ^2. (49) For j=1,…,dj=1,…,d, the chain rule and Equation (48) give |∂zjgi(z)|=2R|∂xjuQ,i⋆(TR(z))|≤8Rσ2| _z_jg_i(z)|=2R | _x_ju_Q,i (T_R(z)) |≤ 8Rσ^2, so we get |gi(z)−gi(z′)| |g_i(z)-g_i(z )| ≤[∑j=1d+1supz~∈[0,1]d+1|∂zjgi(z~)|]‖z−z′‖∞≤[8Rdσ2+2+8d(R+1)σ2]‖z−z′‖∞ ≤ [ _j=1^d+1 _ z∈[0,1]^d+1| _z_jg_i( z)| ]\|z-z \|_∞≤ [ 8Rdσ^2+2+ 8d(R+1)σ^2 ]\|z-z \|_∞ =[2+8d(2R+1)σ2]‖z−z′‖∞=Λu‖z−z′‖∞. = [2+ 8d(2R+1)σ^2 ]\|z-z \|_∞= _u\|z-z \|_∞. ∎ Network Construction and Approximation Guarantee. Let ε>0 >0, and define ηu _u =σ2εdΛu. = σ^2 d\, _u. (50) Proposition 7 (Approximation Error for the Score Oracle Under the Conditional Mean Parametrization). Assume Assumption 2. Let R>1R>1, q=d+1q=d+1, and ε>0 >0 satisfy 0<ηu=σ2ε/(dΛu)<10< _u=σ^2 /( d\, _u)<1 and 8d2σ4exp(−(R−1)22σ2) 8d^2σ^4 (- (R-1)^22σ^2 ) ≤ε2. ≤ ^2. (51) Use the same architecture bounds KLip(q,ηu)K_Lip(q, _u), LLip(q,ηu)L_Lip(q, _u), pLip(q,ηu)p_Lip(q, _u), JLip(q,ηu)J_Lip(q, _u), and κLip(q,ηu) _Lip(q, _u) as in Proposition 6, now evaluated at ηu _u. In particular, LLipL_Lip counts total affine layers, pLip≥qp_Lip≥ q, and κLip≥2 _Lip≥ 2. For every i=1,…,di=1,…,d, construct uiapp(x,t) u_i^app(x,t) =clip1(Λuh^i(SR(x,t))+gi(0)), =clip_1 ( _u h_i(S_R(x,t))+g_i(0) ), where h^i h_i ∈scal(q,LLip(q,ηu),pLip(q,ηu),JLip(q,ηu),KLip(q,ηu),κLip(q,ηu),10q), ∈ N_scal (q,L_Lip(q, _u),p_Lip(q, _u),J_Lip(q, _u),K_Lip(q, _u), _Lip(q, _u),10q ), and gi(z)=uQ,i⋆(TR(z))g_i(z)=u_Q,i (T_R(z)). Define uapp=(u1app,…,udapp)u^app=(u_1^app,…,u_d^app) and sapp(x,t)=uapp(x,t)−x/σ2s^app(x,t)=\u^app(x,t)-x\/σ^2. Then 1N∑k=0N−1[‖sapp(Xtk,tk)−sQ⋆(Xtk,tk)‖2] 1N _k=0^N-1E [ \|s^app(X_t_k,t_k)-s_Q (X_t_k,t_k) \|^2 ] ≤2ε2. ≤ 2 ^2. Proof. The proof has two subclaims: uniform approximation on the cube [−R,R]d×[0,1][-R,R]^d×[0,1], followed by control of the Gaussian tail outside the cube. For the first subclaim, define fi(z)=gi(z)−gi(0)Λu,z∈[0,1]d+1.f_i(z)= g_i(z)-g_i(0) _u, z∈[0,1]^d+1. Lemma 18 implies that fi(0)=0f_i(0)=0 and that fif_i is 11-Lipschitz in the ℓ∞ _∞ norm. Since ‖z−z′‖∞≤‖z−z′‖2\|z-z \|_∞≤\|z-z \|_2, it is also 11-Lipschitz in the Euclidean norm required by the scalar approximation result invoked in Proposition 6. That result provides h^i h_i in the displayed network class such that supz∈[0,1]d+1|h^i(z)−fi(z)|≤ηu. _z∈[0,1]^d+1| h_i(z)-f_i(z)|≤ _u. If x∈[−R,R]dx∈[-R,R]^d, then TR(SR(x,t))=(x,t)T_R(S_R(x,t))=(x,t), and hence |Λuh^i(SR(x,t))+gi(0)−uQ,i⋆(x,t)|≤Λuηu=σ2εd. | _u h_i(S_R(x,t))+g_i(0)-u_Q,i (x,t) |≤ _u _u= σ^2 d. Both uQ,i⋆(x,t)u_Q,i (x,t) and the clipped approximation lie in [−1,1][-1,1], so projection onto this interval cannot increase the error. Consequently, supx∈[−R,R]d,t∈[0,1]‖sapp(x,t)−sQ⋆(x,t)‖2≤ε2. _x∈[-R,R]^d,\ t∈[0,1]\|s^app(x,t)-s_Q (x,t)\|^2≤ ^2. For the second subclaim, St∈[−1,1]dS_t∈[-1,1]^d almost surely, and a coordinatewise Gaussian tail bound followed by a union bound gives, uniformly in t, ℙXt∉[−R,R]d≤2dexp(−(R−1)22σ2).P\X_t∉[-R,R]^d\≤ 2d \! (- (R-1)^22σ^2 ). On this event, the two conditional mean remain in [−1,1]d[-1,1]^d, so ‖sapp(Xt,t)−sQ⋆(Xt,t)‖2=‖uapp(Xt,t)−uQ⋆(Xt,t)‖2σ4≤4dσ4.\|s^app(X_t,t)-s_Q (X_t,t)\|^2= \|u^app(X_t,t)-u_Q (X_t,t)\|^2σ^4≤ 4dσ^4. The tail contribution is therefore at most 8d2σ4exp(−(R−1)22σ2)≤ε2 8d^2σ^4 \! (- (R-1)^22σ^2 )≤ ^2 by Equation (51). Adding the cube and tail contributions proves the claimed 2ε22 ^2 bound at every grid time, and averaging over k=0,…,N−1k=0,…,N-1 completes the proof. ∎ The preceding proposition motivates n _n =u:ℝd×[0,1]→ℝd:u(x,t)=(u1(x,t),…,ud(x,t)), = \u:R^d×[0,1] ^d:u(x,t)= (u_1(x,t),…,u_d(x,t) ), (52) ui(x,t)=clip1(Λuhi(SR(x,t))+bi),i=1,…,d, u_i(x,t)=clip_1 ( _uh_i(S_R(x,t))+b_i ), i=1,…,d, hi∈scal(q,LLip(q,ηu),pLip(q,ηu),JLip(q,ηu),KLip(q,ηu),κLip(q,ηu),10q), h_i∈ N_scal (q,L_Lip(q, _u),p_Lip(q, _u),J_Lip(q, _u),K_Lip(q, _u), _Lip(q, _u),10q ), |bi|≤1,i=1,…,d. |b_i|≤ 1, i=1,…,d \. The induced score class is n _n =su:ℝd×[0,1]→ℝd:su(x,t)=u(x,t)−xσ2,u∈n. = \s_u:R^d×[0,1] ^d:s_u(x,t)= u(x,t)-xσ^2, u _n \. (53) Proposition 7 implies infs∈n1N∑k=0N−1[‖s(Xtk,tk)−sQ⋆(Xtk,tk)‖2]≤2ε2. _s _n 1N _k=0^N-1E [ \|s(X_t_k,t_k)-s_Q (X_t_k,t_k) \|^2 ]≤ 2 ^2. Covering Number Evaluation. The conditional-mean class is an instance of Lemma 15 with clipping level ω=1ω=1, scale Λu _u, and scalar-network accuracy ηu=σ2ε/(dΛu) _u=σ^2 /( d\, _u). Its scalar architecture orders are LLip(q,ηu) L_Lip(q, _u) =O(logdΛuσ2ε+d), =O\! ( d\, _uσ^2 +d ), (54) pLip(q,ηu) p_Lip(q, _u) =O(d+1+(dΛuσ2ε)d+1), =O\! (d+1+ ( d\, _uσ^2 )^d+1 ), JLip(q,ηu) J_Lip(q, _u) =O((dΛuσ2ε)d+1[logdΛuσ2ε+d]). =O\! ( ( d\, _uσ^2 )^d+1 [ d\, _uσ^2 +d ] ). For r>0r>0, define ℭu(r) C_u(r) :=dJLip(q,ηu)log(1+8LLip(q,ηu)2(pLip(q,ηu)+2)κLip(q,ηu)LLip(q,ηu)pLip(q,ηu)LLip(q,ηu)+1Λudr) :=dJ_Lip(q, _u) \! (1+ 8L_Lip(q, _u)^2(p_Lip(q, _u)+2) _Lip(q, _u)^L_Lip(q, _u)p_Lip(q, _u)^L_Lip(q, _u)+1 _u dr ) +dlog(1+4dr). +d \! (1+ 4 dr ). (55) Therefore, for every δcov>0 _cov>0, log(δcov,n,∥⋅∥∞,[−R,R]d×[0,1])≤ℭu(δcov). ( _cov,U_n,\|·\|_∞,[-R,R]^d×[0,1] )≤ C_u( _cov). (56) For u,u~∈nu, u _n, su(x,t)−su~(x,t) s_u(x,t)-s_ u(x,t) =u(x,t)−u~(x,t)σ2. = u(x,t)- u(x,t)σ^2. Therefore, supx∈[−R,R]d,t∈[0,1]‖su(x,t)−su~(x,t)‖=1σ2supx∈[−R,R]d,t∈[0,1]‖u(x,t)−u~(x,t)‖. _x∈[-R,R]^d,\;t∈[0,1]\|s_u(x,t)-s_ u(x,t)\|= 1σ^2 _x∈[-R,R]^d,\;t∈[0,1]\|u(x,t)- u(x,t)\|. It follows that (δcov,n,∥⋅∥∞,[−R,R]d×[0,1])≤(σ2δcov,n,∥⋅∥∞,[−R,R]d×[0,1]). ( _cov,S_n,\|·\|_∞,[-R,R]^d×[0,1] ) (σ^2 _cov,U_n,\|·\|_∞,[-R,R]^d×[0,1] ). Hence log(δcov,n,∥⋅∥∞,[−R,R]d×[0,1])≤ℭu(σ2δcov). ( _cov,S_n,\|·\|_∞,[-R,R]^d×[0,1] )≤ C_u(σ^2 _cov). (57) The fixed affine term −x/σ2-x/σ^2 does not increase the covering number, because it is identical for every element of nS_n and cancels when two induced score fields are subtracted. Uniform Bounds Used in the Error Analysis. Lemma 19 (Uniform Bounds for the Conditional-Mean Class and Induced Score Field). Every u∈nu _n satisfies supx∈ℝd,t∈[0,1]‖u(x,t)‖ _x ^d,\;t∈[0,1]\|u(x,t)\| ≤d, ≤ d, (58) ‖u(x,t)−u(x′,t)‖ \|u(x,t)-u(x ,t)\| ≤γx,R,εu‖x−x′‖∞, ≤ _x,R, ^u\|x-x \|_∞, (59) ‖u(x,t)−u(x,r)‖ \|u(x,t)-u(x,r)\| ≤γt,R,εu|t−r|, ≤ _t,R, ^u|t-r|, (60) where γx,R,εu _x,R, ^u =5(d+1)dΛuR,γt,R,εu=10(d+1)dΛu. = 5(d+1) d\, _uR, _t,R, ^u=10(d+1) d\, _u. The induced score field satisfies ‖su(x,t)−su(x′,t)‖ \|s_u(x,t)-s_u(x ,t)\| ≤γx,R,εu+dσ2‖x−x′‖∞, ≤ _x,R, ^u+ dσ^2\|x-x \|_∞, (61) ‖su(x,t)−su(x,r)‖ \|s_u(x,t)-s_u(x,r)\| ≤γt,R,εuσ2|t−r|. ≤ _t,R, ^uσ^2|t-r|. (62) No finite uniform bound on supx∈ℝd,t∈[0,1]‖su(x,t)‖ _x ^d,\;t∈[0,1]\|s_u(x,t)\| is asserted. Proof. The proof has two parts. First, clipping each coordinate to [−1,1][-1,1] immediately gives ‖u(x,t)‖≤d\|u(x,t)\|≤ d. For regularity, each scalar subnetwork hih_i is 10q=10(d+1)10q=10(d+1)-Lipschitz in the ℓ∞ _∞ norm. The spatial part of SRS_R is 1/(2R)1/(2R)-Lipschitz, while its time coordinate is 11-Lipschitz. Since the outer clipping map is 11-Lipschitz, |ui(x,t)−ui(x′,t)| |u_i(x,t)-u_i(x ,t)| ≤5(d+1)ΛuR‖x−x′‖∞, ≤ 5(d+1) _uR\|x-x \|_∞, |ui(x,t)−ui(x,r)| |u_i(x,t)-u_i(x,r)| ≤10(d+1)Λu|t−r|. ≤ 10(d+1) _u|t-r|. Summing the squared coordinate bounds proves Equation (59) and Equation (60). Finally, su(x,t)−su(x′,t)=u(x,t)−u(x′,t)−(x−x′)σ2.s_u(x,t)-s_u(x ,t)= u(x,t)-u(x ,t)-(x-x )σ^2. Using ‖x−x′‖≤d‖x−x′‖∞\|x-x \|≤ d\,\|x-x \|_∞ gives Equation (61); the time bound follows because the affine term −x/σ2-x/σ^2 is independent of time. The score itself is not uniformly bounded because of this affine term, which is why the subsequent loss analysis uses its exact cancellation with the denoising target. ∎ Lemma 20 (Uniform Bounds and Lipschitz Properties of the Denoising Score Loss). For every u∈nu _n, yt∈[−1,1]dy_t∈[-1,1]^d, t∈[0,1]t∈[0,1], and g∈ℝdg ^d, 0≤‖su(yt+σg,t)+gσ‖2 0≤ \|s_u(y_t+σ g,t)+ gσ \|^2 ≤4dσ4. ≤ 4dσ^4. (63) For every yt,yt′∈[−1,1]dy_t,y_t ∈[-1,1]^d, using the same g in both evaluation points, |‖su(yt+σg,t)+gσ‖2−‖su(yt′+σg,t)+gσ‖2| | \|s_u(y_t+σ g,t)+ gσ \|^2- \|s_u(y_t +σ g,t)+ gσ \|^2 | ≤4dσ4(1+5(d+1)ΛuR)‖yt−yt′‖∞. ≤ 4dσ^4 (1+ 5(d+1) _uR )\|y_t-y_t \|_∞. (64) Finally, for every u,u~∈nu, u _n, |‖su(yt+σg,t)+gσ‖2−‖su~(yt+σg,t)+gσ‖2| | \|s_u(y_t+σ g,t)+ gσ \|^2- \|s_ u(y_t+σ g,t)+ gσ \|^2 | ≤4dσ4‖u(yt+σg,t)−u~(yt+σg,t)‖. ≤ 4 dσ^4 \|u(y_t+σ g,t)- u(y_t+σ g,t) \|. (65) Proof. By su(x,t)=u(x,t)−x/σ2s_u(x,t)=\u(x,t)-x\/σ^2, and because the same g appears in the evaluation point yt+σgy_t+σ g and in the denoising target g/σg/σ, su(yt+σg,t)+gσ s_u(y_t+σ g,t)+ gσ =u(yt+σg,t)−(yt+σg)σ2+gσ=u(yt+σg,t)−ytσ2. = u(y_t+σ g,t)-(y_t+σ g)σ^2+ gσ= u(y_t+σ g,t)-y_tσ^2. Since u(yt+σg,t)∈[−1,1]du(y_t+σ g,t)∈[-1,1]^d and yt∈[−1,1]dy_t∈[-1,1]^d, ‖su(yt+σg,t)+gσ‖ \|s_u(y_t+σ g,t)+ gσ \| =‖u(yt+σg,t)−yt‖σ2≤2dσ2. = \|u(y_t+σ g,t)-y_t\|σ^2≤ 2 dσ^2. Squaring proves Equation (63). For yt,yt′∈[−1,1]dy_t,y_t ∈[-1,1]^d, applying the cancellation identity at each evaluation point gives su(yt+σg,t)+gσ s_u(y_t+σ g,t)+ gσ =u(yt+σg,t)−ytσ2,su(yt′+σg,t)+gσ=u(yt′+σg,t)−yt′σ2. = u(y_t+σ g,t)-y_tσ^2, s_u(y_t +σ g,t)+ gσ= u(y_t +σ g,t)-y_t σ^2. Using |‖a‖2−‖b‖2|≤(‖a‖+‖b‖)‖a−b‖ |\|a\|^2-\|b\|^2 |≤(\|a\|+\|b\|)\|a-b\|, |‖su(yt+σg,t)+gσ‖2−‖su(yt′+σg,t)+gσ‖2| | \|s_u(y_t+σ g,t)+ gσ \|^2- \|s_u(y_t +σ g,t)+ gσ \|^2 | ≤4dσ4[‖u(yt+σg,t)−u(yt′+σg,t)‖+‖yt−yt′‖] ≤ 4 dσ^4 [ \|u(y_t+σ g,t)-u(y_t +σ g,t) \|+\|y_t-y_t \| ] ≤4dσ4(γx,R,εu+d)‖yt−yt′‖∞ ≤ 4 dσ^4 ( _x,R, ^u+ d )\|y_t-y_t \|_∞ =4dσ4(1+5(d+1)ΛuR)‖yt−yt′‖∞. = 4dσ^4 (1+ 5(d+1) _uR )\|y_t-y_t \|_∞. where note that ‖(yt+σg)−(yt′+σg)‖∞ \|(y_t+σ g)-(y_t +σ g)\|_∞ =‖yt−yt′‖∞. =\|y_t-y_t \|_∞. This proves Equation (64). Finally, for u,u~∈nu, u _n, su(yt+σg,t)+gσ s_u(y_t+σ g,t)+ gσ =u(yt+σg,t)−ytσ2,su~(yt+σg,t)+gσ=u~(yt+σg,t)−ytσ2. = u(y_t+σ g,t)-y_tσ^2, s_ u(y_t+σ g,t)+ gσ= u(y_t+σ g,t)-y_tσ^2. Therefore, |‖su(yt+σg,t)+gσ‖2−‖su~(yt+σg,t)+gσ‖2| | \|s_u(y_t+σ g,t)+ gσ \|^2- \|s_ u(y_t+σ g,t)+ gσ \|^2 | ≤4dσ4‖u(yt+σg,t)−u~(yt+σg,t)‖, ≤ 4 dσ^4 \|u(y_t+σ g,t)- u(y_t+σ g,t) \|, which proves Equation (65). ∎ Using the same g in the two evaluation points is valid in the later population comparison because both losses are integrated with respect to the same Gaussian measure. In particular, G[‖su(yt+σG,t)+Gσ‖2]−G[‖su(yt′+σG,t)+Gσ‖2] _G [ \|s_u(y_t+σ G,t)+ Gσ \|^2 ]-E_G [ \|s_u(y_t +σ G,t)+ Gσ \|^2 ] =G[‖su(yt+σG,t)+Gσ‖2−‖su(yt′+σG,t)+Gσ‖2]. =E_G [ \|s_u(y_t+σ G,t)+ Gσ \|^2- \|s_u(y_t +σ G,t)+ Gσ \|^2 ]. This is only a common-variable representation of two expectations under the same Gaussian law. It does not impose any additional probabilistic relationship between yty_t and yt′y_t . For each term, the same Gaussian G is used in its noisy evaluation point and its denoising target. I.2.3 Completion of the Score Bound Recall that for u∈nu _n, we defined Φuscore(y0,y1,k,g) _u^score(y_0,y_1,k,g) =‖su(y0+tk(y1−y0)+σg,tk)+gσ‖2. = \|s_u (y_0+t_k(y_1-y_0)+σ g,t_k )+ gσ \|^2. (66) We equip nU_n with du(u,u~)=supx∈[−R,R]d,t∈[0,1]‖u(x,t)−u~(x,t)‖.d_u(u, u)= _x∈[-R,R]^d,\;t∈[0,1]\|u(x,t)- u(x,t)\|. Lemma 21 (Constants for the score loss). The loss in Equation (66) satisfies Equation (20)–Equation (22) with Bscore B_score =4dσ4, = 4dσ^4, LY,score L_Y,score =4dσ4(1+5(d+1)ΛuR), = 4dσ^4 (1+ 5(d+1) _uR ), LΘ,score L_ ,score =4dσ4. = 4 dσ^4. (67) Moreover, log(r,n,du) (r,U_n,d_u) ≤ℭu(r). ≤ C_u(r). (68) Proof. Let St=(1−t)y0+ty1S_t=(1-t)y_0+ty_1. The cancellation su(St+σg,t)+gσ=u(St+σg,t)−Stσ2s_u(S_t+σ g,t)+ gσ= u(S_t+σ g,t)-S_tσ^2 and the bounds u(St+σg,t),St∈[−1,1]du(S_t+σ g,t),S_t∈[-1,1]^d give Φuscore≤4d/σ4 _u^score≤ 4d/σ^4. For a second endpoint pair, write St′=(1−t)y0′+ty1′S_t =(1-t)y_0 +ty_1 . The pointwise bound Equation (64) gives |Φuscore(y0,y1,k,g)−Φuscore(y0′,y1′,k,g)| | _u^score(y_0,y_1,k,g)- _u^score(y_0 ,y_1 ,k,g) | ≤4dσ4(1+5(d+1)ΛuR)‖Stk−Stk′‖∞ ≤ 4dσ^4 (1+ 5(d+1) _uR )\|S_t_k-S_t_k \|_∞ ≤LY,score(‖y0−y0′‖+‖y1−y1′‖). ≤ L_Y,score (\|y_0-y_0 \|+\|y_1-y_1 \| ). Averaging this inequality over K,GK,G proves the endpoint condition. The parameter perturbation bound Equation (65) gives |Φuscore(y0,y1,k,g)−Φu~score(y0,y1,k,g)|≤4dσ4‖u(Stk+σg,tk)−u~(Stk+σg,tk)‖. | _u^score(y_0,y_1,k,g)- _ u^score(y_0,y_1,k,g) |≤ 4 dσ^4\|u(S_t_k+σ g,t_k)- u(S_t_k+σ g,t_k)\|. Every u∈nu _n depends on its spatial argument through clipRclip_R, so the last norm is at most du(u,u~)d_u(u, u). Finally, Equation (68) follows from Equation (56). ∎ Proof of the score bound in Proposition 1. Choose ε=n−1/(d+5),R=1+2σ[log(8d2σ4ε2)]1/2. =n^-1/(d+5), R=1+ 2\,σ [ ( 8d^2σ^4 ^2 ) ]^1/2. The tail condition in Proposition 7 then holds. Write s^M=su^M s_M=s_ u_M for the empirical minimizer, with u^M∈n u_M _n. By Equation (45) and the approximation proposition, infu∈nLa,Qscore(su)−La,Qscore(sQ⋆) _u _n \L_a,Q^score(s_u)-L_a,Q^score(s_Q ) \ ≤2ε2. ≤ 2 ^2. (69) For Φuscore _u^score, La,QΦ(u)=La,Qscore(su)L_a,Q (u)=L_a,Q^score(s_u), and L^actualΦ(u) L_actual (u) is the implemented score objective. Lemma 21 verifies the hypotheses of Proposition 5 with Θ=n,dΘ=du,BΦ=Bscore,LY=LY,score,LΘ=LΘ,score. =U_n, _ =d_u, B_ =B_score, L_Y=L_Y,score, L_ =L_ ,score. For the selected R, ΛuR _uR =2R+8d(2R+1)Rσ2=O(σ−2), = 2R+ 8d(2R+1)Rσ^2=O(σ^-2), Λu _u =O~(σ−2),ηu−1=dΛuσ2ε=O~(1σ4ε). = O(σ^-2), _u^-1= d\, _uσ^2 = O ( 1σ^4 ). (70) Substituting the architecture orders Equation (54) into ℭu C_u, at the three radii in Proposition 5, yields logNDRscore+logNobsscore+logNauxscore N_DR^score+ N_obs^score+ N_aux^score =O~[(1σ4ε)d+1]. = O [ ( 1σ^4 )^d+1 ]. (71) Since Bscore=4d/σ4B_score=4d/σ^4, combining Equation (23) and Equation (71), with δ=n−1δ=n^-1, give ℜscore(n−1) R_score(n^-1) ≲1+2πmin−1σ2d+6n−2/(d+5). 1+2 _ ^-1σ^2d+6n^-2/(d+5). (72) Before the substitution ε=n−1/(d+5) =n^-1/(d+5), the leading term is O~((1+2πmin−1)σ−(2d+6)ε−(d+1)/2n−1/2). O ((1+2 _ ^-1)σ^-(2d+6) ^-(d+1)/2n^-1/2 ). The bounds Equation (67) and Equation (70) give LY,score L_Y,score =O(σ−6). =O(σ^-6). (73) On the event in Corollary 3, ΔQ≲εOT−1[n−1/(d+2)+εpred,train+(nMQ)−1/(d+2)]. _Q _OT^-1 [n^-1/(d+2)+ _pred,train+(nM_Q)^-1/(d+2) ]. Substitute this display, Equation (69), Equation (72), and Equation (73) into Equation (24). The inequalities (nMQ)−1/(d+2)≤n−1/(d+2)(nM_Q)^-1/(d+2)≤ n^-1/(d+2) and 0<σ≤10<σ≤ 1 give La,Qscore(s^M)−La,Qscore(sQ⋆) L_a,Q^score( s_M)-L_a,Q^score(s_Q ) ≲εOT−1σ6[n−1/(d+2)+εpred,train] _OT^-1σ^6 [n^-1/(d+2)+ _pred,train ] +1+εOT−1σ6εαεpred+1+2πmin−1σ2d+6n−2/(d+5). + 1+ _OT^-1σ^6 _α _pred+ 1+2 _ ^-1σ^2d+6n^-2/(d+5). Finally, Equation (42) converts this excess risk into the grid-averaged score error in Equation (5). The coupling and conditional statistical events fail with probability at most 2/n2/n and 1/n1/n, respectively, while ℙ(ℰac)≤e−nπminP(E_a^c)≤ e^-n _ . A final union bound with the velocity event proves that both conclusions of Proposition 1 hold with probability 1−(n−1)−e−nπmin1-O(n^-1)-e^-n _ . ∎ Appendix J Proof of Theorem 2 Proof. Work on the event where the population coupling Q is defined and the two bounds in Proposition 1 hold. Since D5≤D¯5<∞D_5≤ D_5<∞, Lemma 1 supplies a weak solution Z satisfying Zt∼qtZ_t q_t for every t∈[0,1]t∈[0,1]. Moreover, the grid-frozen learned drifts are continuous by continuity of the fitted neural networks. Thus the standard sampling conditions hold. Since h=1/Nh=1/N, each grid-averaged field bound in Proposition 1 is exactly the corresponding integrated estimation error in Assumption 1. Thus that proposition may be substituted into Theorem 1. In the stated regime, the velocity part of Proposition 1 gives εvel2 _vel^2 =O~(2+εOT−1σ2n−1/(d+2)+1+2πmin−1σd+1n−2/(d+5)), = O ( 2+ _OT^-1σ^2n^-1/(d+2)+ 1+2 _ ^-1σ^d+1n^-2/(d+5) ), while its score part gives εscore2 _score^2 =O~(1σ6εOTn−1/(d+2)+1+2πmin−1σ2d+6n−2/(d+5)). = O ( 1σ^6 _OTn^-1/(d+2)+ 1+2 _ ^-1σ^2d+6n^-2/(d+5) ). Since τ2=σD5¯/dτ^2=σ D_5/d, εvel2τ2 _vel^2τ^2 =O~(d(2+εOT−1)σ3D5¯n−1/(d+2)+d(1+2πmin−1)σd+2D5¯n−2/(d+5)), = O ( d (2+ _OT^-1 )σ^3 D_5n^-1/(d+2)+ d (1+2 _ ^-1 )σ^d+2 D_5n^-2/(d+5) ), (74) and τ2εscore2 τ^2 _score^2 =O~(D5¯dεOTσ5n−1/(d+2)+(1+2πmin−1)D5¯dσ2d+5n−2/(d+5)). = O ( D_5d _OTσ^5n^-1/(d+2)+ (1+2 _ ^-1 ) D_5dσ^2d+5n^-2/(d+5) ). (75) By Corollary 1, under hτ22σ2≤1 hτ^22σ^2≤ 1 and σ≤1σ≤ 1, W1(∫ℙY∣C=c,A=adℙC(c),ℒ(Z^N)) W_1 ( _Y C=c,A=a\,dP_C(c),L( Z_N) ) ≲σ+KL(ℒ(Y1+σG)∥ℒ(Z^N))1/4. σ+KL (L(Y_1+σ G) \|L( Z_N) )^1/4. Taking σ=Θ(εtarget)σ= ( _target), it is sufficient to require KL(ℒ(Y1+σG)∥ℒ(Z^N)) (L(Y_1+σ G) \|L( Z_N) ) ≲εtarget4. _target^4. To make each estimation contribution in Equation (74) and Equation (75) at most of order εtarget4 _target^4, it is sufficient that n n ≳[(2+εOT−1)D5¯]d+2εtarget−7(d+2),n≳[(1+2πmin−1)D5¯](d+5)/2εtarget−(d+6)(d+5)/2, [ (2+ _OT^-1 ) D_5 ]^d+2 _target^-7(d+2), n [ (1+2 _ ^-1 ) D_5 ]^(d+5)/2 _target^-(d+6)(d+5)/2, n n ≳[D5¯εOT]d+2εtarget−9(d+2),n≳[(1+2πmin−1)D5¯](d+5)/2εtarget−(2d+9)(d+5)/2, [ D_5 _OT ]^d+2 _target^-9(d+2), n [ (1+2 _ ^-1 ) D_5 ]^(d+5)/2 _target^-(2d+9)(d+5)/2, where σ=Θ(εtarget)σ= ( _target) has been substituted. Note that the last condition determines the asymptotic dependence on εtarget _target. It remains to control the discretization terms in Theorem 1. The choice h≤ch(dD5¯2εtarget4σ2+dD5¯σ)−1 h≤ c_h ( d D_5^2 _target^4σ^2+ d D_5σ )^-1 implies hdD5¯2σ2≲εtarget4 hd D_5^2σ^2 _target^4 and hdD5¯σ≲1 hd D_5σ 1. Consequently, KL(ℒ(ZtN)∥ℒ(Z^N))≲εtarget4KL\! (L(Z_t_N) \|L( Z_N) ) _target^4. Moreover, hτ22σ2 hτ^22σ^2 =hD5¯2dσ≤12d2hdD5¯σ=hD5¯2dσ≤chD5¯2dσdD5¯2εtarget4σ2+dD5¯σ≤ch2d2≤1, = h D_52dσ≤ 12d^2 hd D_5σ= h D_52dσ≤ c_h D_52dσ d D_5^2 _target^4σ^2+ d D_5σ≤ c_h2d^2≤ 1, so the required stability condition holds. ∎