Primal-Dual Flow Matching for Sample-Wise Constrained Generation

# Primal-Dual Flow Matching for Sample-Wise Constrained Generation
Source: [https://arxiv.org/html/2610.12699](https://arxiv.org/html/2610.12699)
Zhengyan HuanAffiliation:Department of Electrical and Computer Engineering, Tufts UniversityAffiliation:NSF AI Institute for Artificial Intelligence and Fundamental InteractionsEmail:[Zhengyan\.Huan@tufts\.edu](mailto:[email protected])Peter Y\. LuAffiliation:Department of Electrical and Computer Engineering, Tufts UniversityAffiliation:NSF AI Institute for Artificial Intelligence and Fundamental InteractionsEmail:[Peter\.Lu@tufts\.edu](mailto:[email protected])Shuchin AeronAffiliation:Department of Electrical and Computer Engineering, Tufts UniversityAffiliation:NSF AI Institute for Artificial Intelligence and Fundamental InteractionsEmail:[Shuchin\.Aeron@tufts\.edu](mailto:[email protected])

###### Abstract

We consider the problem of learning to generate data while satisfying a prescribed*sample\-wise*constraint with a user\-specified probability of constraint satisfaction\. Starting with a population\-level constrained negative log\-likelihood minimization and using a flow\-matching \(FM\) parameterization, under regularity assumptions and sufficient expressivity of the velocity field family, we derive guarantees on the probability of constraint satisfaction for the corresponding primal\-dual optimization algorithm\. We provide a practical implementation of this approach, called Primal\-Dual Flow Matching \(PDFM\), which has direct control over constraint satisfaction rate and requires only binary membership oracle feedback for constraint satisfaction, avoiding the need for differentiable distances, projections, or special structure of the constraint set, such as convexity\. We show that PDFM improves constraint satisfaction over vanilla FM and achieves competitive performance against constraint\-aware baselines while maintaining distributional fidelity\. Our code is available at[https://github\.com/ZhengyanHuan/PDFM](https://github.com/ZhengyanHuan/PDFM)\.

## 1Introduction

The utility of generative models for scientific applications highly depends on their ability to generate samples that obey scientific laws or satisfy an application\-specific criterion, modeled via constraints that can be validated numerically\. For example, flow\- and diffusion\-based generative models are increasingly used to emulate chaotic spatiotemporal systems\([Lienen et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib24);[Du et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib30);[Price et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib27)\), where the chaotic dynamics result in high sensitivity to initial conditions and therefore unpredictable trajectories that can be interpreted and modeled probabilistically\([Medio and Lines, 2001](https://arxiv.org/html/2610.12699#bib.bib31)\)\. However, in these settings, distributional fidelity alone is rarely sufficient: a generated sample is useful only if it is also*physically consistent*, e\.g\., satisfying boundary conditions, respecting conservation laws, or following a governing PDE\([Beucler et al\., 2021](https://arxiv.org/html/2610.12699#bib.bib25);[Utkarsh et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib10)\)\. Another important example comes from molecular generation\([Hoogeboom et al\., 2022](https://arxiv.org/html/2610.12699#bib.bib34);[Song et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib32)\), where one wants to generate new molecules such that they conform to structures with chemically useful properties\([Bickerton et al\., 2012](https://arxiv.org/html/2610.12699#bib.bib35);[Giaginis et al\., 2018](https://arxiv.org/html/2610.12699#bib.bib36)\)that are numerically quantifiable\.

#### Related work:

For constrained generation, one may divide existing literature into \(a\) Type of constraints, e\.g\., distributional constraints\([Khalafi et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib21)\)or sample\-wise constraints \[this work\]\. Among sample\-wise constraints, one may have constraints with topological specification, such as convexity\([Liu et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib23)\); constraints with known boundaries\([Lou and Ermon, 2023](https://arxiv.org/html/2610.12699#bib.bib5);[Fishman et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib3)\); constraints with differentiable distances to them\([Feng et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib12);[Utkarsh et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib10)\); or constraints that are known only via a membership oracle\([Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\); and \(b\) Methods used – whether it is inference\-time\([Christopher et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib9)\)\([Utkarsh et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib10)\)\([Li et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib8)\)\([Zhu et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib6)\)or training\-time \[this work\],\([Lou and Ermon, 2023](https://arxiv.org/html/2610.12699#bib.bib5);[Fishman et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib3)\)\([Xie et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib13)\)\([Feng et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib12)\)\([Giannone et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib11)\)\([Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\)\. A more detailed review of these related works is provided in Appendix[A](https://arxiv.org/html/2610.12699#A1)and throughout the exposition we will highlight other closely related works\.

For sample\-wise constrained generation when the constraints are accessible via a membership oracle, we make the followingmain contributions:

1. 1\.We develop a theoretically grounded approach, with provable guarantees on the probability of constraint satisfaction\. In particular, leveraging an FM parameterization of a constrained negative log\-likelihood minimization \(Sections[2](https://arxiv.org/html/2610.12699#S2),[3](https://arxiv.org/html/2610.12699#S3)\), in Theorem[3\.5](https://arxiv.org/html/2610.12699#S3.Thmtheorem5)we show that under mild assumptions the Total Variation \(TV\) norm between the optimal solution and the FM parameterized solution can be controlled in terms of approximation error between the flows due to parametric modeling, which in turn leads to overall control on constraint satisfaction\. We then show that a*theoretical*Primal\-Dual Flow Matching \(PDFM\) Alg\.[1](https://arxiv.org/html/2610.12699#alg1)achieves this bound\.
2. 2\.We then propose a practical implementation of PDFM in Alg\.[2](https://arxiv.org/html/2610.12699#alg2), with theoretical guarantees discussed in Sec\.[4\.3](https://arxiv.org/html/2610.12699#S4.SS3)\. Key innovations in this algorithm are motivated by deterministic policy gradient methods, where we build differentiable surrogates for constraint satisfaction probability under the FM parameterization\. This enables the method to work with*only membership oracle access to the constraint set\.*We note that this type of access to the constraint is the most flexible when the constraints are complex\.
3. 3\.We provide an extensive evaluation of PDFM and show that compared to existing methods, it achieves competitive constraint satisfaction rates and distribution fidelity\. For scientific applications, we consider the problem of learning to generate PDE\-constrained spatiotemporal dynamics of chaotic systems and show that our method achieves superior performance while requiring only binary feedback\. In a molecule generation task, PDFM improves molecule stability and distributional fidelity over FM over FM, without task\-specific designs\.

## 2Constrained Generation Setup

All notations used are summarized in Table[3](https://arxiv.org/html/2610.12699#A2.T3)in Appendix[B](https://arxiv.org/html/2610.12699#A2)for reference\. LetQQbe the training\-data distribution onℝd\\mathbb\{R\}^\{d\}with densityqqand letPPbe the distribution onℝd\\mathbb\{R\}^\{d\}to be modeled, with the corresponding densitypp\. Let𝒞⊂ℝd\\mathcal\{C\}\\subset\\mathbb\{R\}^\{d\}be a closed, application\-specific constraint set, for example, the set of particle trajectories that satisfy a prescribed conservation\-law tolerance\. We denote𝟏𝒞​\(x\)\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)by the indicator function of the constraint set𝒞\\mathcal\{C\}that takes the value11ifx∈𝒞x\\in\\mathcal\{C\}and00otherwise\. Given a target constraint satisfaction rateb∈\(0,1\)b\\in\(0,1\), consider the following population\-level problem:

minimizeP𝔼x∼Q​\[−log⁡p⁡\(x\)\]​s\.t\.b−∫𝒞p⁡\(x\)​𝑑x≤0\.\\operatorname\*\{minimize\}\_\{P\}~~\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\(x\)\]~~\\mathrm\{s\.t\.\}~~b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\leq 0\.\(OPT\)\([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\) requires that under the modeled distributionPP, the probability of constraint satisfaction is at leastbb\. In other words𝔼x∼P​\[𝟏𝒞​\(x\)\]=∫𝒞p⁡\(x\)​𝑑x≥b\\mathbb\{E\}\_\{x\\sim P\}\[\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\)\]=\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\geq b\. Assume additionally that𝔼x∼Q​\[log⁡q​\(x\)\]\\mathbb\{E\}\_\{x\\sim Q\}\[\\log q\(x\)\]is finite, the problem can equivalently be restated as:

minimizePKL\(Q\|\|P\)s\.t\.b−∫𝒞p\(x\)dx≤0,\\operatorname\*\{minimize\}\_\{P\}~~\\mathrm\{KL\}\(Q\|\|P\)~~\\mathrm\{s\.t\.\}~~b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\leq 0,\(1\)whereKL\(Q\|\|P\)=𝔼x∼Q\[logq⁡\(x\)p⁡\(x\)\]\\mathrm\{KL\}\(Q\|\|P\)=\\mathbb\{E\}\_\{x\\sim Q\}\[\\log\\frac\{q\(x\)\}\{p\(x\)\}\]\.

DefineQ⁡\(𝒞\):=∫𝒞q⁡\(x\)​𝑑x,P⁡\(𝒞\):=∫𝒞p⁡\(x\)​𝑑xQ\(\\mathcal\{C\}\):=\\int\_\{\\mathcal\{C\}\}q\(x\)dx,P\(\\mathcal\{C\}\):=\\int\_\{\\mathcal\{C\}\}p\(x\)dx\. Let us consider two regimes: 1\.0<Q⁡\(𝒞\)<b0<Q\(\\mathcal\{C\}\)<b, and 2\.b≤Q⁡\(𝒞\)≤1b\\leq Q\(\\mathcal\{C\}\)\\leq 1\. For the first regime, a quick observation shows that the solutionP=QP=Qis infeasible due to the constraint\. We additionally note that at the boundaryb=1b=1: ifQ⁡\(𝒞\)<1Q\(\\mathcal\{C\}\)<1, then every feasiblePPmust satisfyP⁡\(𝒞\)=1P\(\\mathcal\{C\}\)=1andP⁡\(𝒞c\)=0P\(\\mathcal\{C\}^\{c\}\)=0, whileQ⁡\(𝒞c\)\>0Q\(\\mathcal\{C\}^\{c\}\)\>0leads to infinite objective value\. For the second regime, even thoughP=QP=Qis the optimal solution, this problem remains practically non\-trivial since the learned generative model may still violate the constraint due to the finite model capacity and approximation via limited samples\.

One can immediately note that[OPT](https://arxiv.org/html/2610.12699#S2.Ex1)is a convex optimization problem\. Introducing a multiplierλ≥0\\lambda\\geq 0, the Lagrangian is given by

L⁡\(p,λ\)=𝔼x∼Q​\[−log⁡p⁡\(x\)\]\+λ⁡\(b−∫𝒞p⁡\(x\)​𝑑x\)\.L\(p,\\lambda\)=\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\(x\)\]\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\right\)\.\(2\)
###### Assumption 2\.1\.

Assume that0<Q⁡\(𝒞\)<b<10<Q\(\\mathcal\{C\}\)<b<1, that there exists a strictly feasible distributionPPsatisfyingb−∫𝒞p⁡\(x\)​𝑑x<0b\-\\int\_\{\\mathcal\{C\}\}p\(x\)\\,dx<0, and that, for everyλ≥0\\lambda\\geq 0considered in the analysis, the minimum ofL⁡\(⋅,λ\)L\(\\cdot,\\lambda\)is attained\. We denote the corresponding minimizer byp\(λ\)∈arg⁡minp⁡L⁡\(p,λ\)p^\{\(\\lambda\)\}\\in\\arg\\min\_\{p\}L\(p,\\lambda\), and letP\(λ\)P^\{\(\\lambda\)\}be the distribution with densityp\(λ\)p^\{\(\\lambda\)\}\.

###### Lemma 2\.2\.

Under Assumption[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),

p\(λ∗\)​\(x\)=b​q​\(x\)Q⁡\(𝒞\)​𝟏𝒞​\(x\)\+\(1−b\)​q​\(x\)1−Q⁡\(𝒞\)​𝟏𝒞c​\(x\),λ∗=1−Q⁡\(𝒞\)1−b−Q⁡\(𝒞\)b\.p^\{\(\\lambda^\{\*\}\)\}\(x\)=\\frac\{bq\(x\)\}\{Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{\(1\-b\)q\(x\)\}\{1\-Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\),~~\\lambda^\{\*\}=\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{1\-b\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{b\}\.\(3\)

This lemma, which will be proved in Appendix[C](https://arxiv.org/html/2610.12699#A3), states that the optimal population\-level solution corresponds to the re\-weighting of the density ofQQinside and outside of𝒞\\mathcal\{C\}\.

In the regime0<Q⁡\(𝒞\)<b<10<Q\(\\mathcal\{C\}\)<b<1, a simple alternative to enforce constraint satisfaction is to discard infeasible samples and train on the truncated distributionQ′=Q\(⋅∣𝒞\)Q^\{\\prime\}=Q\(\\cdot\\mid\\mathcal\{C\}\), whose density isq′​\(x\)=q⁡\(x\)​𝟏𝒞​\(x\)Q⁡\(𝒞\)\.q^\{\\prime\}\(x\)=\\frac\{q\(x\)\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\)\}\{Q\(\\mathcal\{C\}\)\}\.However, practically, especially when the training data are noisy, this rejection step can substantially reduce the effective sample size and degrade the training distribution\. Moreover, even with sufficient truncated samples, finite\-capacity generative models may still fail to achieve the target constraint satisfaction rate\. This will be shown in Sec\.[5\.1](https://arxiv.org/html/2610.12699#S5.SS1)and[5\.3](https://arxiv.org/html/2610.12699#S5.SS3)\.

### 2\.1Problem Regimes and Learning Objective

We can now state our problem regimes and the learning objective\. Letx\(j\)​∼i\.i\.d\.​Qx^\{\(j\)\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}Q,j∈\[n\]j\\in\[n\], be the training samples, and letb∈\(0,1\)b\\in\(0,1\)be the target constraint satisfaction rate\. When0<Q⁡\(𝒞\)<b<10<Q\(\\mathcal\{C\}\)<b<1, the goal is to train an FM model whose generated distribution approximates the optimal constrained distributionP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}characterized in Lemma[2\.2](https://arxiv.org/html/2610.12699#S2.Thmtheorem2)with the desired constraint satisfaction rate\. This regime is studied both theoretically and empirically\. When0<b≤Q⁡\(𝒞\)≤10<b\\leq Q\(\\mathcal\{C\}\)\\leq 1, the distribution\-level optimum isQQ, and therefore the objective is to train an FM model that matchesQQwhile achieving the desired constraint satisfaction rate, and we study this setting empirically\. The constraint set𝒞\\mathcal\{C\}is specified only through a membership oracle: given a sample, the oracle determines whether it belongs to𝒞\\mathcal\{C\}\. Differentiable distances, projections, and structural properties such as convexity are unavailable\.

## 3Constraint Guarantees with FM Parameterization

\([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\) yields a convex distribution\-level problem with a unique optimizer given by Lemma[2\.2](https://arxiv.org/html/2610.12699#S2.Thmtheorem2)\. In practical generative modeling through FM, the distribution is represented with a velocity field, whose approximation error controls KL divergence through Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1)\. The development in this section analyzes the total variation \(TV\) between the FM\-parameterized solution and the distribution\-level optimum\. The following Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1), proved in Appendix[C](https://arxiv.org/html/2610.12699#A3), translates TV into a bound on the constraint satisfaction rate\.

###### Lemma 3\.1\.

LetP1,P2P\_\{1\},P\_\{2\}be probability measures defined on the same measurable space\. If the TV distance betweenP1P\_\{1\}andP2P\_\{2\}is given as‖P1−P2‖TV=δ\\\|P\_\{1\}\-P\_\{2\}\\\|\_\{\\mathrm\{TV\}\}=\\delta, then

max⁡\{0,P2​\(𝒞\)−δ\}≤P1​\(𝒞\)≤min⁡\{1,P2​\(𝒞\)\+δ\}\.\\max\\\{0,P\_\{2\}\(\\mathcal\{C\}\)\-\\delta\\\}\\leq P\_\{1\}\(\\mathcal\{C\}\)\\leq\\min\\\{1,P\_\{2\}\(\\mathcal\{C\}\)\+\\delta\\\}\.\(4\)

Letθ∈Θ\\theta\\in\\Thetadenote the parameters of the FM model, and letuθ:ℝd×\[0,1\]→ℝdu\_\{\\theta\}:\\mathbb\{R\}^\{d\}\\times\[0,1\]\\to\\mathbb\{R\}^\{d\}be the corresponding parameterized velocity field\. The induced velocity field class is defined asUΘ:=\{uθ:θ∈Θ\}U\_\{\\Theta\}:=\\\{u\_\{\\theta\}:\\theta\\in\\Theta\\\}\. Given an initial distributionP0P\_\{0\}, the velocity fielduθu\_\{\\theta\}defines a flow mapΦuθ,t\\Phi\_\{u\_\{\\theta\},t\}, leading to the generated distributionPuθP\_\{u\_\{\\theta\}\},

Φuθ,t​\(x\)=x\+∫0tuθ​\(Φuθ,s​\(x\),s\)​𝑑s,Puθ:=\(Φuθ,1\)\#​P0,\\Phi\_\{u\_\{\\theta\},t\}\(x\)=x\+\\int\_\{0\}^\{t\}u\_\{\\theta\}\(\\Phi\_\{u\_\{\\theta\},s\}\(x\),s\)ds,\\quad P\_\{u\_\{\\theta\}\}:=\(\\Phi\_\{u\_\{\\theta\},1\}\)\_\{\\\#\}P\_\{0\},\(5\)where for any mapTT,T⁡\(x\)T\(x\)denotes the transported sample starting fromxx, while\(T\)\#​P0\(T\)\_\{\\\#\}P\_\{0\}denotes the pushforward distribution obtained by transportingP0P\_\{0\}throughTT, i\.e\., ifx0∼P0x\_\{0\}\\sim P\_\{0\}thenΦuθ,t​\(x0\)∼\(Φuθ,t\)\#​P0\\Phi\_\{u\_\{\\theta\},t\}\(x\_\{0\}\)\\sim\(\\Phi\_\{u\_\{\\theta\},t\}\)\_\{\\\#\}P\_\{0\}\. The FM\-parameterized Lagrangian is defined as

L¯​\(uθ,λ\):=L⁡\(puθ,λ\)=𝔼x∼Q​\[−log⁡puθ​\(x\)\]\+λ⁡\(b−∫𝒞puθ​\(x\)​𝑑x\)\.\\bar\{L\}\(u\_\{\\theta\},\\lambda\):=L\(p\_\{u\_\{\\theta\}\},\\lambda\)=\\mathbb\{E\}\_\{x\\sim Q\}\\big\[\-\\log p\_\{u\_\{\\theta\}\}\(x\)\\big\]\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\_\{u\_\{\\theta\}\}\(x\)\\,dx\\right\)\.\(6\)The corresponding parameterized dual function isg¯​\(λ\):=infu∈UΘL¯​\(u,λ\)\\bar\{g\}\(\\lambda\):=\\inf\_\{u\\in U\_\{\\Theta\}\}\\bar\{L\}\(u,\\lambda\), and we denote an optimal dual variable byλ¯∗∈arg⁡maxλ≥0​g¯​\(λ\)\\bar\{\\lambda\}^\{\*\}\\in\\arg\\max\_\{\\lambda\\geq 0\}\\bar\{g\}\(\\lambda\)\. For each fixedλ≥0\\lambda\\geq 0, letuθ\(λ\)∈arg⁡minu∈UΘ​L¯​\(u,λ\)u\_\{\\theta\}^\{\(\\lambda\)\}\\in\\arg\\min\_\{u\\in U\_\{\\Theta\}\}\\bar\{L\}\(u,\\lambda\)denote a minimized parameterized velocity field, and letPuθ\(λ\):=\(Φuθ\(λ\),1\)\#​P0P\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}:=\(\\Phi\_\{u\_\{\\theta\}^\{\(\\lambda\)\},1\}\)\_\{\\\#\}P\_\{0\}denote the distribution induced byuθ\(λ\)u\_\{\\theta\}^\{\(\\lambda\)\}, with densitypuθ\(λ\)p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\.L¯​\(uθ,λ\)\\bar\{L\}\(u\_\{\\theta\},\\lambda\)is generally nonconvex in bothθ\\thetaanduθu\_\{\\theta\}\.

Defining the velocity fieldu\(λ\)u^\{\(\\lambda\)\}forP\(λ\)P^\{\(\\lambda\)\}in Assumption[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1)requires specifying both a coupling and an interpolation strategy; although many choices are possible, we consider the following construction\. Letπ\(λ\)\\pi^\{\(\\lambda\)\}be a coupling between the initial distributionP0P\_\{0\}andP\(λ\)P^\{\(\\lambda\)\}\. For\(x0,x1\)∼π\(λ\)\(x\_\{0\},x\_\{1\}\)\\sim\\pi^\{\(\\lambda\)\}, define the linear interpolantxt=\(1−t\)​x0\+t​x1,t∈\[0,1\]x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\},t\\in\[0,1\]\. The corresponding true velocity field is defined byu\(λ\)​\(x,t\):=𝔼\(x0,x1\)∼π\(λ\)​\[x1−x0∣xt=x\]u^\{\(\\lambda\)\}\(x,t\):=\\mathbb\{E\}\_\{\(x\_\{0\},x\_\{1\}\)\\sim\\pi^\{\(\\lambda\)\}\}\\left\[x\_\{1\}\-x\_\{0\}\\mid x\_\{t\}=x\\right\]\.

###### Assumption 3\.2\.

Assume that, for every dual variableλ\\lambdaconsidered in the analysis, the target probability path, the corresponding true velocity fieldu\(λ\)u^\{\(\\lambda\)\}, and the parameterized velocity fielduθ\(λ\)u\_\{\\theta\}^\{\(\\lambda\)\}satisfy the regularity conditions of Assumptions 3\.2 and 3\.3 in[Su et al\. \(2025\)](https://arxiv.org/html/2610.12699#bib.bib15): the target path has uniformly bounded score and score Jacobian, and the true and parameterized velocity fields are sufficiently smooth with bounded velocity norms, Jacobians, divergence discrepancy, and gradients of divergence\. In addition, the parameter familyΘ\\Thetais sufficiently rich so that the FM parameterization error is bounded byν\>0\\nu\>0: for everyλ\\lambdaconsidered in the analysis,

𝔼t,xt​\[‖uθ\(λ\)​\(xt,t\)−u\(λ\)​\(xt,t\)‖22\]≤ν2\.\\displaystyle\\mathbb\{E\}\_\{t,x\_\{t\}\}\\\!\\left\[\\\|u^\{\(\\lambda\)\}\_\{\\theta\}\(x\_\{t\},t\)\-u^\{\(\\lambda\)\}\(x\_\{t\},t\)\\\|\_\{2\}^\{2\}\\right\]\\leq\\nu^\{2\}\.\(7\)

###### Assumption 3\.3\.

There exists a constantλmax<∞\\lambda\_\{\\max\}<\\inftysuch thatλ∗,λ¯∗∈\[0,λmax\]\\lambda^\{\*\},\\bar\{\\lambda\}^\{\*\}\\in\[0,\\lambda\_\{\\max\}\]\.

###### Assumption 3\.4\.

For everyλ\\lambdaconsidered and everyxxsuch thatq⁡\(x\)\>0q\(x\)\>0, bothpuθ\(λ\)​\(x\)\>0p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\>0andp\(λ\)​\(x\)\>0p^\{\(\\lambda\)\}\(x\)\>0hold\.

###### Theorem 3\.5\.

For problem \([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\) with FM parameterization, under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2),[3\.3](https://arxiv.org/html/2610.12699#S3.Thmtheorem3),[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4),

∥P\(λ¯∗\)uθ−P\(λ∗\)∥TV≤ℰpara,Puθ\(λ¯∗\)\(𝒞\)≥b−ℰpara,\\displaystyle\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\\theta\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{para\}\},\\quad P\_\{u\_\{\\theta\}\}^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\(\\mathcal\{C\}\)\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{para\}\},\(8\)ℰpara=1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​2k​μ​\(λ¯∗\)​\(A1​ν\+A2​ν2\)\+12​\(A1​ν\+A2​ν2\)\\mathcal\{E\}\_\{\\mathrm\{para\}\}=\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\\sqrt\{\\frac\{2\}\{k\}\\mu\(\\bar\{\\lambda\}^\{\*\}\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\+\\sqrt\{\\frac\{1\}\{2\}\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\(9\)in whichA1,A2\>0A\_\{1\},A\_\{2\}\>0are constants that will be specified in Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1),kkis the strong concavity constant ofg⁡\(λ\)g\(\\lambda\), which will be specified in Lemma[D\.3](https://arxiv.org/html/2610.12699#A4.Thmtheorem3), andμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\.

Theorem[3\.5](https://arxiv.org/html/2610.12699#S3.Thmtheorem5)shows that, provided the FM parameterization errorν\\nuis sufficiently small, the FM\-parameterized constrained solution induces a distributionPuθ\(λ¯∗\)P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\\theta\}\}that remains close in total variation to the distribution\-level optimal constrained distributionP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}\. This bound separates two main sources of error: the FM approximation error, controlled byA1​ν\+A2​ν2A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}, and the dual mismatch betweenλ¯∗\\bar\{\\lambda\}^\{\*\}andλ∗\\lambda^\{\*\}, which is implicitly induced by the FM approximation error and is controlled through the strong concavity constantkk\. Consequently,Puθ\(λ¯∗\)P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\\theta\}\}satisfies the target constraint ratebbup to an explicit error term that vanishes asν\\nudecreases\. The proof of Theorem[3\.5](https://arxiv.org/html/2610.12699#S3.Thmtheorem5)is deferred to Appendix[D](https://arxiv.org/html/2610.12699#A4)\.

## 4Primal\-Dual Flow Matching \(PDFM\) for Constrained Generation

The analysis in the previous section assumes access to the optimal parameterized velocity fielduθ\(λ\)u\_\{\\theta\}^\{\(\\lambda\)\}for each fixed dual variableλ\\lambda\. In practice, however, both the velocity field and the dual variable must be learned through an implementable optimization procedure\. In this section, we first introduce a theoretical PDFM algorithm for solving[OPT](https://arxiv.org/html/2610.12699#S2.Ex1), then propose practical implementations of PDFM, and finally analyze the errors in the practical PDFM algorithm\.

### 4\.1Theoretical PDFM Algorithm

To solve the problem practically, we begin with an idealized primal\-dual procedure shown in Alg\.[1](https://arxiv.org/html/2610.12699#alg1)that serves as the theoretical template for PDFM\. Primal\-dual methods are a standard framework for constrained optimization\([Boyd and Vandenberghe, 2004](https://arxiv.org/html/2610.12699#bib.bib22)\)and have also been applied to generative modeling with distributional constraints, i\.e\., requiring the generated distribution to remain within a prescribed divergence from reference distributions\([Khalafi et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib21)\)\. At each iteration,

Input:

bb, learning rate

ηi\\eta\_\{i\}
λ⁡\(0\)←0\\lambda\(0\)\\leftarrow 0
for

i=0,1,⋯i=0,1,\\cdotsdo

uθ\(λ⁡\(i\)\)←arg⁡minuθ∈UΘ​L¯​\(uθ,λ⁡\(i\)\)u^\{\(\\lambda\(i\)\)\}\_\{\\theta\}\\leftarrow\\arg\\min\_\{u\_\{\\theta\}\\in U\_\{\\Theta\}\}\\bar\{L\}\(u\_\{\\theta\},\\lambda\(i\)\)
λ⁡\(i\+1\)←\[λ⁡\(i\)\+ηi​\(b−Puθ\(λ⁡\(i\)\)​\(𝒞\)\)\]\+\\lambda\(i\+1\)\\leftarrow\[\\lambda\(i\)\+\\eta\_\{i\}\(b\-P^\{\(\\lambda\(i\)\)\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\]\_\{\+\}
endfor

Algorithm 1Theoretical PDFMthe primal step aims to minimize the parameterized LagrangianL¯​\(uθ,λ\)\\bar\{L\}\(u\_\{\\theta\},\\lambda\)overUΘU\_\{\\Theta\}for the current dual variableλ\\lambda, while the dual step updatesλ\\lambdaaccording to∇λL¯​\(uθ,λ\)=b−Puθ​\(𝒞\)\\nabla\_\{\\lambda\}\\bar\{L\}\(u\_\{\\theta\},\\lambda\)=b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\), i\.e\., the constraint satisfaction gap\. SinceL¯​\(uθ,λ\)\\bar\{L\}\(u\_\{\\theta\},\\lambda\)is affine inλ\\lambdafor every fixeduθu\_\{\\theta\},g¯\\bar\{g\}is concave as the pointwise infimum of affine functions\. Suppose that the infimum definingg¯​\(λ\)\\bar\{g\}\(\\lambda\)is attained for allλ\\lambdaconsidered in the analysis and that the learning rateηi\\eta\_\{i\}satisfiesηi\>0\\eta\_\{i\}\>0,∑iηi=∞\\sum\_\{i\}\\eta\_\{i\}=\\infty, and∑iηi2<∞\\sum\_\{i\}\\eta\_\{i\}^\{2\}<\\infty\. And note that the supergradients∇λL¯​\(uθ\(λ\),λ\)=b−Puθ\(λ\)​\(𝒞\)\\nabla\_\{\\lambda\}\\bar\{L\}\(u^\{\(\\lambda\)\}\_\{\\theta\},\\lambda\)=b\-P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)are uniformly bounded in\[b−1,b\]\[b\-1,b\]\. The dual update is a projected supergradient ascent method for maximizing the concave functiong¯\\bar\{g\}\. Thus, under Alg\.[1](https://arxiv.org/html/2610.12699#alg1)with exact primal minimization and exact dual supergradient updates,λ\\lambdaconverges to someλ¯∗∈arg⁡maxλ≥0​g¯​\(λ\)\\bar\{\\lambda\}^\{\*\}\\in\\arg\\max\_\{\\lambda\\geq 0\}\\bar\{g\}\(\\lambda\)\. If, in addition, the induced primal solution atλ¯∗\\bar\{\\lambda\}^\{\*\}is unique and the primal solution correspondence is continuous atλ¯∗\\bar\{\\lambda\}^\{\*\}, then the corresponding primal distributions converge to the parameterized solutionPuθ\(λ¯∗\)P\_\{u\_\{\\theta\}\}^\{\(\\bar\{\\lambda\}^\{\*\}\)\}characterized in Sec\.[3](https://arxiv.org/html/2610.12699#S3)\.

### 4\.2Practical Implementation of PDFM

Input:

bb, parameter learning rate

η\(θ\),η\(ϕ\)\\eta^\{\(\\theta\)\},\\eta^\{\(\\phi\)\}, dual variable learning rate

ηi\\eta\_\{i\}
λ′​\(0\)←0\\lambda^\{\\prime\}\(0\)\\leftarrow 0
repeat

θ←θ−η\(θ\)​∇θ𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\\theta\\leftarrow\\theta\-\\eta^\{\(\\theta\)\}\\nabla\_\{\\theta\}\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]

untilconverged

repeat

ϕ←ϕ−η\(ϕ\)​gradϕ\\phi\\leftarrow\\phi\-\\eta^\{\(\\phi\)\}\\text\{grad\}\_\{\\phi\}
untilconverged

λ′​\(1\)←\[λ′​\(0\)\+η0​\(b−Puθ​\(𝒞\)\)\]\+\\lambda^\{\\prime\}\(1\)\\leftarrow\[\\lambda^\{\\prime\}\(0\)\+\\eta\_\{0\}\(b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\]\_\{\+\}
for

i=1,2,⋯i=1,2,\\cdotsdo

repeat

θ←θ−η\(θ\)​gradθ\\theta\\leftarrow\\theta\-\\eta^\{\(\\theta\)\}\\text\{grad\}\_\{\\theta\}
ϕ←ϕ−η\(ϕ\)​gradϕ\\phi\\leftarrow\\phi\-\\eta^\{\(\\phi\)\}\\text\{grad\}\_\{\\phi\}
untilconverged

λ′​\(i\+1\)←\[λ′​\(i\)\+ηi​\(b−Puθ​\(𝒞\)\)\]\+\\lambda^\{\\prime\}\(i\+1\)\\leftarrow\[\\lambda^\{\\prime\}\(i\)\+\\eta\_\{i\}\(b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\]\_\{\+\}
endfor

Note:

gradϕ,gradθ\\mathrm\{grad\}\_\{\\phi\},\\mathrm\{grad\}\_\{\\theta\}are given in \([17](https://arxiv.org/html/2610.12699#S4.E17)\), \([18](https://arxiv.org/html/2610.12699#S4.E18)\)\.

Algorithm 2PDFMFor implementation, we replace the likelihood objective𝔼x∼Q​\[−log⁡puθ​\(x\)\]\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\_\{u\_\{\\theta\}\}\(x\)\]with the conditional\-FM \(CFM\) objective\. This is motivated by the fact that CFM is highly tractable\([Lipman et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib18)\), and a recent result by[Su et al\. \(2025\)](https://arxiv.org/html/2610.12699#bib.bib15), where it is shown that under smoothness assumptions and sufficient spatio\-temporal regularity of the velocity fields, a bounded velocity error yields a bounded KL divergence between the target distribution and the learned distribution\.

With the linear interpolant and random coupling strategy, i\.e\.,x0∼P0,x1∼Q,xt=t​x1\+\(1−t\)​x0x\_\{0\}\\sim P\_\{0\},x\_\{1\}\\sim Q,x\_\{t\}=tx\_\{1\}\+\(1\-t\)x\_\{0\}, in whichx0x\_\{0\}andx1x\_\{1\}are independent, we define the surrogate Lagrangian as follows:

L′​\(uθ,λ\)=w​𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\+λ⁡\(b−Puθ​\(𝒞\)\),\\begin\{aligned\} L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)=&w\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]\\\\ &\+\\lambda\(b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\),\\end\{aligned\}

\(10\)in whicht∼Unif⁡\[0,1\]t\\sim\\mathrm\{Unif\}\[0,1\]andw\>0w\>0is a weighting constant that calibrates the scale of the velocity error term relative to𝔼x∼Q​\[−log⁡puθ​\(x\)\]\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\_\{u\_\{\\theta\}\}\(x\)\]inL¯\\bar\{L\}\.

To minimizeL′L^\{\\prime\}, a natural approach is to updateθ\\thetaby gradient descent, which requires estimable gradients for each term in \([10](https://arxiv.org/html/2610.12699#S4.E10)\)\. The gradient of𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]follows from the standard FM optimization process\. However, estimating−∇θPuθ​\(𝒞\)\-\\nabla\_\{\\theta\}P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)requires further development, especially in the case where the constraints are available only via a membership oracle\.

In this paper, we propose a gradient estimation strategy that avoids Gaussian randomization of the velocity field proposed in[Huan et al\. \(2026\)](https://arxiv.org/html/2610.12699#bib.bib16), drawing intuition from deterministic policy gradient methods\([Silver et al\., 2014](https://arxiv.org/html/2610.12699#bib.bib14)\)\.

We consider anNN\-step forward Euler discretization of the flow\. LetΔ​t=1/N≪1\\Delta t=1/N\\ll 1be the selected interval for discretization and restrict time to the grid\{t=iΔt:i=0,…,N\}\\\{t=i\\Delta t:i=0,\\ldots,N\\\}\. Starting fromx0∼P0x\_\{0\}\\sim P\_\{0\}, the forward Euler discretization is:xt\+Δ​t=xt\+uθ​\(xt,t\)​Δ​tx\_\{t\+\\Delta t\}=x\_\{t\}\+u\_\{\\theta\}\(x\_\{t\},t\)\\Delta t\. Equivalently,

xi​Δ​t=Φuθ,i​Δ​tΔ​t​\(x0\)=\{x0,i=0,x0\+∑j=0i−1uθ​\(xj​Δ​t,j​Δ​t\)​Δ​t,i=1,2,⋯,N\.x\_\{i\\Delta t\}=\\Phi^\{\\Delta t\}\_\{u\_\{\\theta\},i\\Delta t\}\(x\_\{0\}\)=\\begin\{cases\}x\_\{0\},&i=0,\\\\ x\_\{0\}\+\\sum\_\{j=0\}^\{i\-1\}u\_\{\\theta\}\(x\_\{j\\Delta t\},j\\Delta t\)\\Delta t,&i=1,2,\\cdots,N\.\\end\{cases\}\(11\)
Note thatPuθ​\(𝒞\)=limΔ​t→0∫p0​\(x0\)​𝟏𝒞​\[Φuθ,1Δ​t​\(x0\)\]​d​x0P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)=\\lim\_\{\\Delta t\\to 0\}\\int p\_\{0\}\(x\_\{0\}\)\\mathbf\{1\}\_\{\\mathcal\{C\}\}\\left\[\\Phi^\{\\Delta t\}\_\{u\_\{\\theta\},1\}\(x\_\{0\}\)\\right\]dx\_\{0\}\. For sufficiently smallΔ​t\\Delta t, we use the finite\-step quantity inside the limit to approximatePuθ​\(𝒞\)P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\. We now note the main difficulty\. Direct path\-wise differentiation ofPuθ​\(𝒞\)P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)with respect toθ\\thetarequires differentiating the indicator function𝟏𝒞\\mathbf\{1\}\_\{\\mathcal\{C\}\}, for which the gradient is00almost everywhere and undefined on the constraint boundary\. To address this issue, assume there exists a differentiable functionf\(uθ\):ℝd×\[0,1\]→\[0,1\]f^\{\(u\_\{\\theta\}\)\}:\\mathbb\{R\}^\{d\}\\times\[0,1\]\\rightarrow\[0,1\]with the following properties: \(a\)\[Bellman consistency\]For everyi∈\{0,1,…,N−1\}i\\in\\\{0,1,\\ldots,N\-1\\\}andx∈ℝdx\\in\\mathbb\{R\}^\{d\},

f\(uθ\)​\(x,i​Δ​t\)=f\(uθ\)​\(x\+uθ​\(x,i​Δ​t\)​Δ​t,\(i\+1\)​Δ​t\);f^\{\(u\_\{\\theta\}\)\}\(x,i\\Delta t\)=f^\{\(u\_\{\\theta\}\)\}\\left\(x\+u\_\{\\theta\}\(x,i\\Delta t\)\\Delta t,\(i\+1\)\\Delta t\\right\);\(12\)\(b\)\[No explicit terminal parameter dependence\]∇θf\(uθ\)​\(x,1\)=0\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x,1\)=0; and \(c\)\[A good estimator of the indicator\]𝔼x0∼P0​\[\|f\(uθ\)​\(x0,0\)−𝟏𝒞​\[Φuθ,1​\(x0\)\]\|\]≤ρ\\mathbb\{E\}\_\{x\_\{0\}\\sim P\_\{0\}\}\\left\[\\left\|f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)\-\\mathbf\{1\}\_\{\\mathcal\{C\}\}\\left\[\\Phi\_\{u\_\{\\theta\},1\}\(x\_\{0\}\)\\right\]\\right\|\\right\]\\leq\\rho, in whichρ\>0\\rho\>0is a small approximation tolerance\. The main idea is that the functionf\(uθ\)f^\{\(u\_\{\\theta\}\)\}gives a surrogate forPuθ​\(𝒞\)P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)viaℱuθ​\(𝒞\):=∫p0​\(x0\)​f\(uθ\)​\(x0,0\)​d​x0\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\):=\\int p\_\{0\}\(x\_\{0\}\)f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)dx\_\{0\}\. Indeed,

\|Puθ​\(𝒞\)−ℱuθ​\(𝒞\)\|≤∫p0​\(x0\)​\|f\(uθ\)​\(x0,0\)−𝟏𝒞​\[Φuθ,1​\(x0\)\]\|​d​x0=𝔼x0∼P0​\[\|f\(uθ\)​\(x0,0\)−𝟏𝒞​\[Φuθ,1​\(x0\)\]\|\]≤ρ,\|P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\-\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\|\\leq\\int p\_\{0\}\(x\_\{0\}\)\\left\|f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)\-\\mathbf\{1\}\_\{\\mathcal\{C\}\}\\left\[\\Phi\_\{u\_\{\\theta\},1\}\(x\_\{0\}\)\\right\]\\right\|dx\_\{0\}=\\mathbb\{E\}\_\{x\_\{0\}\\sim P\_\{0\}\}\\left\[\\left\|f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)\-\\mathbf\{1\}\_\{\\mathcal\{C\}\}\\left\[\\Phi\_\{u\_\{\\theta\},1\}\(x\_\{0\}\)\\right\]\\right\|\\right\]\\leq\\rho,

\(13\)leads to

L′​\(uθ,λ\)≤w​𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\+λ⁡\(b−ℱuθ​\(𝒞\)\)\+λ​ρ\.L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)\\leq w\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]\+\\lambda\(b\-\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\+\\lambda\\rho\.\(14\)We therefore minimize the objective on RHS of \([14](https://arxiv.org/html/2610.12699#S4.E14)\) to controlL′​\(uθ,λ\)L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)which is a good approximation whenρ\\rhois small\.

###### Proposition 4\.1\.

Under the assumption that∇θf\(uθ\)​\(xi​Δ​t,i​Δ​t\)\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\},i\\Delta t\)exists and is uniformly bounded for alli∈\{0,…,N−1\}i\\in\\\{0,\\ldots,N\-1\\\},i∼Unif​\{0,⋯,N−1\}i\\sim\\text\{Unif\}\\\{0,\\cdots,N\-1\\\}andxi​Δ​t∼\(Φuθ,i​Δ​tΔ​t\)\#​P0x\_\{i\\Delta t\}\\sim\(\\Phi^\{\\Delta t\}\_\{u\_\{\\theta\},i\\Delta t\}\)\_\{\\\#\}P\_\{0\},

∇θℱuθ​\(𝒞\)=N​𝔼i,xi​Δ​t​\[∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\],\\nabla\_\{\\theta\}\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)=N\\mathbb\{E\}\_\{i,x\_\{i\\Delta t\}\}\\left\[\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\\right\],

\(15\)in which∇θuθ​\(xt,t\)\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{t\},t\)is a Jacobian matrix such that each column is the gradient∇θ\[uθ​\(xt,t\)\]j\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{j\}of thejj\-th dimension of thedd\-dimensional velocity with respect to the velocity parameters, i\.e\.,

∇θuθ​\(xt,t\)=\[∇θ\[uθ​\(xt,t\)\]1∇θ\[uθ​\(xt,t\)\]2⋯∇θ\[uθ​\(xt,t\)\]d\]\.\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{t\},t\)=\[\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{1\}\\quad\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{2\}\\quad\\cdots\\quad\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{d\}\]\.\(16\)

Proposition[4\.1](https://arxiv.org/html/2610.12699#S4.Thmtheorem1)with its proofs in Appendix[E](https://arxiv.org/html/2610.12699#A5), states that one can approximate∇θℱuθ​\(𝒞\)\\nabla\_\{\\theta\}\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)via Monte Carlo\. But we still needf\(uθ\)f^\{\(u\_\{\\theta\}\)\}that satisfies properties \(a\), \(b\), and \(c\) above\. For this we propose to trainfϕf\_\{\\phi\}, an NN parameterized byϕ\\phito approximatef\(uθ\)f^\{\(u\_\{\\theta\}\)\}\. It is trained via supervised learning on sampled trajectories\. Specifically, we samplex0∼P0x\_\{0\}\\sim P\_\{0\}, generate an Euler trajectory\{xi​Δ​t\}i∈\{0,1,⋯,N\}\\\{x\_\{i\\Delta t\}\\\}\_\{i\\in\\\{0,1,\\cdots,N\\\}\}under the current velocity fielduθu\_\{\\theta\}, and assign the terminal labely=𝟏𝒞​\(xN​Δ​t\)y=\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\_\{N\\Delta t\}\)\. Thenfϕf\_\{\\phi\}is trained to predictyyfrom intermediate states by minimizing the binary cross\-entropy loss, thereby the gradient ofϕ\\phican be given as

gradϕ=∇ϕ𝔼i,xi​Δ​t​\[−y​log⁡fϕ​\(xi​Δ​t,i​Δ​t\)−\(1−y\)​log⁡\(1−fϕ​\(xi​Δ​t,i​Δ​t\)\)\]\.\\mathrm\{grad\}\_\{\\phi\}=\\nabla\_\{\\phi\}\\mathbb\{E\}\_\{i,x\_\{i\\Delta t\}\}\\left\[\-y\\log f\_\{\\phi\}\(x\_\{i\\Delta t\},i\\Delta t\)\-\(1\-y\)\\log\\big\(1\-f\_\{\\phi\}\(x\_\{i\\Delta t\},i\\Delta t\)\\big\)\\right\]\.\(17\)At each sampled trajectory, this structure encourages\{fϕ​\(xi​Δ​t,i​Δ​t\)\}i∈\{0,1,⋯,N\}\\\{f\_\{\\phi\}\(x\_\{i\\Delta t\},i\\Delta t\)\\\}\_\{i\\in\\\{0,1,\\cdots,N\\\}\}to target at the same terminal labely=𝟏𝒞​\(xN​Δ​t\)y=\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\_\{N\\Delta t\}\), which is held fixed during theϕ\\phi\-update, thereby encouraging to fulfill property \(a\),\(b\),\(c\)\.fϕf\_\{\\phi\}serves as a differentiable predictor of terminal constraint satisfaction from\(xt,t\)\(x\_\{t\},t\)\.

Practically, we aim to minimize1w​L′​\(uθ,λ\)=𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\+λ′​\(b−Puθ​\(𝒞\)\)\\frac\{1\}\{w\}L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)=\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]\+\\lambda^\{\\prime\}\(b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)withλ′=λ/w\\lambda^\{\\prime\}=\\lambda/w, which has the same minimizer asL′​\(uθ,λ\)L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)\. Therefore, by substituting the learned predictorfϕf\_\{\\phi\}forf\(uθ\)f^\{\(u\_\{\\theta\}\)\}in Proposition[4\.1](https://arxiv.org/html/2610.12699#S4.Thmtheorem1), the gradient used in practice to updateθ\\thetais given by

gradθ=∇θ𝔼t,x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]−λ′​N​𝔼i,xi​Δ​t​\[∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇ufϕ​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\]\.\\begin\{aligned\} \\mathrm\{grad\}\_\{\\theta\}=&\\nabla\_\{\\theta\}\\mathbb\{E\}\_\{t,x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]\\\\ &\-\\lambda^\{\\prime\}N\\mathbb\{E\}\_\{i,x\_\{i\\Delta t\}\}\\left\[\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f\_\{\\phi\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\\right\]\.\\end\{aligned\}

\(18\)
Therefore, we can approximately solvearg⁡minθ​1w​L′​\(uθ,λ\)\\arg\\min\_\{\\theta\}\\frac\{1\}\{w\}L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)by alternating between updatingθ\\thetaandϕ\\phi\. Withθ\\thetafixed, we updatefϕf\_\{\\phi\}withgradϕ\\mathrm\{grad\}\_\{\\phi\}in \([17](https://arxiv.org/html/2610.12699#S4.E17)\) to predict terminal constraint satisfaction following the current velocity field\. Withϕ\\phifixed, we updateθ\\thetawithgradθ\\mathrm\{grad\}\_\{\\theta\}in \([18](https://arxiv.org/html/2610.12699#S4.E18)\)\.

Withθ\\thetaupdated, we update the dual variable, for which the gradient is proportional tob−Puθ​\(𝒞\)b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\. In practice,Puθ​\(𝒞\)P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)is estimated by Monte Carlo simulation, i\.e\., by generating terminal samples usinguθu\_\{\\theta\}and computing the probability of generating samples belonging to𝒞\\mathcal\{C\}\. The algorithm for practical PDFM is provided in Alg\.[2](https://arxiv.org/html/2610.12699#alg2)\.

### 4\.3Constraint Guarantees on PDFM

Let the velocity field solved by minimizing1w​L′\\frac\{1\}\{w\}L^\{\\prime\}atλ′=λ/w\\lambda^\{\\prime\}=\\lambda/w, which is analyzed in the previous section be denoted byuθ′\(λ\)u^\{\\prime\(\\lambda\)\}\_\{\\theta\}, the dual variable ofL¯\\bar\{L\}at theii\-th iteration be denoted byλ⁡\(i\)\\lambda\(i\)and the distribution induced by the velocity fielduθ′\(λ\)u^\{\\prime\(\\lambda\)\}\_\{\\theta\}be denoted byPuθ′\(λ\):=\(Φu′\(λ\)θ,1\)\#​P0P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}:=\(\\Phi\_\{u^\{\\prime\(\\lambda\)\_\{\\theta\}\},1\}\)\_\{\\\#\}P\_\{0\}\. In this subsection, we analyze the asymptotic gap between the distributionPuθ′\(λ\(i\)\)P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda^\{\(i\)\}\)\}produced by the practical primal\-dual algorithm and the optimal distributionP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}in \([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\) asi→∞i\\to\\infty\.

###### Assumption 4\.2\.

Note the scaling relationshipλ=w​λ′\\lambda=w\\lambda^\{\\prime\}\. For every iteration of Alg\.[2](https://arxiv.org/html/2610.12699#alg2), we assume that there exists a constantλmax<∞\\lambda\_\{\\max\}<\\inftysuch that

∀i,λ′​\(i\)≤λmax/w,equivalently,λ⁡\(i\)≤λmax\.\\forall i,\\lambda^\{\\prime\}\(i\)\\leq\\lambda\_\{\\max\}/w,\\quad\\mathrm\{equivalently\},\\quad\\lambda\(i\)\\leq\\lambda\_\{\\max\}\.\(19\)

We next make the following assumption to quantify the optimization error introduced by the approximation ofPuθ\(λ\)P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}via the strategy in Sec\.[4\.2](https://arxiv.org/html/2610.12699#S4.SS2)\.

###### Assumption 4\.3\.

For every fixed0≤λ≤λmax0\\leq\\lambda\\leq\\lambda\_\{\\max\},Puθ′\(λ\)P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}isε\\varepsilon\-suboptimal with respect to the FM\-parameterized optimum,

L⁡\(puθ′\(λ\),λ\)−g¯​\(λ\)≤ε,equivalently,L⁡\(puθ′\(λ\),λ\)−infu∈​UΘL¯​\(u,λ\)≤ε\.L\\left\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\\right\)\-\\bar\{g\}\(\\lambda\)\\leq\\varepsilon,\\quad\\mathrm\{equivalently\},\\quad L\\left\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\\right\)\-\\inf\_\{u\_\{\\in\}U\_\{\\Theta\}\}\\bar\{L\}\(u,\\lambda\)\\leq\\varepsilon\.\(20\)

For the dual update, we need to evaluateb−Puθ′​\(𝒞\)b\-P\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\), wherePuθ′​\(𝒞\)P\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\)is estimated in practice by Monte Carlo simulation\. To simplify the analysis, we make the following assumption\.

###### Assumption 4\.4\.

The empirical estimator ofb−Puθ′​\(𝒞\)b\-P\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\)is exact at every dual update\.

If this assumption is relaxed to a weaker condition, such as unbiased Monte Carlo estimation, then the convergence guarantee should be stated in expectation\. In particular, one can generally boundlim infi→∞𝔼⁡\[Puθ′\(λ⁡\(i\)\)​\(𝒞\)\]\\liminf\_\{i\\to\\infty\}\\mathbb\{E\}\\\!\\left\[P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\\right\]rather than the pathwise quantitylim infi→∞Puθ′\(λ⁡\(i\)\)​\(𝒞\)\\liminf\_\{i\\to\\infty\}P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\. The corresponding statements and proofs are provided in Appendix[G](https://arxiv.org/html/2610.12699#A7)\.

###### Theorem 4\.5\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2)\-[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4), and[4\.2](https://arxiv.org/html/2610.12699#S4.Thmtheorem2)\-[4\.4](https://arxiv.org/html/2610.12699#S4.Thmtheorem4), with∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty,

lim supi→∞‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV≤ℰPDFM,lim infi→∞Puθ′\(λ⁡\(i\)\)​\(𝒞\)≥b−ℰPDFM,\\limsup\_\{i\\to\\infty\}\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\\quad\\liminf\_\{i\\to\\infty\}P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\(21\)ℰPDFM=emax\+2​emaxk​\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3,e⁡\(λ\):=ε\+μ⁡\(λ\)​\(A1​ν\+A2​ν2\)2​\(μ​\(λ\)−λ\),\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}=e\_\{\\max\}\+\\frac\{2e\_\{\\max\}\}\{k\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\},\\quad e\(\\lambda\):=\\sqrt\{\\frac\{\\varepsilon\+\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\{2\(\\mu\(\\lambda\)\-\\lambda\)\}\},\(22\)in whichμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)andemax=maxλ∈\[0,λmax\]⁡e⁡\(λ\)e\_\{\\max\}=\\max\_\{\\lambda\\in\[0,\\lambda\_\{\\max\}\]\}e\(\\lambda\)\.

Theorem[4\.5](https://arxiv.org/html/2610.12699#S4.Thmtheorem5)gives the asymptotic guarantee for the practical primal\-dual algorithm in Alg\.[2](https://arxiv.org/html/2610.12699#alg2), with detailed proofs in Appendix[F](https://arxiv.org/html/2610.12699#A6)\. Under the stated boundedness, approximation, and exact dual\-gradient assumptions,Puθ′\(λ⁡\(i\)\)P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}remains within an explicit TV neighborhood of the distribution\-level optimumP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}\. The radius of this neighborhood is controlled byemaxe\_\{\\max\}, which is determined by the optimization errorε\\varepsilon, the FM parameterization errorA1​ν\+A2​ν2A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}, and their worst\-case behavior over the bounded dual range\[0,λmax\]\[0,\\lambda\_\{\\max\}\]\. Consequently, the asymptotic constraint satisfaction rate is at least the target levelbbminus the same TV error; the gap decreases as the practical optimization errorε\\varepsilonand FM parameterization errorν\\nudecrease\.

## 5Experiments

### 5\.1Illustrative22\-D Constrained Generation \(Q⁡\(𝒞\)<bQ\(\\mathcal\{C\}\)<b\)

We begin with a synthetic22\-D constrained generation example\. Letz=\(z1,z2\)∈ℝ2z=\(z\_\{1\},z\_\{2\}\)\\in\\mathbb\{R\}^\{2\}denote a sample\. The clean distribution is a uniform distribution over two disconnected regions,

𝒞=\{\(z1,z2\):1≤\|z1\|≤5,1≤\|z2\|≤5,z1z2\>0\}\.\\mathcal\{C\}=\\\{\(z\_\{1\},z\_\{2\}\):1\\leq\|z\_\{1\}\|\\leq 5,\\;1\\leq\|z\_\{2\}\|\\leq 5,\\;z\_\{1\}z\_\{2\}\>0\\\}\.Samples from the clean distribution are corrupted by additive zero\-mean Gaussian noise with variance0\.010\.01, yielding the training distributionQQ, for whichQ⁡\(𝒞\)≈96%Q\(\\mathcal\{C\}\)\\approx 96\\%\. Following the formulation in \([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\), the goal is to learn a distribution that remains as close as possible toQQwhile achieving a constraint satisfaction rate of at leastb=99\.5%b=99\.5\\%\.

We compare PDFM against two FM\-based baselines: \(1\) vanilla FM trained on samples fromQQ; and \(2\) FM trained on the truncated distributionQ′=Q\(⋅∣𝒞\)Q^\{\\prime\}=Q\(\\cdot\\mid\\mathcal\{C\}\), obtained by discarding infeasible training samples, with densityq′​\(x\)=q⁡\(x\)​𝟏𝒞​\(x\)/Q⁡\(𝒞\)q^\{\\prime\}\(x\)=\{q\(x\)\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\)\}/\{Q\(\\mathcal\{C\}\)\}\. The visualization of the clean distribution,QQ,P\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}\(computed for this simple case via Lemma[2\.2](https://arxiv.org/html/2610.12699#S2.Thmtheorem2)\), the distributions induced by baselines \(1\), \(2\), and PDFM are shown in Appendix[H\.1](https://arxiv.org/html/2610.12699#A8.SS1)Fig\.[1](https://arxiv.org/html/2610.12699#A8.F1)\. We evaluate two metrics: the constraint satisfaction rate and the Sliced Wasserstein Distance \(SWD\)\([Bonneel et al\., 2014](https://arxiv.org/html/2610.12699#bib.bib2)\)between samples from the trained model and samples fromP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}\. SWD measures distributional similarity, with smaller values indicating better similarity\. Reported metrics use5000050000generated samples, averaged over100100trials\.

Baseline \(1\) shows that vanilla FM trained directly onQQproduces a distribution with constraint satisfaction rate96\.37%96\.37\\%, close toQ⁡\(𝒞\)Q\(\\mathcal\{C\}\), but below the target levelb=99\.5%b=99\.5\\%; its SWD is0\.13840\.1384\. Baseline \(2\) shows that even when FM is trained on the truncated distribution, whose training samples are all feasible, it still fails to fully enforce the constraint, achieving constraint satisfaction rate98\.74%98\.74\\%and SWD0\.11560\.1156\. In this synthetic example, discarding infeasible samples removes only about4%4\\%of the data;*in more realistic settings, as shown in Sec\.[5\.3](https://arxiv.org/html/2610.12699#S5.SS3), this rejection step can discard a much larger fraction of the training set and thereby degrade distributional matching*\. In contrast, PDFM achieves the target constraint level, with a constraint satisfaction rate99\.51%99\.51\\%, while also attaining the lowest SWD,0\.10960\.1096\. We therefore conclude that, with samples fromQQ, PDFM achieves better constraint satisfaction than FM baselines trained on eitherQQor the truncated distributionQ′Q^\{\\prime\}, while attaining superior distributional fidelity\.

### 5\.2Constrained Generation withℓ2\\ell\_\{2\}\-Ball Constraints \(Q⁡\(𝒞\)≥bQ\(\\mathcal\{C\}\)\\geq b\)

We next evaluate PDFM on higher\-dimensional synthetic tasks in this regime, together with several baselines, MDM\([Liu et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib23)\), NAMM\([Feng et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib12)\), PDM\([Christopher et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib9)\), PCFM\([Utkarsh et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib10)\), and FM\-RE\([Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\), introduced in Appendix[A](https://arxiv.org/html/2610.12699#A1)\.

The target distributionQQis a Gaussian mixture model, with a constraint defined as anℓ2\\ell\_\{2\}ball constraint𝒞=\{x∈ℝd\|‖x‖2≤1\}\\mathcal\{C\}=\\left\\\{x\\in\\mathbb\{R\}^\{d\}\\Big\|\\\|x\\\|\_\{2\}\\leq 1\\right\\\}\. Specifically, we considerddisotropic Gaussians, each with variance0\.050\.05, centered at each of theddstandard unit vectors, and reject samples outside𝒞\\mathcal\{C\}, i\.e\.,Q⁡\(𝒞\)=1Q\(\\mathcal\{C\}\)=1\. This setting is also considered in[Liu et al\. \(2023\)](https://arxiv.org/html/2610.12699#bib.bib23)\.

FM achievesP⁡\(𝒞\)=97\.633%P\(\\mathcal\{C\}\)=97\.633\\%and90\.918%90\.918\\%in the 8\-D and 20\-D cases, respectively, showing that constrained generation remains nontrivial even whenQ⁡\(𝒞\)≥bQ\(\\mathcal\{C\}\)\\geq b, in which case the population\-level optimum isQQ\. For a direct comparison with FM\-RE\([Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\), which also relies on membership oracle feedback, we set the target constraint satisfaction ratebbto match theP⁡\(𝒞\)P\(\\mathcal\{C\}\)attained by FM\-RE, namelyb=99\.923%b=99\.923\\%for the88\-D case andb=99\.748%b=99\.748\\%for the2020\-D case\. The results in Table[1](https://arxiv.org/html/2610.12699#S5.T1)show that PDFM achievesP⁡\(𝒞\)P\(\\mathcal\{C\}\)close to the targetbbwhile yielding a smaller SWD than FM\-RE, demonstrating superior performance in both constraint satisfaction and distributional matching\. It is worth noting that PDFM can be configured to match the constraint satisfaction level achieved by FM\-RE, whereas the reverse is not straightforward\. The reason is that PDFM directly controlsP⁡\(𝒞\)P\(\\mathcal\{C\}\)throughbb, while FM\-RE can only influenceP⁡\(𝒞\)P\(\\mathcal\{C\}\)indirectly by adjusting the weight of the constraint term\. Therefore, compared with FM\-RE, PDFM substantially reduces the burden of hyperparameter tuning\.

Compared with MDM, which requires convexity and a special structure of𝒞\\mathcal\{C\}, NAMM and PCFM, which both require a differentiable distance to𝒞\\mathcal\{C\}, and PDM, which requires projection onto𝒞\\mathcal\{C\},*PDFM works with arguably the most flexible requirement on the constraint set specification: access via a membership oracle\.*Despite this weaker requirement, PDFM achieves strong constraint satisfaction while maintaining competitive distributional fidelity\.

Table 1:Results on88\-D and2020\-D constrained generation tasks\.P⁡\(𝒞\)P\(\\mathcal\{C\}\)denotes the mean constrained\-satisfaction rate of1000010000samples, averaged over100100generation trials\. SWD is computed with1000010000generated samples and samples fromQQ, averaged over100100generation trials\.FMMDMNAMMPDMPCFMFM\-REPDFM88\-Dℓ2\\ell\_\{2\}ballP⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)97\.633%97\.633\\%100%100\\%99\.986%99\.986\\%100%100\\%100%100\\%99\.923%99\.923\\%99\.906%99\.906\\%SWD\(↓\)\(\\downarrow\)0\.01930\.01930\.02170\.02170\.02280\.02280\.02000\.02000\.02840\.02840\.01940\.01940\.01720\.01722020\-Dℓ2\\ell\_\{2\}ballP⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)90\.918%90\.918\\%100%100\\%99\.711%99\.711\\%100%100\\%100%100\\%99\.748%99\.748\\%99\.755%99\.755\\%SWD\(↓\)\(\\downarrow\)0\.00870\.00870\.01440\.01440\.02050\.02050\.02570\.02570\.01480\.01480\.01320\.01320\.01010\.0101

### 5\.3Unconditional PDE\-Constrained Generation of Spatiotemporal Dynamics

A central challenge in applying generative modeling to applications involving dynamical systems, including chaotic systems, is generating samples that respect known physical laws such as the underlying governing equations\. We illustrate PDFM on the Kuramoto–Sivashinsky \(KS\) equation, a canonical model of spatiotemporal chaos\([Kuramoto and Tsuzuki, 1976](https://arxiv.org/html/2610.12699#bib.bib28);[Sivashinsky, 1977](https://arxiv.org/html/2610.12699#bib.bib29)\), where the constraint set𝒞\\mathcal\{C\}is defined by the residual of generated space\-time patches against the numerical PDE solver\. This setting is representative of generative emulation tasks in which a forward solver can evaluate whether a sample obeys the dynamics\.

Given a dataset of non\-overlapping space–time patchesx∈ℝ64×64x\\in\\mathbb\{R\}^\{64\\times 64\}, the objective is to learn a generative model that produces patches satisfying the KS dynamics up to a prescribed tolerance with a user\-specified probability\. Specifically, a forward solver, which is assumed to benon\-differentiable, provides the residual feedbackδLn\\delta\_\{L\_\{n\}\}, whereδLn=0\\delta\_\{L\_\{n\}\}=0indicates strict alignment with the KS dynamics up to solver precision\. A sample is considered feasible ifδLn<τ\\delta\_\{L\_\{n\}\}<\\tau, whereτ\\taudenotes the prescribed error tolerance\. Details of the dataset construction and residual computation are provided in Appendix[H\.3](https://arxiv.org/html/2610.12699#A8.SS3)\. Two generation cases are considered\.

#### Case 1:

The clean patchesx∈ℝ64×64x\\in\\mathbb\{R\}^\{64\\times 64\}withδLn=0\\delta\_\{L\_\{n\}\}=0from the solver are used as the training data\. The threshold isτ=0\.033\\tau=0\.033, and the target constraint satisfaction rate is98%98\\%\. In this case, all samples from the training distribution satisfy the constraints, indicatingQ⁡\(𝒞\)=1Q\(\\mathcal\{C\}\)=1\.

#### Case 2:

We create degraded samples from the space–time patches, simulating measurements from a sensor with reduced spatial resolution, where only part of the spatial grid is directly observed and the missing locations are reconstructed by simple linear interpolation\. Specifically, for each sample, along the spatial dimension, every99\-th position is replaced by a linearly interpolated value between the adjacent values\. This leads to an averageδLn=0\.0555\\delta\_\{L\_\{n\}\}=0\.0555for the training samples\. Setting the threshold asτ=0\.05\\tau=0\.05, the training samples constraint satisfaction rate is39\.47%39\.47\\%, i\.e\.,Q⁡\(𝒞\)≈39\.47%Q\(\\mathcal\{C\}\)\\approx 39\.47\\%\. The target constraint satisfaction rate is90%90\\%\.

Table 2:Comparison across the considered methods for Case 1 and Case 2\. Results are means over1010trials, using20002000generated samples per method per trial\.MethodCase 1\(Q⁡\(𝒞\)=1\)\(Q\(\\mathcal\{C\}\)=1\)Case 2\(Q⁡\(𝒞\)≈39\.47%\)\(Q\(\\mathcal\{C\}\)\\approx 39\.47\\%\)P⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)AvgδLn\\delta\_\{L\_\{n\}\}\(↓\)\(\\downarrow\)HE\(↓\)\(\\downarrow\)P⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)AvgδLn\\delta\_\{L\_\{n\}\}\(↓\)\(\\downarrow\)HEclean\\mathrm\{HE\}\_\{\\mathrm\{clean\}\}\(↓\)\(\\downarrow\)HEdeg\\mathrm\{HE\}\_\{\\mathrm\{deg\}\}\(↓\)\(\\downarrow\)FM \(QQ\)14\.19%0\.0409\(0\.0087,0\.0453\)9\.82%0\.0622\(0\.0094,0\.0465\)\(0\.0086,0\.0565\)FM \(Q′Q^\{\\prime\}\)N/AN/AN/A33\.24%0\.0539\(0\.0135,0\.0477\)\(0\.0114,0\.0489\)PCFM6\.03%0\.0484\(0\.0156,0\.1547\)0\.06%0\.1022\(0\.0173,0\.0729\)\(0\.0186,0\.0823\)FM\-RE43\.17%0\.0339\(0\.0117,0\.0329\)46\.33%0\.0513\(0\.0118,0\.0558\)\(0\.0097,0\.0570\)PDFM97\.74%0\.0235\(0\.0107,0\.0269\)90\.48%0\.0406\(0\.0118,0\.0435\)\(0\.0103,0\.0457\)

The statistical results are reported in Table[2](https://arxiv.org/html/2610.12699#S5.T2), and visualizations of the generated samples and corresponding histograms are provided in Appendix[H\.3](https://arxiv.org/html/2610.12699#A8.SS3)\. In Table[2](https://arxiv.org/html/2610.12699#S5.T2),P⁡\(𝒞\)P\(\\mathcal\{C\}\)denotes the empirical constraint satisfaction rate\. To evaluate the distribution matching quality, we follow[Jiang et al\. \(2023\)](https://arxiv.org/html/2610.12699#bib.bib20), and provide the mean histogram error \(denoted asHE\\mathrm\{HE\}\) – the mean absolute difference between the generated and reference summary\-statistic histograms over the bin grids, averaged over20002000samples\.HE\\mathrm\{HE\}in Table[2](https://arxiv.org/html/2610.12699#S5.T2)is in the format\(HE\(1\),HE\(2\)\)\(\\mathrm\{HE\}^\{\(1\)\},\\mathrm\{HE\}^\{\(2\)\}\), whereHE\(1\)\\mathrm\{HE\}^\{\(1\)\}corresponds to the histogram error of∂u/∂t\\partial u/\\partial tvs\.∂u/∂x\\partial u/\\partial xandHE\(2\)\\mathrm\{HE\}^\{\(2\)\}corresponds to the histogram error of∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}vs\.∂u/∂x\\partial u/\\partial x\. A lower histogram error denotes a better match in distribution\. In Case 2, “HEclean\\mathrm\{HE\}\_\{\\mathrm\{clean\}\}” and “HEdeg\\mathrm\{HE\}\_\{\\mathrm\{deg\}\}” denote the error relative to the clean samples and the degraded training samples, respectively\.

In Case 1, we compare PDFM with FM trained onQQ, denoted by FM\(Q\)\(Q\), PCFM, and FM\-RE\. In Case 2, a natural alternative is to discard infeasible training samples and train FM on the truncated distributionQ′Q^\{\\prime\}, with densityq′​\(x\)=q⁡\(x\)​𝟏𝒞​\(x\)Q⁡\(𝒞\)\.q^\{\\prime\}\(x\)=\\frac\{q\(x\)\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\)\}\{Q\(\\mathcal\{C\}\)\}\.We additionally include this baseline, denoted by FM\(Q′\)\(Q^\{\\prime\}\)\. This rejection step removes approximately60%60\\%of the training data\. Although FM\(Q′\)\(Q^\{\\prime\}\)improves the constraint satisfaction rate from9\.82%9\.82\\%for FM\(Q\)\(Q\)to33\.24%33\.24\\%, it remains far below the target rate of90%90\\%\. Moreover, fewer training samples lead to a degraded generated distribution, as reflected by the increased histogram error\.

The results in Table[2](https://arxiv.org/html/2610.12699#S5.T2)illustrate four properties of PDFM that are particularly relevant in this setting\. First, even when the training distribution itself satisfies the constraint perfectly \(Case 1,Q⁡\(𝒞\)=1Q\(\\mathcal\{C\}\)=1\), vanilla FM only achievesP⁡\(𝒞\)=14\.19%P\(\\mathcal\{C\}\)=14\.19\\%, confirming that local consistency does not transfer for free from the training distribution to the trained model; PDFM raises this rate to97\.74%97\.74\\%at the prescribed target while keeping the marginal summary statistics close to the training data\. Second, in the more realistic Case 2 with degraded training data \(Q⁡\(𝒞\)≈39\.47%Q\(\\mathcal\{C\}\)\\approx 39\.47\\%\), PDFM closes the gap from9\.82%/33\.24%9\.82\\%/33\.24\\%to90\.48%90\.48\\%, demonstrating that the chance\-constrained formulation lets the generative model produce samples of higher dynamical quality than the data it was trained on, by leveraging the solver\-based residual oracle to suppress non\-physical regions of the learned distribution\. Third, while PCFM and NAMM, requiring the fully differentiable residual, fail in this task, PDFM reaches these satisfaction rates without differentiating through the numerical stepper, which is essential when the stepper is non\-differentiable, written in a legacy language, or otherwise available only as a black box\. Fourth, compared with FM\-RE, which also requires only membership oracle feedback, PDFM provides direct control over the constraint satisfaction rate and can therefore target a prescribed feasibility level rather than relying on an implicitly tuned penalty weight\.

### 5\.4Additional Applications

Fingerprint image generation\.We further evaluate PDFM on SOCOFing fingerprint\([Shehu et al\., 2018](https://arxiv.org/html/2610.12699#bib.bib19)\)generation with a ridge\-connectivity constraint, assessed by a non\-differentiable image\-processing oracle\. The oracle applies thresholding and skeletonization, then checks whether the number of connected ridge components is below a prescribed threshold\. PDFM achieves a mean constraint satisfaction rate of99\.28%99\.28\\%, meeting the targetb=99%b=99\\%, compared with72\.74%72\.74\\%for FM and88\.14%88\.14\\%for FM\-RE\. Meanwhile, PDFM maintains NLLs comparable to FM and preserves visual fidelity\. Experimental details and sample visualizations are provided in Appendix[H\.4](https://arxiv.org/html/2610.12699#A8.SS4)\.

33\-D molecule generation\.Trained on QM9\([Ramakrishnan et al\., 2014](https://arxiv.org/html/2610.12699#bib.bib7)\), we aim to generate samples satisfying a molecule\-stability constraint requiring all atoms to satisfy valid valency rules after bond inference\. Using only binary feasibility feedback for constraint satisfaction, PDFM improves the mean molecule stability rate from32\.77%32\.77\\%to48\.75%48\.75\\%and reduces NLL from−19\.74\-19\.74to−20\.70\-20\.70relative to FM\. Although the stability rate remains below the targetb=60%b=60\\%, these results demonstrate both improved feasibility and improved likelihood quality, without task\-specific designs\. Further details are provided in Appendix[H\.5](https://arxiv.org/html/2610.12699#A8.SS5)\.

## 6Conclusion

In this paper, we present PDFM, a primal\-dual framework for constrained generation that requires only membership oracle feedback, provides direct control over the constraint satisfaction rate, and admits theoretical guarantees\. We also propose a non\-stochastic constraint\-guided approach for updating the velocity field, enabling efficient use of oracle feedback during primal optimization\. One of the limitations of PDFM arises from the need to obtain sufficient positive constraint satisfaction feedback during training to learnfϕf\_\{\\phi\}reliably\. Further details are discussed in Appendix[I](https://arxiv.org/html/2610.12699#A9)\.

## Acknowledgment

Zhengyan Huan is supported via NSF under Cooperative Agreement PHY\-2019786, PHY\-2525568, and DOE DE\-SC002396\. Peter Y\. Lu and Shuchin Aeron are supported via NSF under Cooperative Agreement PHY\-2019786, PHY\-2525568\. Shuchin Aeron would also like to acknowledge funding via NSF DMS 2309519\.

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## Appendix ARelated work

Constrained generative modeling has been studied from several complementary perspectives, depending on whether the constraint is imposed at the distribution level or at the level of individual generated samples\. We refer to these two settings asdistributional constraintsandsample\-wise constraints, respectively\. This paper investigatessample\-wise constraintswith the objective of generating samples within the constraint set𝒞\\mathcal\{C\}\.

A closely related work of constrained generative models in the domain ofdistributional constraintsis[Khalafi et al\. \[2024\]](https://arxiv.org/html/2610.12699#bib.bib21), which requires the generated distribution to remain close in KL divergence, to one or more reference distributions\. While[Khalafi et al\. \[2024\]](https://arxiv.org/html/2610.12699#bib.bib21)and this work both utilize primal\-dual frameworks for constrained generative modeling, their scopes and theoretical guarantees differ substantially due to different constraint settings\. Algorithmically, this work’s optimization strategy designed for sample\-wise constraints in Sec\.[4\.2](https://arxiv.org/html/2610.12699#S4.SS2)deviates from the score/flow loss in classical generative models\.

For sample\-wise constraints, existing approaches can be broadly divided intotraining\-timeandinference\-time methods\. Training\-time methods incorporate constraint information during model training, for example, by modifying the objective, architecture, or learned dynamics\. This typically requires additional training effort, but they can generally use standard/slightly\-modified sampling procedures at inference time\. In contrast, inference\-time methods can often be applied to regularly trained generative models, but they must steer, project, or correct the sampling trajectory during generation, which usually increases inference\-time computational cost\. This paper is atraining\-time method, that informs the FM model of the constraints during training and uses standard Euler sampling with the learned velocity field\.

### A\.1Training\-Time Methods

Among training\-time methods assumingQ⁡\(𝒞\)=1Q\(\\mathcal\{C\}\)=1, approaches that enforce strict feasibility of the generated samples, i\.e\.,P⁡\(𝒞\)=1P\(\\mathcal\{C\}\)=1, generally require stronger structures and knowledge of the constraint set𝒞\\mathcal\{C\}\. Reflection\-based methods, such as the reflected diffusion model \(RDM\)\[[Lou and Ermon, 2023](https://arxiv.org/html/2610.12699#bib.bib5),[Fishman et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib3)\]and reflected flow matching \(RFM\)\[[Xie et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib13)\]enforce constraint satisfaction via reflected generation trajectories within𝒞\\mathcal\{C\}, requiring convexity of𝒞\\mathcal\{C\}and explicit knowledge of the constraint boundary∂𝒞\\partial\\mathcal\{C\}\. Mirror\-based approaches, such as mirror diffusion model \(MDM\)\[[Liu et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib23)\], enforce constraint satisfaction by performing DDPM training in the domain of an unconstrained mapping of𝒞\\mathcal\{C\}, and obtaining generated samples from the inverse mapping of the unconstrained domain\. MDM requires convexity of𝒞\\mathcal\{C\}and an available closed\-form bijective mapping between𝒞\\mathcal\{C\}and the unconstrained space\.

Convexity of𝒞\\mathcal\{C\}, boundary information, and tractable bijective maps are often unavailable for complex constraints in practical settings\. To broaden applicability, several existing methods relax strict feasibility guarantees and instead aim to improve, but not necessarily ensure, constraint satisfaction\. Neural approximate mirror map \(NAMM\)\[[Feng et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib12)\]aims to approximate the bijective projection in MDM via NNs, relaxing the requirement of convexity and a closed\-form bijective map to a differentiable distance to𝒞\\mathcal\{C\}\. Trajectory Alignment\[[Giannone et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib11)\]trains diffusion models to align their sampling paths with the intermediate iterates of a physics\-based optimizer, improving constrained design generation but requiring access to optimizer\-generated trajectories during training\. Flow Matching with Random Exploration \(FM\-RE\)\[[Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\]improves feasibility by adding an explicit constraint\-aware term to the FM training objective and injecting Gaussian noise into the generation trajectories to enable likelihood\-ratio gradient estimation, thereby requiring only membership oracle feedback for𝒞\\mathcal\{C\}\. However, FM\-RE’sP⁡\(𝒞\)P\(\\mathcal\{C\}\)is only implicitly controlled by the weight of the constraint\-aware term, and FM\-RE does not provide a theoretical guarantee for feasibility\. PDFM improves upon FM\-RE while retaining the same weak requirement of𝒞\\mathcal\{C\}, i\.e\., membership oracle access, and further provides theoretical guarantees and direct control over the constraint satisfaction rateP⁡\(𝒞\)P\(\\mathcal\{C\}\)\.

### A\.2Inference\-Time Methods

Inference\-time methods improve or enforce constraint satisfaction by modifying the sampling procedure of a pretrained generative model, typically through guidance, projection, or trajectory correction during generation\. Projected diffusion model \(PDM\)\[[Christopher et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib9)\]enforces strict feasibility by repeatedly projecting stochastic diffusion updates onto𝒞\\mathcal\{C\}at each step, requiring an available projection function, and potentially causing deviations from the training distribution\. Physics\-constrained flow matching \(PCFM\)\[[Utkarsh et al\., 2025](https://arxiv.org/html/2610.12699#bib.bib10)\]enforces constraints through forward\-shooting and Gauss–Newton correction of predicted terminal states, which requires a differentiable constraint residual and access to its Jacobian\. HardFlow\[[Li et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib8)\]enforces hard constraints by reformulating FM sampling as a trajectory\-optimization problem, requiring an explicit constraint function and a tractable constrained\-optimization solver during sampling\. Doob\-Oriented Inference\-time Transformation \(DOIT\)\[[Zhu et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib6)\]improves constraint satisfaction by applying Doob’shh\-transform to add a dynamic correction to the diffusion sampling process, which can handle non\-differentiable reward feedback but requires Monte Carlo reward\-query estimates during sampling\. While inference\-time methods avoid retraining, they typically require a nontrivial design of the guidance, correction, or optimization procedure and impose additional computational burden during sampling\.

## Appendix BNotations

Table 3:Definitions and descriptions of the notations used in this paper\.SymbolDefinitionDescriptionP,QP,QMeasuresp,qp,qProbability densitiesP⁡\(𝒞\),Q⁡\(𝒞\)P\(\\mathcal\{C\}\),Q\(\\mathcal\{C\}\)P⁡\(𝒞\)=∫𝒞p⁡\(x\)​𝑑x,Q⁡\(𝒞\)=∫𝒞q⁡\(x\)​𝑑xP\(\\mathcal\{C\}\)=\\int\_\{\\mathcal\{C\}\}p\(x\)dx,Q\(\\mathcal\{C\}\)=\\int\_\{\\mathcal\{C\}\}q\(x\)dxconstraint satisfaction rates ofPPandQQL⁡\(p,λ\)L\(p,\\lambda\)L⁡\(p,λ\)=𝔼x∼Q​\[−log⁡p⁡\(x\)\]\+λ⁡\(b−∫𝒞p⁡\(x\)​𝑑x\)L\(p,\\lambda\)=\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\(x\)\]\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\right\)The LagrangianuuThe velocity fieldg⁡\(λ\)g\(\\lambda\)g⁡\(λ\)=infpL⁡\(p,λ\)g\(\\lambda\)=\\inf\_\{p\}L\(p,\\lambda\)The dual functionλ∗\\lambda^\{\*\}λ∗∈arg⁡maxλ≥0⁡g⁡\(λ\)\\lambda^\{\*\}\\in\\arg\\max\_\{\\lambda\\geq 0\}g\(\\lambda\)The optimal dual variablep\(λ\)p^\{\(\\lambda\)\}p\(λ\)∈arg⁡minp⁡L⁡\(p,λ\)p^\{\(\\lambda\)\}\\in\\arg\\min\_\{p\}L\(p,\\lambda\)Distribution\-level optimal solution atλ\{\\lambda\}u\(λ\)u^\{\(\\lambda\)\}Letπ\(λ\)\\pi^\{\(\\lambda\)\}be the coupling ofP0P\_\{0\}andP\(λ\)P^\{\(\\lambda\)\}, i\.e\.,\(x0,x1\)∼π\(λ\),xt=\(1−t\)​x0\+t​x1\.\(x\_\{0\},x\_\{1\}\)\\sim\\pi^\{\(\\lambda\)\},x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\}\.u\(λ\)​\(x,t\):=𝔼\(x0,x1\)∼π\(λ\)​\[x1−x0∣xt=x\]u^\{\(\\lambda\)\}\(x,t\):=\\mathbb\{E\}\_\{\(x\_\{0\},x\_\{1\}\)\\sim\\pi^\{\(\\lambda\)\}\}\\left\[x\_\{1\}\-x\_\{0\}\\mid x\_\{t\}=x\\right\]The velocity field ofP\(λ\)P^\{\(\\lambda\)\}uθu\_\{\\theta\}The velocity field parameterized byθ\\thetaΦu,t\\Phi\_\{u,t\}Φu,t​\(x\)=x\+∫0tu⁡\(Φu,s​\(x\),s\)​𝑑s\\Phi\_\{u,t\}\(x\)=x\+\\int\_\{0\}^\{t\}u\(\\Phi\_\{u,s\}\(x\),s\)dsPushforward operatorPuP\_\{u\}Pu:=\(Φu,1\)\#​P0P\_\{u\}:=\(\\Phi\_\{u,1\}\)\_\{\\\#\}P\_\{0\}Distribution induced byuuΘ,UΘ\\Theta,U\_\{\\Theta\}UΘ:=\{uθ:θ∈Θ\}U\_\{\\Theta\}:=\\\{u\_\{\\theta\}:\\theta\\in\\Theta\\\}The parameter familyand the image of the parameterization\.L¯​\(uθ,λ\)\\bar\{L\}\(u\_\{\\theta\},\\lambda\)L¯​\(uθ,λ\)=L⁡\(puθ,λ\)\\bar\{L\}\(u\_\{\\theta\},\\lambda\)=L\(p\_\{u\_\{\\theta\}\},\\lambda\)The Lagrangian after parameterization\.g¯​\(λ\)\\bar\{g\}\(\\lambda\)g¯​\(λ\)=infu∈​UΘL¯​\(u,λ\)\\bar\{g\}\(\\lambda\)=\\inf\_\{u\_\{\\in\}U\_\{\\Theta\}\}\\bar\{L\}\(u,\\lambda\)The dual function after parameterization\.λ¯∗\\bar\{\\lambda\}^\{\*\}λ¯∗∈arg⁡maxλ≥0​g¯​\(λ\)\\bar\{\\lambda\}^\{\*\}\\in\\arg\\max\_\{\\lambda\\geq 0\}\\bar\{g\}\(\\lambda\)uθ\(λ\)u\_\{\\theta\}^\{\(\\lambda\)\}uθ\(λ\)∈arg⁡minu∈UΘ​L¯​\(u,λ\)u\_\{\\theta\}^\{\(\\lambda\)\}\\in\\arg\\min\_\{u\\in U\_\{\\Theta\}\}\\bar\{L\}\(u,\\lambda\)Puθ\(λ\)P\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}Puθ\(λ\)=\(Φuθ\(λ\),1\)\#​P0P\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}=\(\\Phi\_\{u\_\{\\theta\}^\{\(\\lambda\)\},1\}\)\_\{\\\#\}P\_\{0\}Distribution induced byuθ\(λ\)u\_\{\\theta\}^\{\(\\lambda\)\}Φuθ,tΔ​t\\Phi^\{\\Delta t\}\_\{u\_\{\\theta\},t\}Defined in \([11](https://arxiv.org/html/2610.12699#S4.E11)\)Euler samplerL′​\(uθ,λ\)L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)L′​\(uθ,λ\)=λ⁡\(b−Puθ​\(𝒞\)\)L^\{\\prime\}\(u\_\{\\theta\},\\lambda\)=\\lambda\(b\-P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\+w​𝔼t∼Unif⁡\[0,1\],x0,x1​\[‖uθ​\(xt,t\)−\(x1−x0\)‖2\]\+w\\mathbb\{E\}\_\{t\\sim\\mathrm\{Unif\}\[0,1\],x\_\{0\},x\_\{1\}\}\\left\[\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\(x\_\{1\}\-x\_\{0\}\)\\\|^\{2\}\\right\]Surrogate Lagrangianfor practical optimizationuθ′\(λ\)u^\{\\prime\(\\lambda\)\}\_\{\\theta\}The learned velocity field bythe iterative method in Sec\.[4\.2](https://arxiv.org/html/2610.12699#S4.SS2)Puθ′\(λ\)P^\{\(\\lambda\)\}\_\{u^\{\\prime\}\_\{\\theta\}\}Puθ′\(λ\)=\(Φu′θ\(λ\),1\)\#​P0P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}=\(\\Phi\_\{\{u^\{\\prime\}\}\_\{\\theta\}^\{\(\\lambda\)\},1\}\)\_\{\\\#\}P\_\{0\}The distribution induced byu′θ\(λ\)\{u^\{\\prime\}\}^\{\(\\lambda\)\}\_\{\\theta\}

## Appendix CProofs for Lemma[2\.2](https://arxiv.org/html/2610.12699#S2.Thmtheorem2)and Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1)

#### Statement of Lemma[2\.2](https://arxiv.org/html/2610.12699#S2.Thmtheorem2)\.

*Under Assumption[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),*

p\(λ∗\)​\(x\)=b​q​\(x\)Q⁡\(𝒞\)​𝟏𝒞​\(x\)\+\(1−b\)​q​\(x\)1−Q⁡\(𝒞\)​𝟏𝒞c​\(x\),λ∗=1−Q⁡\(𝒞\)1−b−Q⁡\(𝒞\)b\.p^\{\(\\lambda^\{\*\}\)\}\(x\)=\\frac\{bq\(x\)\}\{Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{\(1\-b\)q\(x\)\}\{1\-Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\),~~\\lambda^\{\*\}=\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{1\-b\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{b\}\.\(23\)
###### Proof\.

Under Assumption[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1), the primal problem is convex and satisfies Slater’s condition\. Therefore, strong duality holds and the duality gap is zero\.

The optimization is over probability densitiesp≥0p\\geq 0satisfying∫p⁡\(x\)​𝑑x=1\\int p\(x\)dx=1\. Every optimizer with finite objective satisfiesp\>0p\>0,QQ\-almost everywhere\. We introduce the Lagrangian:

𝒥\(p,μ\)=∫−q\(x\)logp\(x\)dx\+λ\(b−∫𝒞p\(x\)dx\)\+μ\(∫p\(x\)dx−1\),\\mathcal\{J\}\(p,\\mu\)=\\int\-q\(x\)\\log p\(x\)d\{x\}\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\right\)\+\\mu\\left\(\\int p\(x\)dx\-1\\right\),\(24\)in whichλ\\lambdais a fixed multiplier for the inequality constraint andμ\\muis the multiplier for the equality constraint\. The KKT condition shows

∂𝒥∂p⁡\(x\)=−q⁡\(x\)p⁡\(x\)−λ​𝟏𝒞​\(x\)\+μ=0\.\\frac\{\\partial\\mathcal\{J\}\}\{\\partial p\(x\)\}=\-\\frac\{q\(x\)\}\{p\(x\)\}\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\mu=0\.\(25\)The solution ofp⁡\(x\)p\(x\)is

p⁡\(x\)=q⁡\(x\)μ−λ​𝟏𝒞​\(x\),equivalently,p⁡\(x\)=q⁡\(x\)μ−λ​𝟏𝒞​\(x\)\+q⁡\(x\)μ​𝟏𝒞c​\(x\)\.p\(x\)=\\frac\{q\(x\)\}\{\\mu\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\},\\quad\\text\{equivalently\},\\quad p\(x\)=\\frac\{q\(x\)\}\{\\mu\-\\lambda\}\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{q\(x\)\}\{\\mu\}\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\)\.\(26\)SinceQ⁡\(𝒞\)<bQ\(\\mathcal\{C\}\)<b, the constraint is active \(otherwisep=qp=qwould be the solution\), therefore

\{∫𝒞p∗​\(x\)​dx=Q⁡\(𝒞\)μ−λ=b,∫p∗​\(x\)​dx=Q⁡\(𝒞\)μ−λ\+1−Q⁡\(𝒞\)μ=1,⇒\{λ∗=1−Q⁡\(𝒞\)1−b−Q⁡\(𝒞\)b,μ∗=1−Q⁡\(𝒞\)1−b\.\\begin\{cases\}\\int\_\{\\mathcal\{C\}\}p^\{\*\}\(x\)dx=\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\-\\lambda\}=b,\\\\ \\int p^\{\*\}\(x\)dx=\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\-\\lambda\}\+\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{\\mu\}=1,\\end\{cases\}\\Rightarrow\\begin\{cases\}\\lambda^\{\*\}=\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{1\-b\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{b\},\\\\ \\mu^\{\*\}=\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{1\-b\}\.\\end\{cases\}\(27\)Givenλ∗,μ∗\\lambda^\{\*\},\\mu^\{\*\}, we have

p\(λ∗\)​\(x\)=b​q​\(x\)Q⁡\(𝒞\)​𝟏𝒞​\(x\)\+\(1−b\)​q​\(x\)1−Q⁡\(𝒞\)​𝟏𝒞c​\(x\)\.p^\{\(\\lambda^\{\*\}\)\}\(x\)=\\frac\{bq\(x\)\}\{Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{\(1\-b\)q\(x\)\}\{1\-Q\(\\mathcal\{C\}\)\}\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\)\.\(28\)Since the feasible set is convex and the objective∫−q\(x\)logp\(x\)dx\\int\-q\(x\)\\log p\(x\)dxis strictly convex inppon the set whereq⁡\(x\)\>0q\(x\)\>0, the KKT solution is unique on\{x:q⁡\(x\)\>0\}\\\{x:q\(x\)\>0\\\}and the optimizer is unique\.

∎

#### Statement of Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1)\.

*LetP1,P2P\_\{1\},P\_\{2\}be probability measures defined on the same measurable space\(Ω,𝒜\)\(\\Omega,\\mathcal\{A\}\)with‖P1−P2‖TV=δ\\\|P\_\{1\}\-P\_\{2\}\\\|\_\{\\mathrm\{TV\}\}=\\delta,*

max⁡\{0,P2​\(𝒞\)−δ\}≤P1​\(𝒞\)≤min⁡\{1,P2​\(𝒞\)\+δ\}\.\\max\\\{0,P\_\{2\}\(\\mathcal\{C\}\)\-\\delta\\\}\\leq P\_\{1\}\(\\mathcal\{C\}\)\\leq\\min\\\{1,P\_\{2\}\(\\mathcal\{C\}\)\+\\delta\\\}\.\(29\)
###### Proof\.

Note that the definition of total variation is

δ=‖P1−P2‖TV=supA∈𝒜\|P1​\(A\)−P2​\(A\)\|≥\|P1​\(𝒞\)−P2​\(𝒞\)\|\.\\delta=\\\|P\_\{1\}\-P\_\{2\}\\\|\_\{\\mathrm\{TV\}\}=\\sup\_\{A\\in\\mathcal\{A\}\}\|P\_\{1\}\(A\)\-P\_\{2\}\(A\)\|\\geq\|P\_\{1\}\(\\mathcal\{C\}\)\-P\_\{2\}\(\\mathcal\{C\}\)\|\.\(30\)ThereforeP1​\(𝒞\)∈\[P2​\(𝒞\)−δ,P2​\(𝒞\)\+δ\]P\_\{1\}\(\\mathcal\{C\}\)\\in\[P\_\{2\}\(\\mathcal\{C\}\)\-\\delta,P\_\{2\}\(\\mathcal\{C\}\)\+\\delta\]\. Also consider thatP1​\(𝒞\)∈\[0,1\]P\_\{1\}\(\\mathcal\{C\}\)\\in\[0,1\]by definition, we can conclude thatmax⁡\{0,P2​\(𝒞\)−δ\}≤P1​\(𝒞\)≤min⁡\{1,P2​\(𝒞\)\+δ\}\\max\\\{0,P\_\{2\}\(\\mathcal\{C\}\)\-\\delta\\\}\\leq P\_\{1\}\(\\mathcal\{C\}\)\\leq\\min\\\{1,P\_\{2\}\(\\mathcal\{C\}\)\+\\delta\\\}\. ∎

## Appendix DProofs for Constraint guarantees with FM Parameterization

To control the total variation distance between the FM\-parameterized solutionPuθ\(λ¯∗\)P^\{\(\{\\bar\{\\lambda\}^\{\*\}\}\)\}\_\{u\_\{\\theta\}\}and the optimal solution to \([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\)P\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}, we decompose this error as

‖Puθ\(λ¯∗\)−P\(λ∗\)‖TV≤‖Puθ\(λ¯∗\)−P\(λ¯∗\)‖TV\+‖P\(λ¯∗\)−P\(λ∗\)‖TV,\\\|P^\{\(\{\\bar\{\\lambda\}^\{\*\}\}\)\}\_\{u\_\{\\theta\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\{\\theta\}\}\}\-P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\+\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\},\(31\)in which the first term on the RHS captures the approximation error introduced by restricting the solution to the FM\-parameterized classUΘU\_\{\\Theta\}at fixed dual variableλ¯∗\\bar\{\\lambda\}^\{\*\}, while the second term captures the distributional error caused by the dual\-variable mismatch betweenλ¯∗\\bar\{\\lambda\}^\{\*\}andλ∗\\lambda^\{\*\}\.

### D\.1Bound‖Puθ\(λ¯∗\)−P\(λ¯∗\)‖TV\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\{\\theta\}\}\}\-P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}

###### Lemma D\.1\.

Under Assumption[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2), for any dual variableλ\\lambdaconsidered in the analysis, there exist uniform constantsA1,A2\>0A\_\{1\},A\_\{2\}\>0such that

KL⁡\(P\(λ\)∥Puθ\(λ\)\)≤A1​ν\+A2​ν2,‖Puθ\(λ\)−P\(λ\)‖TV≤12​\(A1​ν\+A2​ν2\)\.\\mathrm\{KL\}\\left\(P^\{\(\\lambda\)\}\\middle\\\|P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}\\right\)\\leq A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\},\\quad\\left\\\|P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq\\sqrt\{\\frac\{1\}\{2\}\\left\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\\right\)\}\.\(32\)whereA1,A2\>0A\_\{1\},A\_\{2\}\>0are uniform upper bounds on the regularity constants in Theorem 3\.1 of[Su et al\. \[2025\]](https://arxiv.org/html/2610.12699#bib.bib15)over all dual variablesλ\\lambdaconsidered in the analysis\. They depend only on the worst\-case smoothness and boundedness of the corresponding probability paths and velocity fields in Assumption[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2)\.

###### Proof\.

The KL boundKL⁡\(P\(λ\)∥Puθ\(λ\)\)≤A1​ν\+A2​ν2\\mathrm\{KL\}\\left\(P^\{\(\\lambda\)\}\\middle\\\|P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}\\right\)\\leq A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}follows directly from Theorem 3\.1 of[Su et al\. \[2025\]](https://arxiv.org/html/2610.12699#bib.bib15)under Assumption[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2), withA1,A2A\_\{1\},A\_\{2\}chosen as uniform upper bounds over all dual variablesλ\\lambdaconsidered in the analysis\. The total variation bound‖Puθ\(λ\)−P\(λ\)‖TV≤12​\(A1​ν\+A2​ν2\)\\left\\\|P^\{\(\\lambda\)\}\_\{u\_\{\\theta\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq\\sqrt\{\\frac\{1\}\{2\}\\left\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\\right\)\}then follows immediately from Pinsker’s inequality\. ∎

Plugging inλ=λ¯∗\\lambda=\\bar\{\\lambda\}^\{\*\}in Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1)leads to

‖Puθ\(λ¯∗\)−P\(λ¯∗\)‖TV≤12​\(A1​ν\+A2​ν2\)\.\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\{\\theta\}\}\}\-P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\sqrt\{\\frac\{1\}\{2\}\\left\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\\right\)\}\.\(33\)

### D\.2Bound‖P\(λ¯∗\)−P\(λ∗\)‖TV\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}

###### Lemma D\.2\.

Under Assumption[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1), for dual variablesλ1,λ2≥0\\lambda\_\{1\},\\lambda\_\{2\}\\geq 0,‖P\(λ1\)−P\(λ2\)‖TV\\\|P^\{\(\\lambda\_\{1\}\)\}\-P^\{\(\\lambda\_\{2\}\)\}\\\|\_\{\\mathrm\{TV\}\}is bounded in the sense that

‖P\(λ1\)−P\(λ2\)‖TV≤1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​\|λ1−λ2\|\.\\\|P^\{\(\\lambda\_\{1\}\)\}\-P^\{\(\\lambda\_\{2\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\|\\lambda\_\{1\}\-\{\\lambda\_\{2\}\}\|\.\(34\)
###### Proof\.

To compare the TV distance betweenP\(λ1\)P^\{\(\\lambda\_\{1\}\)\}andP\(λ2\)P^\{\(\\lambda\_\{2\}\)\}, we first fixλ\\lambdaand find the solution of

minimize𝔼x∼Q\[−logp\(x\)\]\+λ\(b−∫𝒞p\(x\)dx\),s\.t\.∫p\(x\)dx=1,p\(x\)≥0\.\\mathrm\{minimize\}~\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\(x\)\]\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\right\),~~\\mathrm\{s\.t\.\}~~\\int p\(x\)dx=1,p\(x\)\\geq 0\.\(35\)p⁡\(x\)≥0p\(x\)\\geq 0automatically holds since the primal objective includeslog⁡p⁡\(x\)\\log p\(x\)\. Then we form a Lagrangian:

𝒥\(p,λ,μ\)=∫−q\(x\)logp\(x\)dx\+λ\(b−∫𝒞p\(x\)dx\)\+μ\(∫p\(x\)dx−1\),\\mathcal\{J\}\(p,\\lambda,\\mu\)=\\int\-q\(x\)\\log p\(x\)d\{x\}\+\\lambda\\left\(b\-\\int\_\{\\mathcal\{C\}\}p\(x\)dx\\right\)\+\\mu\\left\(\\int p\(x\)dx\-1\\right\),\(36\)in whichλ\\lambdais a fixed multiplier for the inequality constraint andμ\\muis the multiplier for the equality constraint\. The KKT condition shows

∂𝒥∂p⁡\(x\)=−q⁡\(x\)p⁡\(x\)−λ​𝟏𝒞​\(x\)\+μ=0\.\\frac\{\\partial\\mathcal\{J\}\}\{\\partial p\(x\)\}=\-\\frac\{q\(x\)\}\{p\(x\)\}\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\mu=0\.\(37\)The solution ofp\(λ\)​\(x\)p^\{\(\\lambda\)\}\(x\)is

p\(λ\)​\(x\)=q⁡\(x\)μ−λ​𝟏𝒞​\(x\),equivalently,p\(λ\)​\(x\)=q⁡\(x\)μ−λ​𝟏𝒞​\(x\)\+q⁡\(x\)μ​𝟏𝒞c​\(x\)\.p^\{\(\\lambda\)\}\(x\)=\\frac\{q\(x\)\}\{\\mu\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\},\\quad\\text\{equivalently\},\\quad p^\{\(\\lambda\)\}\(x\)=\\frac\{q\(x\)\}\{\\mu\-\\lambda\}\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{q\(x\)\}\{\\mu\}\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\)\.\(38\)Integrate in𝒞\\mathcal\{C\}and𝒞c\\mathcal\{C\}^\{c\}respectively leads to

\{P\(λ\)​\(𝒞\)=∫𝒞p\(λ\)​\(x\)​dx=Q⁡\(𝒞\)μ−λ,1−P\(λ\)​\(𝒞\)=∫𝒞cp\(λ\)​\(x\)​dx=Q⁡\(𝒞c\)μ\.\\begin\{cases\}P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)=\\int\_\{\\mathcal\{C\}\}p^\{\(\\lambda\)\}\(x\)dx=\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\-\\lambda\},\\\\ 1\-P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)=\\int\_\{\\mathcal\{C\}^\{c\}\}p^\{\(\\lambda\)\}\(x\)dx=\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{\\mu\}\.\\end\{cases\}\(39\)This leads to a relationship betweenλ,μ\\lambda,\\muandP\(λ\)​\(𝒞\)P^\{\(\\lambda\)\}\(\\mathcal\{C\}\):

λ=Q⁡\(𝒞c\)1−P\(λ\)​\(𝒞\)−Q⁡\(𝒞\)P\(λ\)​\(𝒞\),μ=Q⁡\(𝒞c\)1−P\(λ\)​\(𝒞\)\.\\lambda=\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{1\-P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\},~~\\mu=\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{1\-P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\}\.\(40\)Substituting back into \([38](https://arxiv.org/html/2610.12699#A4.E38)\) leads to

p\(λ\)​\(x\)=P\(λ\)​\(𝒞\)Q⁡\(𝒞\)​q​\(x\)​𝟏𝒞​\(x\)\+1−P\(λ\)​\(𝒞\)Q⁡\(𝒞c\)​q​\(x\)​𝟏𝒞c​\(x\)\.p^\{\(\\lambda\)\}\(x\)=\\frac\{P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\}\{Q\(\\mathcal\{C\}\)\}q\(x\)\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\+\\frac\{1\-P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\}\{Q\(\\mathcal\{C\}^\{c\}\)\}q\(x\)\\bm\{1\}\_\{\\mathcal\{C\}^\{c\}\}\(x\)\.\(41\)Now we evaluate the total variation using the expression in \([41](https://arxiv.org/html/2610.12699#A4.E41)\),

‖P\(λ1\)−P\(λ2\)‖TV=12​∫\|p\(λ1\)​\(x\)−p\(λ2\)​\(x\)\|​𝑑x=12​\(∫𝒞\|p\(λ1\)​\(x\)−p\(λ2\)​\(x\)\|​dx\+∫𝒞c\|p\(λ1\)​\(x\)−p\(λ2\)​\(x\)\|​dx\)=12\(∫𝒞\|P\(λ1\)​\(𝒞\)−P\(λ2\)​\(𝒞\)Q⁡\(𝒞\)q\(x\)\|dx\+∫𝒞c\|P\(λ1\)​\(𝒞\)−P\(λ2\)​\(𝒞\)Q⁡\(𝒞cCLOSE\)\)q\(x\)\|dx\)=12​\(\|P\(λ1\)​\(𝒞\)−P\(λ2\)​\(𝒞\)\|\+\|P\(λ1\)​\(𝒞\)−P⁡\(λ2\)​\(𝒞\)\|\)=\|P\(λ1\)​\(𝒞\)−P\(λ2\)​\(𝒞\)\|,\\begin\{aligned\} \\\|P^\{\(\\lambda\_\{1\}\)\}\-P^\{\(\\lambda\_\{2\}\)\}\\\|\_\{\\mathrm\{TV\}\}&=\\frac\{1\}\{2\}\\int\|p^\{\(\\lambda\_\{1\}\)\}\(x\)\-p^\{\(\\lambda\_\{2\}\)\}\(x\)\|dx\\\\ &=\\frac\{1\}\{2\}\\left\(\\int\_\{\\mathcal\{C\}\}\|p^\{\(\\lambda\_\{1\}\)\}\(x\)\-p^\{\(\\lambda\_\{2\}\)\}\(x\)\|dx\+\\int\_\{\\mathcal\{C\}^\{c\}\}\|p^\{\(\\lambda\_\{1\}\)\}\(x\)\-p^\{\(\\lambda\_\{2\}\)\}\(x\)\|dx\\right\)\\\\ &=\\frac\{1\}\{2\}\\left\(\\int\_\{\\mathcal\{C\}\}\\left\|\\frac\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\}\{Q\(\\mathcal\{C\}\)\}q\(x\)\\right\|dx\+\\int\_\{\\mathcal\{C\}^\{c\}\}\\left\|\\frac\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\}\{Q\(\\mathcal\{C\}^\{c\}\}\)\)q\(x\)\\right\|dx\\right\)\\\\ &=\\frac\{1\}\{2\}\\left\(\|P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\|\+\|P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\|\\right\)\\\\ &=\|P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\|,\\end\{aligned\}

\(42\)in which the third “=” is by plugging in \([41](https://arxiv.org/html/2610.12699#A4.E41)\)\.

Based on \([40](https://arxiv.org/html/2610.12699#A4.E40)\), define a functionΛ⁡\(r\)\\Lambda\(r\)as follows:

Λ⁡\(r\):=Q⁡\(𝒞c\)1−r−Q⁡\(𝒞\)r,with derivativeΛ′​\(r\)=Q⁡\(𝒞c\)\(1−r\)2\+Q⁡\(𝒞\)r2\.\\Lambda\(r\):=\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{1\-r\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{r\},\\quad\\text\{with derivative\}\\quad\\Lambda^\{\\prime\}\(r\)=\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{\(1\-r\)^\{2\}\}\+\\frac\{Q\(\\mathcal\{C\}\)\}\{r^\{2\}\}\.\(43\)
Note that∀r∈\(0,1\),Λ′​\(r\)\>0\\forall r\\in\(0,1\),\\Lambda^\{\\prime\}\(r\)\>0, the mean\-value theorem shows

∃ξ∈\(min⁡\{P\(λ1\)​\(𝒞\),P\(λ2\)​\(𝒞\)\},max⁡\{P\(λ1\)​\(𝒞\),P\(λ2\)​\(𝒞\)\}\),\\exists\\xi\\in\\left\(\\min\\\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\),P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\\\},\\max\\\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\),P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\\\}\\right\),

\|P\(λ1\)​\(𝒞\)−P\(λ2\)​\(𝒞\)\|\\displaystyle\|P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\|=\|λ1−λ2\|Λ′​\(ξ\)\\displaystyle=\\frac\{\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\}\{\\Lambda^\{\\prime\}\(\\xi\)\}\(44\)≤\|λ1−λ2\|infr∈\(min⁡\{P\(λ1\)​\(𝒞\),P\(λ2\)​\(𝒞\)\},max⁡\{P\(λ1\)​\(𝒞\),P\(λ2\)​\(𝒞\)\}\)Q⁡\(𝒞c\)\(1−r\)2\+Q⁡\(𝒞\)r2\\displaystyle\\leq\\frac\{\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\}\{\\inf\_\{r\\in\\left\(\\min\\\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\),P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\\\},\\max\\\{P^\{\(\\lambda\_\{1\}\)\}\(\\mathcal\{C\}\),P^\{\(\\lambda\_\{2\}\)\}\(\\mathcal\{C\}\)\\\}\\right\)\}\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{\(1\-r\)^\{2\}\}\+\\frac\{Q\(\\mathcal\{C\}\)\}\{r^\{2\}\}\}≤\|λ1−λ2\|infr∈\(0,1\)Q⁡\(𝒞c\)\(1−r\)2\+Q⁡\(𝒞\)r2\.\\displaystyle\\leq\\frac\{\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\}\{\\inf\_\{r\\in\(0,1\)\}\\frac\{Q\(\\mathcal\{C\}^\{c\}\)\}\{\(1\-r\)^\{2\}\}\+\\frac\{Q\(\\mathcal\{C\}\)\}\{r^\{2\}\}\}\.Let∂Λ′​\(r\)∂r=0\\frac\{\\partial\\Lambda^\{\\prime\}\(r\)\}\{\\partial r\}=0leads torm​i​n=t1\+t,t=\(Q⁡\(𝒞\)1−Q⁡\(𝒞\)\)1/3r\_\{min\}=\\frac\{t\}\{1\+t\},t=\\left\(\\frac\{Q\(\\mathcal\{C\}\)\}\{1\-Q\(\\mathcal\{C\}\)\}\\right\)^\{1/3\},Λ′​\(rm​i​n\)=\(1\+t\)2​\(1−Q⁡\(𝒞\)\+Q⁡\(𝒞\)​t−2\)\\Lambda^\{\\prime\}\(r\_\{min\}\)=\(1\+t\)^\{2\}\(1\-Q\(\\mathcal\{C\}\)\+Q\(\\mathcal\{C\}\)t^\{\-2\}\)\. This leads to the bound

‖P\(λ1\)−P\(λ2\)‖TV≤\|λ1−λ2\|\(1\+t\)2​\(1−Q⁡\(𝒞\)\+Q⁡\(𝒞\)​t−2\)=\|λ1−λ2\|\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3\.\\\|P^\{\(\\lambda\_\{1\}\)\}\-P^\{\(\\lambda\_\{2\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\frac\{\|\\lambda\_\{1\}\-\{\\lambda\_\{2\}\}\|\}\{\(1\+t\)^\{2\}\(1\-Q\(\\mathcal\{C\}\)\+Q\(\\mathcal\{C\}\)t^\{\-2\}\)\}=\\frac\{\|\\lambda\_\{1\}\-\{\\lambda\_\{2\}\}\|\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\.\(45\)∎

Substitutingλ¯∗\\bar\{\\lambda\}^\{\*\}andλ∗\\lambda^\{\*\}into Lemma[D\.2](https://arxiv.org/html/2610.12699#A4.Thmtheorem2), we can obtain

‖P\(λ¯∗\)−P\(λ∗\)‖TV≤1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​\|λ¯∗−λ∗\|\.\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\-P^\{\(\{\\lambda\}^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\|\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\|\.\(46\)

###### Lemma D\.3\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1)and[3\.3](https://arxiv.org/html/2610.12699#S3.Thmtheorem3),g⁡\(λ\)g\(\\lambda\)iskk\-strongly concave inλ∈\[0,λmax\]\\lambda\\in\[0,\\lambda\_\{\\max\}\], in which

k=minλ∈\[0,λmax\]−Q⁡\(𝒞\)\(μ⁡\(λ\)−λ\)2​\(−μ⁡\(λ\)\+λ\+Q⁡\(𝒞\)2​μ​\(λ\)−λ−1\),\\displaystyle k=\\min\_\{\\lambda\\in\[0,\\lambda\_\{\\max\}\]\}\-\\frac\{Q\(\\mathcal\{C\}\)\}\{\(\\mu\(\\lambda\)\-\\lambda\)^\{2\}\}\\left\(\\frac\{\-\\mu\(\\lambda\)\+\\lambda\+Q\(\\mathcal\{C\}\)\}\{2\\mu\(\\lambda\)\-\\lambda\-1\}\\right\),\(47\)μ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\.\\displaystyle\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\.

###### Proof\.

Continue from \([38](https://arxiv.org/html/2610.12699#A4.E38)\)p\(λ\)​\(x\)=q⁡\(x\)μ−λ​𝟏𝒞​\(x\),μ\>λp^\{\(\\lambda\)\}\(x\)=\\frac\{q\(x\)\}\{\\mu\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\},\\mu\>\\lambda\. The requirement∫p⁡\(x\)​𝑑x=1\\int p\(x\)dx=1leads to

Q⁡\(𝒞\)μ−λ\+1−Q⁡\(𝒞\)μ=1,μ2−\(λ\+1\)​μ\+λ⁡\(1−Q⁡\(𝒞\)\)=0\.\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\-\\lambda\}\+\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{\\mu\}=1,\\quad\\mu^\{2\}\-\(\\lambda\+1\)\\mu\+\\lambda\(1\-Q\(\\mathcal\{C\}\)\)=0\.\(48\)The solutions forμ\\muare

μ⁡\(λ\)=12​\(λ\+1±\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\.\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\\pm\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\.\(49\)Note thatμ−λ\>0\\mu\-\\lambda\>0must hold\. Ifμ⁡\(λ\)=12​\(λ\+1−\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\-\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\),

μ⁡\(λ\)−λ=12​\(−λ\+1−\(λ−1\)2\+4​λ​Q​\(𝒞\)\)≤0\.\\displaystyle\\mu\(\\lambda\)\-\\lambda=\\frac\{1\}\{2\}\\left\(\-\\lambda\+1\-\\sqrt\{\(\\lambda\-1\)^\{2\}\+4\\lambda Q\(\\mathcal\{C\}\)\}\\right\)\\leq 0\.\(50\)This root satisfiesμ⁡\(λ\)−λ≤0\\mu\(\\lambda\)\-\\lambda\\leq 0, with equality atλ=0\\lambda=0, and therefore violates the requirementμ\>λ\\mu\>\\lambda\. This eliminates one solution, and the solution toμ\\muis

μ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\.\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\.\(51\)Substitute the solutions forλ,p\(λ\)\\lambda,p^\{\(\\lambda\)\}intog⁡\(λ\)g\(\\lambda\)and note thatλ​Q​\(𝒞\)=\(μ−1\)​\(μ−λ\)\\lambda Q\(\\mathcal\{C\}\)=\(\\mu\-1\)\(\\mu\-\\lambda\)from \([48](https://arxiv.org/html/2610.12699#A4.E48)\),

g⁡\(λ\)\\displaystyle g\(\\lambda\)=𝔼x∼Q​\[−log⁡p\(λ\)​\(x\)\]\+λ⁡\(b−∫𝒞p\(λ\)​\(x\)​dx\)\\displaystyle=\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p^\{\(\\lambda\)\}\(x\)\]\+\\lambda\(b\-\\int\_\{\\mathcal\{C\}\}p^\{\(\\lambda\)\}\(x\)dx\)\(52\)=𝔼x∼Q​\[−log⁡q⁡\(x\)\]\+𝔼x∼Q​\[log⁡\(μ⁡\(λ\)−λ​𝟏𝒞​\(x\)\)\]\+λ​b−λ​Q⁡\(𝒞\)μ⁡\(λ\)−λ\\displaystyle=\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log q\(x\)\]\+\\mathbb\{E\}\_\{x\\sim Q\}\[\\log\(\\mu\(\\lambda\)\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\)\]\+\\lambda b\-\\lambda\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\(\\lambda\)\-\\lambda\}=1\+𝔼x∼Q​\[−log⁡q⁡\(x\)\]\+𝔼x∼Q​\[log⁡\(μ⁡\(λ\)−λ​𝟏𝒞​\(x\)\)\]\+λ​b−μ⁡\(λ\)\.\\displaystyle=1\+\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log q\(x\)\]\+\\mathbb\{E\}\_\{x\\sim Q\}\[\\log\(\\mu\(\\lambda\)\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\)\]\+\\lambda b\-\\mu\(\\lambda\)\.Then we can check the concavity ofgg,

g′​\(λ\)=∂g∂λ\\displaystyle g^\{\\prime\}\(\\lambda\)=\\frac\{\\partial g\}\{\\partial\\lambda\}=𝔼x∼Q​\[∂μ∂λ−𝟏𝒞​\(x\)μ⁡\(λ\)−λ​𝟏𝒞​\(x\)\]\+b−∂μ∂λ\\displaystyle=\\mathbb\{E\}\_\{x\\sim Q\}\\left\[\\frac\{\\frac\{\\partial\\mu\}\{\\partial\\lambda\}\-\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\}\{\\mu\(\\lambda\)\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\}\\right\]\+b\-\\frac\{\\partial\\mu\}\{\\partial\\lambda\}\(53\)=−Q⁡\(𝒞\)μ⁡\(λ\)−λ\+b\+∂μ∂λ​\(Q⁡\(𝒞\)μ⁡\(λ\)−λ\+1−Q⁡\(𝒞\)μ⁡\(λ\)−1\)\\displaystyle=\-\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\(\\lambda\)\-\\lambda\}\+b\+\\frac\{\\partial\\mu\}\{\\partial\\lambda\}\\left\(\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\(\\lambda\)\-\\lambda\}\+\\frac\{1\-Q\(\\mathcal\{C\}\)\}\{\\mu\(\\lambda\)\}\-1\\right\)=−Q⁡\(𝒞\)μ⁡\(λ\)−λ\+b\.\\displaystyle=\-\\frac\{Q\(\\mathcal\{C\}\)\}\{\\mu\(\\lambda\)\-\\lambda\}\+b\.g′′​\(λ\)=∂2g∂λ2\\displaystyle g^\{\\prime\\prime\}\(\\lambda\)=\\frac\{\\partial^\{2\}g\}\{\\partial\\lambda^\{2\}\}=Q⁡\(𝒞\)\(μ⁡\(λ\)−λ\)2​\(∂μ∂λ−1\)\\displaystyle=\\frac\{Q\(\\mathcal\{C\}\)\}\{\(\\mu\(\\lambda\)\-\\lambda\)^\{2\}\}\(\\frac\{\\partial\\mu\}\{\\partial\\lambda\}\-1\)\(54\)=Q⁡\(𝒞\)\(μ⁡\(λ\)−λ\)2​\(12​\(1\+λ\+1−2​\(1−Q​\(𝒞\)\)\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)−1\)\\displaystyle=\\frac\{Q\(\\mathcal\{C\}\)\}\{\(\\mu\(\\lambda\)\-\\lambda\)^\{2\}\}\\left\(\\frac\{1\}\{2\}\\left\(1\+\\frac\{\\lambda\+1\-2\(1\-Q\(\\mathcal\{C\}\)\)\}\{\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\}\\right\)\-1\\right\)=Q⁡\(𝒞\)\(μ⁡\(λ\)−λ\)2​\(12​\(−1\+λ−1\+2​Q​\(𝒞\)2​μ​\(λ\)−λ−1\)\)\\displaystyle=\\frac\{Q\(\\mathcal\{C\}\)\}\{\(\\mu\(\\lambda\)\-\\lambda\)^\{2\}\}\\left\(\\frac\{1\}\{2\}\\left\(\-1\+\\frac\{\\lambda\-1\+2Q\(\\mathcal\{C\}\)\}\{2\\mu\(\\lambda\)\-\\lambda\-1\}\\right\)\\right\)=Q⁡\(𝒞\)\(μ⁡\(λ\)−λ\)2​\(−μ⁡\(λ\)\+λ\+Q⁡\(𝒞\)2​μ​\(λ\)−λ−1\)\.\\displaystyle=\\frac\{Q\(\\mathcal\{C\}\)\}\{\(\\mu\(\\lambda\)\-\\lambda\)^\{2\}\}\\left\(\\frac\{\-\\mu\(\\lambda\)\+\\lambda\+Q\(\\mathcal\{C\}\)\}\{2\\mu\(\\lambda\)\-\\lambda\-1\}\\right\)\.2​μ​\(λ\)−λ−1\>02\\mu\(\\lambda\)\-\\lambda\-1\>0since it equals\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\. Note that0<Q⁡\(𝒞\)<b<10<Q\(\\mathcal\{C\}\)<b<1, \([48](https://arxiv.org/html/2610.12699#A4.E48)\) shows that

μ⁡\(λ\)−λQ⁡\(𝒞\)=μ⁡\(λ\)μ⁡\(λ\)−1\+Q⁡\(𝒞\)\>1\.\\frac\{\\mu\(\\lambda\)\-\\lambda\}\{Q\(\\mathcal\{C\}\)\}=\\frac\{\\mu\(\\lambda\)\}\{\\mu\(\\lambda\)\-1\+Q\(\\mathcal\{C\}\)\}\>1\.\(55\)Therefore−μ⁡\(λ\)\+λ\+Q⁡\(𝒞\)<0\-\\mu\(\\lambda\)\+\\lambda\+Q\(\\mathcal\{C\}\)<0andg′′​\(λ\)<0g^\{\\prime\\prime\}\(\\lambda\)<0\. Moreover,−g′′​\(λ\)\-g^\{\\prime\\prime\}\(\\lambda\)is continuous and strictly positive on the compact interval\[0,λmax\]\[0,\\lambda\_\{\\max\}\]\. Therefore,k:=minλ∈\[0,λmax\]−g′′​\(λ\)\>0k:=\\min\_\{\\lambda\\in\[0,\\lambda\_\{\\max\}\]\}\-g^\{\\prime\\prime\}\(\\lambda\)\>0and this proves thatggiskk\-strongly concave on this interval\. ∎

###### Lemma D\.4\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2)and[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4), for fixedλ\\lambda,

g¯​\(λ\)−g⁡\(λ\)≤μ⁡\(λ\)​\(A1​ν\+A2​ν2\),\\bar\{g\}\(\\lambda\)\-g\(\\lambda\)\\leq\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\),\(56\)in whichμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\.

###### Proof\.

From \([38](https://arxiv.org/html/2610.12699#A4.E38)\) and \([51](https://arxiv.org/html/2610.12699#A4.E51)\), we can obtain

q⁡\(x\)p\(λ\)​\(x\)=μ⁡\(λ\)−λ​𝟏𝒞​\(x\)≤μ⁡\(λ\),\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}=\\mu\(\\lambda\)\-\\lambda\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\)\\leq\\mu\(\\lambda\),\(57\)in whichμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\. Therefore,

g¯​\(λ\)−g​\(λ\)\\displaystyle\\bar\{g\}\(\\lambda\)\-g\(\\lambda\)=∫q⁡\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+λ​∫𝒞\(p\(λ\)​\(x\)−puθ\(λ\)​\(x\)\)​𝑑x\\displaystyle=\\int q\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\lambda\\int\_\{\\mathcal\{C\}\}\(p^\{\(\\lambda\)\}\(x\)\-p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\)dx\(58\)=∫q⁡\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+∫λ​𝟏𝒞​\(x\)​\(p\(λ\)​\(x\)−puθ\(λ\)​\(x\)\)​𝑑x\\displaystyle=\\int q\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\int\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\(p^\{\(\\lambda\)\}\(x\)\-p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\)dx=∫q⁡\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+∫\(μ−q⁡\(x\)p\(λ\)​\(x\)\)​\(p\(λ\)​\(x\)−puθ\(λ\)​\(x\)\)​𝑑x\\displaystyle=\\int q\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\int\\left\(\\mu\-\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\(p^\{\(\\lambda\)\}\(x\)\-p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\)dx\(Sinceλ​𝟏𝒞​\(x\)=μ−q⁡\(x\)p\(λ\)​\(x\)from \([38](https://arxiv.org/html/2610.12699#A4.E38)\)\)\\displaystyle\\text\{\(Since $\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)=\\mu\-\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}$ from \\eqref\{eq:sol1\}\)\}=∫q⁡\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+∫μ⁡\(p\(λ\)​\(x\)−puθ\(λ\)​\(x\)\)​𝑑x−∫\(q⁡\(x\)p\(λ\)​\(x\)\)​\(p\(λ\)​\(x\)−puθ\(λ\)​\(x\)\)​𝑑x\\displaystyle=\\int q\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\int\\mu\(p^\{\(\\lambda\)\}\(x\)\-p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\)dx\-\\int\\left\(\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\(p^\{\(\\lambda\)\}\(x\)\-p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\)dx=∫q⁡\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​dx\+∫q⁡\(x\)​\(puθ\(λ\)​\(x\)p\(λ\)​\(x\)\)​dx−1\\displaystyle=\\int q\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\int q\(x\)\\left\(\\frac\{\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)dx\-1=∫q⁡\(x\)​\[log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+\(puθ\(λ\)​\(x\)p\(λ\)​\(x\)\)−1\]​𝑑x\\displaystyle=\\int q\(x\)\\left\[\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\left\(\\frac\{\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\-1\\right\]dx=∫p\(λ\)​\(x\)​q⁡\(x\)p\(λ\)​\(x\)​\[log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+\(puθ\(λ\)​\(x\)p\(λ\)​\(x\)\)−1\]​𝑑x\\displaystyle=\\int p^\{\(\\lambda\)\}\(x\)\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}\\left\[\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\left\(\\frac\{\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\-1\\right\]dx≤μ⁡\(λ\)​∫p\(λ\)​\(x\)​\[log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+\(puθ\(λ\)​\(x\)p\(λ\)​\(x\)\)−1\]​𝑑x\\displaystyle\\leq\\mu\(\\lambda\)\\int p^\{\(\\lambda\)\}\(x\)\\left\[\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\left\(\\frac\{\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\-1\\right\]dx\(Since \([57](https://arxiv.org/html/2610.12699#A4.E57)\) and∀a\>0,−log⁡a\+a−1≥0\)\\displaystyle\\text\{\(Since \\eqref\{eq:mulambda\} and $\\forall a\>0,\-\\log a\+a\-1\\geq 0$\)\}=μ⁡\(λ\)​∫\[p\(λ\)​\(x\)​log⁡p\(λ\)​\(x\)puθ\(λ\)​\(x\)​𝑑x\+puθ\(λ\)​\(x\)−p\(λ\)​\(x\)\]​𝑑x\\displaystyle=\\mu\(\\lambda\)\\int\\left\[p^\{\(\\lambda\)\}\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\{\{p\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\-p^\{\(\\lambda\)\}\(x\)\\right\]dx=μ\(λ\)\(DKL\(P\(λ\)∥Puθ\(λ\)\)\+1−1\)\\displaystyle=\\mu\(\\lambda\)\\left\(D\_\{\\text\{KL\}\}\(P^\{\(\\lambda\)\}\\\|P\_\{u\_\{\\theta\}\}^\{\(\\lambda\)\}\)\+1\-1\\right\)≤μ\(λ\)\(A1ν\+A2ν2\)\(By Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1)\)\\displaystyle\\leq\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\\quad\\text\{\(By Lemma~\\ref\{lemma:TVgap1\}\)\}∎

###### Lemma D\.5\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2),[3\.3](https://arxiv.org/html/2610.12699#S3.Thmtheorem3),[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4), the dual variable gap between the FM\-parameterized Lagrangian’sλ¯∗\\bar\{\\lambda\}^\{\*\}and the distribution\-level Lagrangian’sλ∗\\lambda^\{\*\}can be bounded by

\|λ¯∗−λ∗\|2≤2k​μ​\(λ¯∗\)​\(A1​ν\+A2​ν2\)\.\|\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\|^\{2\}\\leq\\frac\{2\}\{k\}\\mu\(\\bar\{\\lambda\}^\{\*\}\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\.\(59\)

###### Proof\.

Bykk\-strong concavity,

g⁡\(λ¯∗\)≤g⁡\(λ∗\)\+∇λg​\(λ∗\)​\(λ¯∗−λ∗\)−k2​\|λ¯∗−λ∗\|2\.g\(\\bar\{\\lambda\}^\{\*\}\)\\leq g\(\\lambda^\{\*\}\)\+\\nabla\_\{\\lambda\}g\(\\lambda^\{\*\}\)\(\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\)\-\\frac\{k\}\{2\}\|\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\|^\{2\}\.\(60\)First check∇λg​\(λ∗\)​\(λ¯∗−λ∗\)\\nabla\_\{\\lambda\}g\(\\lambda^\{\*\}\)\(\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\)\. Note that∇λg​\(λ∗\)=0\\nabla\_\{\\lambda\}g\(\\lambda^\{\*\}\)=0, \([60](https://arxiv.org/html/2610.12699#A4.E60)\) becomesg⁡\(λ¯∗\)≤g⁡\(λ∗\)−k2​\|λ¯∗−λ∗\|2g\(\\bar\{\\lambda\}^\{\*\}\)\\leq g\(\\lambda^\{\*\}\)\-\\frac\{k\}\{2\}\|\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\|^\{2\}, i\.e\.,

k2​\|λ¯∗−λ∗\|2\\displaystyle\\frac\{k\}\{2\}\|\\bar\{\\lambda\}^\{\*\}\-\\lambda^\{\*\}\|^\{2\}≤g⁡\(λ∗\)−g⁡\(λ¯∗\)≤g⁡\(λ∗\)−\(g¯​\(λ¯∗\)−μ⁡\(λ¯∗\)​\(A1​ν\+A2​ν2\)\)\.\\displaystyle\\leq g\(\\lambda^\{\*\}\)\-g\(\\bar\{\\lambda\}^\{\*\}\)\\leq g\(\\lambda^\{\*\}\)\-\\left\(\\bar\{g\}\(\\bar\{\\lambda\}^\{\*\}\)\-\\mu\(\\bar\{\\lambda\}^\{\*\}\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\\right\)\.\(61\)
Consider the fact that

g⁡\(λ∗\)≤g¯​\(λ∗\)≤g¯​\(λ¯∗\),equivalently,g⁡\(λ∗\)−g¯​\(λ¯∗\)≤0,g\(\\lambda^\{\*\}\)\\leq\\bar\{g\}\(\\lambda^\{\*\}\)\\leq\\bar\{g\}\(\\bar\{\\lambda\}^\{\*\}\),\\quad\\text\{equivalently,\}\\quad g\(\\lambda^\{\*\}\)\-\\bar\{g\}\(\\bar\{\\lambda\}^\{\*\}\)\\leq 0,\(62\)we can finish this proof\. ∎

### D\.3Proof of Theorem[3\.5](https://arxiv.org/html/2610.12699#S3.Thmtheorem5)

###### Proof\.

Using \([31](https://arxiv.org/html/2610.12699#A4.E31)\), Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1), Lemma[D\.2](https://arxiv.org/html/2610.12699#A4.Thmtheorem2)and Lemma[D\.5](https://arxiv.org/html/2610.12699#A4.Thmtheorem5), the total variation bound can be given by

‖Puθ\(λ¯∗\)−P\(λ∗\)‖TV≤ℰpara,\\\|P^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\_\{u\_\{\\theta\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{para\}\},\(63\)in which

ℰpara=1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​2k​μ​\(λ¯∗\)​\(A1​ν\+A2​ν2\)\+12​\(A1​ν\+A2​ν2\)\\mathcal\{E\}\_\{\\mathrm\{para\}\}=\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\\sqrt\{\\frac\{2\}\{k\}\\mu\(\\bar\{\\lambda\}^\{\*\}\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\+\\sqrt\{\\frac\{1\}\{2\}\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\(64\)
From Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1), the constraint satisfaction ratePuθ\(λ¯∗\)​\(𝒞\)P\_\{u\_\{\\theta\}\}^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\(\\mathcal\{C\}\)relates to the sameℰpara\\mathcal\{E\}\_\{\\mathrm\{para\}\},

Puθ\(λ¯∗\)​\(𝒞\)≥b−ℰpara\.P\_\{u\_\{\\theta\}\}^\{\(\\bar\{\\lambda\}^\{\*\}\)\}\(\\mathcal\{C\}\)\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{para\}\}\.\(65\)∎

## Appendix EProof for Proposition[4\.1](https://arxiv.org/html/2610.12699#S4.Thmtheorem1)

#### Statement of Proposition[4\.1](https://arxiv.org/html/2610.12699#S4.Thmtheorem1)\.

*Under the assumption that∇θf\(uθ\)​\(xi​Δ​t,i​Δ​t\)\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\},i\\Delta t\)exists and is uniformly bounded for alli∈\{0,…,N−1\}i\\in\\\{0,\\ldots,N\-1\\\},i∼Unif​\{0,⋯,N−1\}i\\sim\\text\{Unif\}\\\{0,\\cdots,N\-1\\\}andxi​Δ​t∼\(Φuθ,i​Δ​tΔ​t\)\#​P0x\_\{i\\Delta t\}\\sim\(\\Phi^\{\\Delta t\}\_\{u\_\{\\theta\},i\\Delta t\}\)\_\{\\\#\}P\_\{0\},*

∇θℱuθ​\(𝒞\)=N​𝔼i,xi​Δ​t​\[∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\],\\nabla\_\{\\theta\}\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)=N\\mathbb\{E\}\_\{i,x\_\{i\\Delta t\}\}\\left\[\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\\right\],

\(66\)*in which∇θuθ​\(xt,t\)\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{t\},t\)is a Jacobian matrix such that each column is the gradient∇θ\[uθ​\(xt,t\)\]j\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{j\}of thejj\-th dimension of thedd\-dimensional velocity with respect to the velocity parameters, i\.e\.,*

∇θuθ​\(xt,t\)=\[∇θ\[uθ​\(xt,t\)\]1∇θ\[uθ​\(xt,t\)\]2⋯∇θ\[uθ​\(xt,t\)\]d\]\.\\nabla\_\{\\theta\}u\_\{\\theta\}\(x\_\{t\},t\)=\[\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{1\}\\quad\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{2\}\\quad\\cdots\\quad\\nabla\_\{\\theta\}\[u\_\{\\theta\}\(x\_\{t\},t\)\]\_\{d\}\]\.\(67\)
###### Proof\.

Fort=i​Δ​t,i∈\{0,1,⋯,N−1\}t=i\\Delta t,i\\in\\\{0,1,\\cdots,N\-1\\\},

∇θf\(uθ\)​\(xt,t\)=∇θf\(uθ\)\(xt\+uθ\(xt,t\)Δt,t\+Δt\)\(by property \(a\)\)=∇θuθ​\(xt,t\)​∇uf\(uθ\)​\(xt\+u​Δ​t,t\+Δ​t\)\|u=uθ​\(xt,t\)\+∇θf\(uθ\)​\(xt\+u​Δ​t,t\+Δ​t\)\|u=uθ​\(xt,t\)=∇θuθ​\(xt,t\)​∇uf\(uθ\)​\(xt\+u​Δ​t,t\+Δ​t\)\|u=uθ​\(xt,t\)\+∇θf\(uθ\)​\(xt\+Δ​t,t\+Δ​t\)\.\\begin\{aligned\} &\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\},t\)\\\\ &=\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\}\+u\_\{\\theta\}\(x\_\{t\},t\)\\Delta t,t\+\\Delta t\)\\qquad\\text\{\(by property \(a\)\)\}\\\\ &=\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{t\},t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\}\+u\\Delta t,t\+\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{t\},t\)\}\+\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\}\+u\\Delta t,t\+\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{t\},t\)\}\\\\ &=\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{t\},t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\}\+u\\Delta t,t\+\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{t\},t\)\}\+\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{t\+\\Delta t\},t\+\\Delta t\)\.\\end\{aligned\}

\(68\)Repeating this expansion yields

∇θf\(uθ\)​\(x0,0\)=∑i=0N−1∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\+∇θf\(uθ\)​\(x1,1\)=∑i=0N−1∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\.\(by property \(b\)\)\\begin\{aligned\} &\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)\\\\ &=\\sum\_\{i=0\}^\{N\-1\}\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\+\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{1\},1\)\\\\ &=\\sum\_\{i=0\}^\{N\-1\}\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\.\\qquad\\text\{\(by property \(b\)\)\}\\end\{aligned\}

\(69\)
Under the stated assumptions, by applying the dominated convergence theorem, one can exchange the integration and differentiation,

∇θℱuθ​\(𝒞\)=∇θ∫p0\(x0\)f\(uθ\)\(x0,0\)dx0=∫p0​\(x0\)​∇θf\(uθ\)​\(x0,0\)​d​x0=∫p0​\(x0\)​∑i=0N−1∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)​d​x0=𝔼xi​Δ​t​\[∑i=0N−1∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\]=N​𝔼i,xi​Δ​t​\[∇θuθ​\(xi​Δ​t,i​Δ​t\)​∇uf\(uθ\)​\(xi​Δ​t\+u​Δ​t,\(i\+1\)​Δ​t\)\|u=uθ​\(xi​Δ​t,i​Δ​t\)\]\.\\begin\{aligned\} \\nabla\_\{\\theta\}\\mathcal\{F\}\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)&=\\nabla\_\{\\theta\}\\int p\_\{0\}\(x\_\{0\}\)f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)dx\_\{0\}\\\\ &=\\int p\_\{0\}\(x\_\{0\}\)\\nabla\_\{\\theta\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{0\},0\)dx\_\{0\}\\\\ &=\\int p\_\{0\}\(x\_\{0\}\)\\sum\_\{i=0\}^\{N\-1\}\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}dx\_\{0\}\\\\ &=\\mathbb\{E\}\_\{x\_\{i\\Delta t\}\}\\left\[\\sum\_\{i=0\}^\{N\-1\}\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\\right\]\\\\ &=N\\mathbb\{E\}\_\{i,x\_\{i\\Delta t\}\}\\left\[\\nabla\_\{\\theta\}\\ u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\\nabla\_\{u\}f^\{\(u\_\{\\theta\}\)\}\(x\_\{i\\Delta t\}\+u\\Delta t,\(i\+1\)\\Delta t\)\|\_\{u=u\_\{\\theta\}\(x\_\{i\\Delta t\},i\\Delta t\)\}\\right\]\.\\end\{aligned\}

\(70\)
∎

## Appendix FProofs for Constraint guarantees on PDFM

To assess the TV gap between the distributionPuθ′\(λ⁡\(i\)\)\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}induced byuθ′\(λ⁡\(i\)\)u^\{\\prime\(\\lambda\(i\)\)\}\_\{\\theta\}solved by Alg\.[2](https://arxiv.org/html/2610.12699#alg2)at theii\-th iteration and the optimal solutionP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}to \([OPT](https://arxiv.org/html/2610.12699#S2.Ex1)\), we decompose this gap as

‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV≤‖Puθ′\(λ⁡\(i\)\)−P\(λ⁡\(i\)\)‖TV\+‖P\(λ⁡\(i\)\)−P\(λ∗\)‖TV\.\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda\(i\)\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\+\\left\\\|\{P^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\.\(71\)Similar to the decomposition in Appendix[D](https://arxiv.org/html/2610.12699#A4), the first term on the RHS captures the approximation error introduced by both restricting the solution to the FM\-parameterized classUΘU\_\{\\Theta\}and solving the minimizer with the approach in Sec\.[4\.2](https://arxiv.org/html/2610.12699#S4.SS2), while the second term captures the distributional error caused by the dual\-variable mismatch betweenλ⁡\(i\)\\lambda\(i\)andλ∗\\lambda^\{\*\}\.

### F\.1Bound‖Puθ′\(λ\)−P\(λ\)‖TV\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}

###### Lemma F\.1\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2),[3\.3](https://arxiv.org/html/2610.12699#S3.Thmtheorem3),[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4),[4\.2](https://arxiv.org/html/2610.12699#S4.Thmtheorem2)and[4\.3](https://arxiv.org/html/2610.12699#S4.Thmtheorem3), for allλ∈\[0,λmax\]\\lambda\\in\[0,\\lambda\_\{\\max\}\],

‖Puθ′\(λ\)−P\(λ\)‖TV≤e⁡\(λ\)≤emax,\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq e\(\\lambda\)\\leq e\_\{\\max\},\(72\)in whiche⁡\(λ\):=ε\+μ⁡\(λ\)​\(A1​ν\+A2​ν2\)2​\(μ​\(λ\)−λ\)e\(\\lambda\):=\\sqrt\{\\frac\{\\varepsilon\+\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\{2\(\\mu\(\\lambda\)\-\\lambda\)\}\},μ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\),emax:=maxλ∈\[0,λmax\]⁡e⁡\(λ\)\.e\_\{\\max\}:=\\max\_\{\\lambda\\in\[0,\\lambda\_\{\\max\}\]\}e\(\\lambda\)\.

###### Proof\.

From Assumption[4\.3](https://arxiv.org/html/2610.12699#S4.Thmtheorem3)and Lemma[D\.4](https://arxiv.org/html/2610.12699#A4.Thmtheorem4),

L⁡\(puθ′\(λ\),λ\)−g⁡\(λ\)=L⁡\(puθ′\(λ\),λ\)−g¯​\(λ\)\+g¯​\(λ\)−g⁡\(λ\)≤ε\+μ⁡\(λ\)​\(A1​ν\+A2​ν2\)\.\\begin\{aligned\} L\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\)\-g\(\\lambda\)&=L\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\)\-\\bar\{g\}\(\\lambda\)\+\\bar\{g\}\(\\lambda\)\-g\(\\lambda\)\\leq\\varepsilon\+\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\.\\end\{aligned\}

\(73\)
From \([38](https://arxiv.org/html/2610.12699#A4.E38)\) and \([51](https://arxiv.org/html/2610.12699#A4.E51)\), we can obtain

q⁡\(x\)p\(λ\)​\(x\)=μ−λ​𝟏𝒞​\(x\)≥μ⁡\(λ\)−λ,\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}=\{\\mu\-\\lambda\\bm\{1\}\_\{\\mathcal\{C\}\}\(x\)\}\\geq\\mu\(\\lambda\)\-\\lambda,\(74\)in whichμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)\. Therefore,

L⁡\(puθ′\(λ\),λ\)−g⁡\(λ\)\\displaystyle L\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\)\-g\(\\lambda\)\(75\)=∫p\(λ\)\(x\)q⁡\(x\)p\(λ\)​\(x\)\[logp\(λ\)​\(x\)puθ′\(λ\)​\(x\)dx\+\(puθ′\(λ\)​\(x\)p\(λ\)​\(x\)\)−1\]dx\(With the same procedure as \([58](https://arxiv.org/html/2610.12699#A4.E58)\)\)\\displaystyle=\\int p^\{\(\\lambda\)\}\(x\)\\frac\{q\(x\)\}\{p^\{\(\\lambda\)\}\(x\)\}\\left\[\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\left\(\\frac\{\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\-1\\right\]dx\\quad\\text\{\(With the same procedure as \\eqref\{eq:gbarg\}\)\}≥\(μ⁡\(λ\)−λ\)​∫p\(λ\)​\(x\)​\[log⁡p\(λ\)​\(x\)puθ′\(λ\)​\(x\)​𝑑x\+\(puθ′\(λ\)​\(x\)p\(λ\)​\(x\)\)−1\]​𝑑x\\displaystyle\\geq\(\\mu\(\\lambda\)\-\\lambda\)\\int p^\{\(\\lambda\)\}\(x\)\\left\[\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\\left\(\\frac\{\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\{p^\{\(\\lambda\)\}\(x\)\}\\right\)\-1\\right\]dx=\(μ⁡\(λ\)−λ\)​∫\[p\(λ\)​\(x\)​log⁡p\(λ\)​\(x\)puθ′\(λ\)​\(x\)​𝑑x\+puθ′\(λ\)​\(x\)−p\(λ\)​\(x\)\]​𝑑x\\displaystyle=\(\\mu\(\\lambda\)\-\\lambda\)\\int\\left\[p^\{\(\\lambda\)\}\(x\)\\log\\frac\{p^\{\(\\lambda\)\}\(x\)\}\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}dx\+\{\{p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(x\)\}\}\-p^\{\(\\lambda\)\}\(x\)\\right\]dx=\(μ\(λ\)−λ\)\(DKL\(P\(λ\)∥Puθ′\(λ\)\)\+1−1\)\\displaystyle=\(\\mu\(\\lambda\)\-\\lambda\)\\left\(D\_\{\\text\{KL\}\}\(P^\{\(\\lambda\)\}\\\|P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\)\+1\-1\\right\)≥\(μ⁡\(λ\)−λ\)⋅2​‖Puθ′\(λ\)−P\(λ\)‖TV2\\displaystyle\\geq\(\\mu\(\\lambda\)\-\\lambda\)\\cdot 2\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}^\{2\}=2​\(μ⁡\(λ\)−λ\)​‖Puθ′\(λ\)−P\(λ\)‖TV2\.\\displaystyle=2\(\\mu\(\\lambda\)\-\\lambda\)\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}^\{2\}\.Sinceμ⁡\(λ\)−λ\>0\\mu\(\\lambda\)\-\\lambda\>0ande⁡\(λ\)e\(\\lambda\)is continuous on the compact interval\[0,λmax\]\[0,\\lambda\_\{\\max\}\], its finite maximumemaxe\_\{\\max\}is attained\. ∎

### F\.2Bound‖P\(λ⁡\(i\)\)−P\(λ∗\)‖TV\\\|P^\{\(\\lambda\(i\)\)\}\-P^\{\(\\lambda^\{\*\}\)\}\\\|\_\{\\mathrm\{TV\}\}

###### Lemma F\.2\.

Under Assumption[4\.3](https://arxiv.org/html/2610.12699#S4.Thmtheorem3)and follow Lemma[F\.1](https://arxiv.org/html/2610.12699#A6.Thmtheorem1),

\|∇λL​\(puθ′\(λ\),λ\)−∇λg​\(λ\)\|\\displaystyle\|\\nabla\_\{\\lambda\}L\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\)\-\\nabla\_\{\\lambda\}g\(\\lambda\)\|≤e⁡\(λ\)\.\\displaystyle\\leq e\(\\lambda\)\.\(76\)

###### Proof\.

\|∇λL​\(puθ′\(λ\),λ\)−∇λg​\(λ\)\|\\displaystyle\|\\nabla\_\{\\lambda\}L\(p\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\},\\lambda\)\-\\nabla\_\{\\lambda\}g\(\\lambda\)\|=\|Puθ′\(λ\)​\(𝒞\)−P\(λ\)​\(𝒞\)\|≤‖Puθ′\(λ\)−P\(λ\)‖TV≤e⁡\(λ\)\\displaystyle=\\left\|P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\-P^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\\right\|\\leq\\left\\\|P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\)\}\-P^\{\(\\lambda\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq e\(\\lambda\)\(77\)
∎

###### Lemma F\.3\.

Under Assumption[4\.4](https://arxiv.org/html/2610.12699#S4.Thmtheorem4), with∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty,

lim supi→∞\|λ⁡\(i\)−λ∗\|2≤4​emax2k2,\\limsup\_\{i\\rightarrow\\infty\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\\leq\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\},\(78\)

###### Proof\.

Note thatλ⁡\(0\)=λ′​\(0\)=0\\lambda\(0\)=\\lambda^\{\\prime\}\(0\)=0, and in Alg\.[2](https://arxiv.org/html/2610.12699#alg2),λ′=λ/w\\lambda^\{\\prime\}=\\lambda/wis updated asλ′​\(i\+1\)←\[λ′​\(i\)\+ηi​\(b−Puθ′\(λ⁡\(i\)\)​\(𝒞\)\)\]\+\\lambda^\{\\prime\}\(i\+1\)\\leftarrow\[\\lambda^\{\\prime\}\(i\)\+\\eta\_\{i\}\(b\-P^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\)\)\]\_\{\+\}\. The update inλ\\lambdais equivalent to

λ⁡\(i\+1\)=\[λ⁡\(i\)\+w​ηi​\(b−Puθ′\(λ⁡\(i\)\)​\(𝒞\)\)\]\+=\[λ⁡\(i\)\+w​ηi​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\]\+\.\\lambda\(i\+1\)=\\left\[\\lambda\(i\)\+w\\eta\_\{i\}\\left\(b\-P^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\)\\right\)\\right\]\_\{\+\}=\\left\[\\lambda\(i\)\+w\\eta\_\{i\}\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\\right\]\_\{\+\}\.\(79\)Next we analyze the gap betweenλ⁡\(i\)\\lambda\(i\)andλ∗\\lambda^\{\*\},

\|λ⁡\(i\+1\)−λ∗\|2=\|\[λ⁡\(i\)\+w​ηi​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\]\+−λ∗\|2≤\|λ⁡\(i\)\+w​ηi​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−λ∗\|2​\(Nonexpansiveness of projection\)=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\+\(w​ηi\)2​\[∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\]2≤\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\+\(w​ηi\)2\(Note that\[∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\]2≤1since∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)=b−Puθ′\(λ⁡\(i\)\)​\(𝒞\)∈\[b−1,b\]\)=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​\(∇λg​\(λ⁡\(i\)\)\+∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−∇λg​\(λ⁡\(i\)\)\)\+\(w​ηi\)2=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​∇λg​\(λ⁡\(i\)\)\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​\(∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−∇λg​\(λ⁡\(i\)\)\)\+\(w​ηi\)2=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​\(∇λg​\(λ⁡\(i\)\)−∇λg​\(λ∗\)\)\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​\(∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−∇λg​\(λ⁡\(i\)\)\)\+\(w​ηi\)2\(Since​∇λg​\(λ∗\)=0\)≤\|λ⁡\(i\)−λ∗\|2−k​w​ηi​\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​\(∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−∇λg​\(λ⁡\(i\)\)\)\+\(w​ηi\)2\(k\-strong concavity, Lemma[D\.3](https://arxiv.org/html/2610.12699#A4.Thmtheorem3)\)≤\|λ⁡\(i\)−λ∗\|2−k​w​ηi​\|λ⁡\(i\)−λ∗\|2\+2​w​ηi\|λ⁡\(i\)−λ∗\|\|∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)−∇λg​\(λ⁡\(i\)\)\|\+\(w​ηi\)2≤\|λ⁡\(i\)−λ∗\|2−k​w​ηi​\|λ⁡\(i\)−λ∗\|2\+2​w​ηi\|λ⁡\(i\)−λ∗\|e⁡\(λ⁡\(i\)\)\+\(w​ηi\)2\(Lemma[F\.2](https://arxiv.org/html/2610.12699#A6.Thmtheorem2)\)≤\(1−k​w​ηi\)​\|λ⁡\(i\)−λ∗\|2\+2​w​ηi\|λ⁡\(i\)−λ∗\|emax\+\(w​ηi\)2\(Lemma[F\.1](https://arxiv.org/html/2610.12699#A6.Thmtheorem1)\)≤\(1−k​w​ηi\)​\|λ⁡\(i\)−λ∗\|2\+k2​w​ηi​\|λ⁡\(i\)−λ∗\|2\+2k​w​ηi​emax2\+\(w​ηi\)2≤\(1−k2​w​ηi\)​\|λ⁡\(i\)−λ∗\|2\+w​ηi​2k​emax2\+\(w​ηi\)2\.\\begin\{aligned\} &\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\\\\ &=\|\[\\lambda\(i\)\+w\\eta\_\{i\}\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\]\_\{\+\}\-\{\\lambda\}^\{\*\}\|^\{2\}\\\\ &\\leq\|\\lambda\(i\)\+w\\eta\_\{i\}\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\{\\lambda\}^\{\*\}\|^\{2\}~~\\text\{\(Nonexpansiveness of projection\)\}\\\\ &=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\left\[\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\\right\]^\{2\}\\\\ &\\leq\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\\\ &\\qquad\\left\(\\text\{ Note that $\\left\[\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\\right\]^\{2\}\\leq 1$ since $\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)=b\-P^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\}\(\\mathcal\{C\}\)\\in\[b\-1,b\]$\}\\right\)\\\\ &=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\left\(\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\+\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\\right\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\\\ &=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\left\(\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\\right\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\\\ &=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\(\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda^\{\*\}\)\)\\\\ &~~~~\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\left\(\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\\right\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\quad\(\\text\{Since \}\\nabla\_\{\\lambda\}g\(\\lambda^\{\*\}\)=0\)\\\\ &\\leq\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\-kw\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\left\(\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\\right\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\\\ &\\qquad\\text\{\($k$\-strong concavity, Lemma~\\ref\{lemma:kconcave\}\)\}\\\\ &\\leq\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\-kw\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|\\left\|\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\-\\nabla\_\{\\lambda\}g\(\\lambda\(i\)\)\\right\|\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\\\ &\\leq\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\-kw\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|e\(\\lambda\(i\)\)\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\quad\\text\{\(Lemma~\\ref\{lemma:dualgap2\}\)\}\\\\ &\\leq\(1\-kw\\eta\_\{i\}\)\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|e\_\{\\max\}\+\{\(w\\eta\_\{i\}\)^\{2\}\}\\quad\\text\{\(Lemma~\\ref\{lemma:TVgapthetai\}\)\}\\\\ &\\leq\(1\-kw\\eta\_\{i\}\)\\left\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\\right\|^\{2\}\+\\frac\{k\}\{2\}w\\eta\_\{i\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+\\frac\{2\}\{k\}w\\eta\_\{i\}e\_\{\\max\}^\{2\}\+\(w\\eta\_\{i\}\)^\{2\}\\\\ &\\leq\(1\-\\frac\{k\}\{2\}w\\eta\_\{i\}\)\\left\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\\right\|^\{2\}\+w\\eta\_\{i\}\\frac\{2\}\{k\}e\_\{\\max\}^\{2\}\+\(w\\eta\_\{i\}\)^\{2\}\.\\end\{aligned\}

\(80\)
Rearranging the terms,

\|λ⁡\(i\+1\)−λ∗\|2−4​emax2k2≤\(1−k2​w​ηi\)​\(\|λ⁡\(i\)−λ∗\|2−4​emax2k2\)\+\(w​ηi\)2\.\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\(1\-\\frac\{k\}\{2\}w\\eta\_\{i\}\)\(\\left\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\\right\|^\{2\}\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\)\+\(w\\eta\_\{i\}\)^\{2\}\.\(81\)Unrolling this recursion yields

\|λ⁡\(i\+1\)−λ∗\|2−4​emax2k2≤\(\|λ⁡\(0\)−λ∗\|2−4​emax2k2\)​∏j=0i\(1−k2​w​ηj\)\+∑n=0i\(∏j=n\+1i\(1−k2​w​ηj\)\)​\(w​ηn\)2,\\displaystyle\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\\left\(\|\\lambda\(0\)\-\\lambda^\{\*\}\|^\{2\}\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\right\)\\prod\_\{j=0\}^\{i\}\(1\-\\frac\{k\}\{2\}w\\eta\_\{j\}\)\+\\sum\_\{n=0\}^\{i\}\\left\(\\prod\_\{j=n\+1\}^\{i\}\(1\-\\frac\{k\}\{2\}w\\eta\_\{j\}\)\\right\)\(w\\eta\_\{n\}\)^\{2\},

\(82\)and we define the empty product∏j=i\+1i\(⋅\)=1\\prod\_\{j=i\+1\}^\{i\}\(\\cdot\)=1for the last term\.

With∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty, \([82](https://arxiv.org/html/2610.12699#A6.E82)\) implies that

lim supi→∞\|λ⁡\(i\)−λ∗\|2≤4​emax2k2\.\\displaystyle\\limsup\_\{i\\rightarrow\\infty\}\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\\leq\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\.\(83\)∎

### F\.3Proof of Theorem[4\.5](https://arxiv.org/html/2610.12699#S4.Thmtheorem5)

###### Proof\.

By Lemma[F\.1](https://arxiv.org/html/2610.12699#A6.Thmtheorem1)and Lemma[D\.2](https://arxiv.org/html/2610.12699#A4.Thmtheorem2),

‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV\\displaystyle\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}≤‖Puθ′\(λ⁡\(i\)\)−P\(λ⁡\(i\)\)‖TV\+‖P\(λ⁡\(i\)\)−P\(λ∗\)‖TV\\displaystyle\\leq\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda\(i\)\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\+\\left\\\|\{P^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\(84\)≤e⁡\(λ⁡\(i\)\)\+1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​\|λ⁡\(i\)−λ∗\|\.\\displaystyle\\leq e\(\\lambda\(i\)\)\+\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\|\{\\lambda\(i\)\}\-\{\\lambda^\{\*\}\}\|\.By Lemma[F\.3](https://arxiv.org/html/2610.12699#A6.Thmtheorem3),

lim supi→∞‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV≤ℰPDFM,ℰPDFM=emax\+2​emaxk​\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3\.\\limsup\_\{i\\to\\infty\}\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\\quad\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}=e\_\{\\max\}\+\\frac\{2e\_\{\\max\}\}\{k\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\.\(85\)and by Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1)the constraint satisfaction rate satisfies

lim infi→∞Puθ′\(λ⁡\(i\)\)​\(𝒞\)≥b−ℰPDFM\.\\liminf\_\{i\\to\\infty\}P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}\.\(86\)
∎

## Appendix GConvergence under Unbiased Constraint Estimation

Assumption[4\.4](https://arxiv.org/html/2610.12699#S4.Thmtheorem4)simplifies the analysis by treating the empirical estimate of the constraint satisfaction gap as exact at every dual update\. In practice, however,Puθ​\(𝒞\)P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)is estimated using a finite number of generated samples and is therefore stochastic\. We relax Assumption[4\.4](https://arxiv.org/html/2610.12699#S4.Thmtheorem4)by requiring the Monte Carlo estimator to be conditionally unbiased given the current dual variable\.

###### Assumption G\.1\(Unbiased Monte Carlo estimation\)\.

For each fixed dual variableλ\\lambda, letuθ′\(λ\)u\_\{\\theta\}^\{\\prime\(\\lambda\)\}denote the corresponding velocity field\. The dual gradient∇λL​\(puθ′\(λ\),λ\)=b−Puθ′\(λ\)​\(𝒞\)\\nabla\_\{\\lambda\}L\\\!\\left\(p\_\{u\_\{\\theta\}^\{\\prime\}\}^\{\(\\lambda\)\},\\lambda\\right\)=b\-P\_\{u\_\{\\theta\}^\{\\prime\}\}^\{\(\\lambda\)\}\(\\mathcal\{C\}\)can be estimated via Monte Carlo approximation\. Letgrad⁡\(λ\)\\mathrm\{grad\}\(\\lambda\)denote the resulting estimator\. At iterationii, defineℱi=\(λ⁡\(i\),uθ′\(λ⁡\(i\)\)\)\\mathcal\{F\}\_\{i\}=\(\\lambda\(i\),u\_\{\\theta\}^\{\\prime\(\\lambda\(i\)\)\}\), which consists of the current dual variable and the corresponding velocity field\. We assume that the estimator is conditionally unbiased givenℱi\\mathcal\{F\}\_\{i\}, i\.e\.,

𝔼⁡\[grad⁡\(λ⁡\(i\)\)\|ℱi\]=∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)=b−Puθ′\(λ⁡\(i\)\)​\(𝒞\)\.\\mathbb\{E\}\\left\[\\mathrm\{grad\}\(\\lambda\(i\)\)\\,\\middle\|\\,\\mathcal\{F\}\_\{i\}\\right\]=\\nabla\_\{\\lambda\}L\(p\_\{u\_\{\\theta\}^\{\\prime\}\}^\{\(\\lambda\(i\)\)\},\\lambda\(i\)\)=b\-P\_\{u\_\{\\theta\}^\{\\prime\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\.\(87\)

###### Lemma G\.2\.

Under Assumption[G\.1](https://arxiv.org/html/2610.12699#A7.Thmtheorem1)instead of Assumption[4\.4](https://arxiv.org/html/2610.12699#S4.Thmtheorem4), with∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty,

lim supi→∞𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\]≤4​emax2k2\.\\displaystyle\\limsup\_\{i\\rightarrow\\infty\}\\mathbb\{E\}\[\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\]\\leq\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\.\(88\)

###### Proof\.

Note thatλ⁡\(0\)=λ′​\(0\)=0\\lambda\(0\)=\\lambda^\{\\prime\}\(0\)=0, and in Alg\.[2](https://arxiv.org/html/2610.12699#alg2),λ′=λ/w\\lambda^\{\\prime\}=\\lambda/wis updated in the wayλ′​\(i\+1\)←\[λ′​\(i\)\+ηi​grad​\(λ⁡\(i\)\)\]\+\\lambda^\{\\prime\}\(i\+1\)\\leftarrow\[\\lambda^\{\\prime\}\(i\)\+\\eta\_\{i\}\\mathrm\{grad\}\(\\lambda\(i\)\)\]\_\{\+\}\. The update inλ\\lambdais equivalent to

λ⁡\(i\+1\)=\[λ⁡\(i\)\+w​ηi​grad​\(λ⁡\(i\)\)\]\+\.\\lambda\(i\+1\)=\\left\[\\lambda\(i\)\+w\\eta\_\{i\}\\mathrm\{grad\}\(\\lambda\(i\)\)\\right\]\_\{\+\}\.\(89\)Therefore,

\|λ⁡\(i\+1\)−λ∗\|2\\displaystyle\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}=\|\[λ⁡\(i\)\+w​ηi​grad​\(λ⁡\(i\)\)\]\+−λ∗\|2\\displaystyle=\|\[\\lambda\(i\)\+w\\eta\_\{i\}\\mathrm\{grad\}\(\\lambda\(i\)\)\]\_\{\+\}\-\{\\lambda\}^\{\*\}\|^\{2\}\(90\)≤\|λ⁡\(i\)\+w​ηi​grad​\(λ⁡\(i\)\)−λ∗\|2\\displaystyle\\leq\|\\lambda\(i\)\+w\\eta\_\{i\}\\text\{grad\}\(\\lambda\(i\)\)\-\{\\lambda\}^\{\*\}\|^\{2\}=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​grad​\(λ⁡\(i\)\)\+\(w​ηi\)2​grad​\(λ⁡\(i\)\)2\\displaystyle=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\mathrm\{grad\}\(\\lambda\(i\)\)\+\(w\\eta\_\{i\}\)^\{2\}\\mathrm\{grad\}\(\\lambda\(i\)\)^\{2\}
Take the expectation conditioning onℱi\\mathcal\{F\}\_\{i\}\. After applying the bounds from the proof of Lemma F\.3, we condition the resulting inequality onλ⁡\(i\)\\lambda\(i\)using the tower property to obtain the recurrence below\.

𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\|ℱi\]\\displaystyle\\mathbb\{E\}\[\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\|\\mathcal\{F\}\_\{i\}\]\(91\)≤𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​grad​\(λ⁡\(i\)\)\+\(w​ηi\)2​grad​\(λ⁡\(i\)\)2\|ℱi\]\\displaystyle\\leq\\mathbb\{E\}\\left\[\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\text\{grad\}\(\\lambda\(i\)\)\+\(w\\eta\_\{i\}\)^\{2\}\\text\{grad\}\(\\lambda\(i\)\)^\{2\}\|\\mathcal\{F\}\_\{i\}\\right\]=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​𝔼​\[grad​\(λ⁡\(i\)\)\|ℱi\]\+\(w​ηi\)2​𝔼​\[grad​\(λ⁡\(i\)\)2\|ℱi\]\\displaystyle=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\mathbb\{E\}\[\\text\{grad\}\(\\lambda\(i\)\)\|\\mathcal\{F\}\_\{i\}\]\+\(w\\eta\_\{i\}\)^\{2\}\\mathbb\{E\}\[\\text\{grad\}\(\\lambda\(i\)\)^\{2\}\|\\mathcal\{F\}\_\{i\}\]=\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\+\(w​ηi\)2​𝔼​\[grad​\(λ⁡\(i\)\)2\|ℱi\]\\displaystyle=\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\+\(w\\eta\_\{i\}\)^\{2\}\\mathbb\{E\}\[\\text\{grad\}\(\\lambda\(i\)\)^\{2\}\|\\mathcal\{F\}\_\{i\}\]\(According to Assumption[G\.1](https://arxiv.org/html/2610.12699#A7.Thmtheorem1)\)\\displaystyle\\text\{\(According to Assumption~\\ref\{ass:unbiasedMC\}\)\}≤\|λ⁡\(i\)−λ∗\|2\+2​w​ηi​\(λ⁡\(i\)−λ∗\)​∇λL​\(puθ′\(λ⁡\(i\)\),λ⁡\(i\)\)\+\(w​ηi\)2\\displaystyle\\leq\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\+2w\\eta\_\{i\}\(\\lambda\(i\)\-\{\\lambda\}^\{\*\}\)\\nabla\_\{\\lambda\}L\(p^\{\(\\lambda\(i\)\)\}\_\{u^\{\\prime\}\_\{\\theta\}\},\\lambda\(i\)\)\+\(w\\eta\_\{i\}\)^\{2\}\(grad​\(λ⁡\(i\)\)2≤1since it is a Monte Carlo estimate of\(b−Puθ′\(λ\)​\(𝒞\)\)2≤max⁡\{b2,\(b−1\)2\}\)\\displaystyle\\text\{\($\\text\{grad\}\(\\lambda\(i\)\)^\{2\}\\leq 1$ since it is a Monte Carlo estimate of $\\left\(b\-P\_\{u\_\{\\theta\}^\{\\prime\}\}^\{\(\\lambda\)\}\(\\mathcal\{C\}\)\\right\)^\{2\}\\leq\\max\\\{b^\{2\},\(b\-1\)^\{2\}\\\}$\)\}
Note that the last line of \([91](https://arxiv.org/html/2610.12699#A7.E91)\) arrives at the fifth line of \([80](https://arxiv.org/html/2610.12699#A6.E80)\)\. With the same procedure as in the proof of Lemma[F\.3](https://arxiv.org/html/2610.12699#A6.Thmtheorem3), we first obtain the recursion conditioned onℱi\\mathcal\{F\}\_\{i\}\. Sinceλ⁡\(i\)\\lambda\(i\)is a component ofℱi\\mathcal\{F\}\_\{i\}and the right\-hand side depends only onλ⁡\(i\)\\lambda\(i\), conditioning this recursion onλ⁡\(i\)\\lambda\(i\)and applying the tower property gives

𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\|λ⁡\(i\)\]−4​emax2k2≤\(1−k2​w​ηi\)​\(\|λ⁡\(i\)−λ∗\|2−4​emax2k2\)\+\(w​ηi\)2\.\\mathbb\{E\}\\left\[\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\|\\lambda\(i\)\\right\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\(1\-\\frac\{k\}\{2\}w\\eta\_\{i\}\)\(\\left\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\\right\|^\{2\}\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\)\+\(w\\eta\_\{i\}\)^\{2\}\.\(92\)Taking expectation on both sides of \([92](https://arxiv.org/html/2610.12699#A7.E92)\) and applying the tower property gives

𝔼⁡\[𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\|λ⁡\(i\)\]\]−4​emax2k2≤\(1−k2​w​ηi\)​\(𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\]−4​emax2k2\)\+\(w​ηi\)2\.\\mathbb\{E\}\\left\[\\mathbb\{E\}\\left\[\|\\lambda\(i\+1\)\-\\lambda^\{\*\}\|^\{2\}\\,\\middle\|\\,\\lambda\(i\)\\right\]\\right\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\\left\(1\-\\frac\{k\}\{2\}w\\eta\_\{i\}\\right\)\\left\(\\mathbb\{E\}\\left\[\|\\lambda\(i\)\-\\lambda^\{\*\}\|^\{2\}\\right\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\right\)\+\(w\\eta\_\{i\}\)^\{2\}\.\(93\)By the law of total expectation,

𝔼⁡\[𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\|λ⁡\(i\)\]\]=𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\]\.\\mathbb\{E\}\\left\[\\mathbb\{E\}\\left\[\|\\lambda\(i\+1\)\-\\lambda^\{\*\}\|^\{2\}\\,\\middle\|\\,\\lambda\(i\)\\right\]\\right\]=\\mathbb\{E\}\\left\[\|\\lambda\(i\+1\)\-\\lambda^\{\*\}\|^\{2\}\\right\]\.\(94\)Therefore,

𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\]−4​emax2k2≤\(1−k2​w​ηi\)​\(𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\]−4​emax2k2\)\+\(w​ηi\)2\.\\mathbb\{E\}\\left\[\|\\lambda\(i\+1\)\-\\lambda^\{\*\}\|^\{2\}\\right\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\\left\(1\-\\frac\{k\}\{2\}w\\eta\_\{i\}\\right\)\\left\(\\mathbb\{E\}\\left\[\|\\lambda\(i\)\-\\lambda^\{\*\}\|^\{2\}\\right\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\right\)\+\(w\\eta\_\{i\}\)^\{2\}\.\(95\)
Unrolling this recursion yields

𝔼⁡\[\|λ⁡\(i\+1\)−λ∗\|2\]−4​emax2k2≤\(𝔼⁡\[\|λ⁡\(0\)−λ∗\|2\]−4​emax2k2\)​∏j=0i\(1−k2​w​ηj\)\+∑n=0i\(∏j=n\+1i\(1−k2​w​ηj\)\)​\(w​ηn\)2,\\displaystyle\\mathbb\{E\}\[\|\\lambda\(i\+1\)\-\{\\lambda\}^\{\*\}\|^\{2\}\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\leq\\left\(\\mathbb\{E\}\[\|\\lambda\(0\)\-\\lambda^\{\*\}\|^\{2\}\]\-\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\\right\)\\prod\_\{j=0\}^\{i\}\(1\-\\frac\{k\}\{2\}w\\eta\_\{j\}\)\+\\sum\_\{n=0\}^\{i\}\\left\(\\prod\_\{j=n\+1\}^\{i\}\(1\-\\frac\{k\}\{2\}w\\eta\_\{j\}\)\\right\)\(w\\eta\_\{n\}\)^\{2\},

\(96\)and we define the empty product∏j=i\+1i\(⋅\)=1\\prod\_\{j=i\+1\}^\{i\}\(\\cdot\)=1for the last term\.

With∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty, \([96](https://arxiv.org/html/2610.12699#A7.E96)\) implies that

lim supi→∞𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\]≤4​emax2k2\.\\displaystyle\\limsup\_\{i\\rightarrow\\infty\}\\mathbb\{E\}\[\|\\lambda\(i\)\-\{\\lambda\}^\{\*\}\|^\{2\}\]\\leq\\frac\{4e\_\{\\max\}^\{2\}\}\{k^\{2\}\}\.\(97\)∎

###### Theorem G\.3\.

\(Theorem[4\.5](https://arxiv.org/html/2610.12699#S4.Thmtheorem5)under unbiased Monte Carlo assumption instead of exact Monte Carlo assumption\) Under Assumptions[2\.1](https://arxiv.org/html/2610.12699#S2.Thmtheorem1),[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2)\-[3\.4](https://arxiv.org/html/2610.12699#S3.Thmtheorem4),[4\.2](https://arxiv.org/html/2610.12699#S4.Thmtheorem2),[4\.3](https://arxiv.org/html/2610.12699#S4.Thmtheorem3), and[G\.1](https://arxiv.org/html/2610.12699#A7.Thmtheorem1), with∀i,0≤ηi≤2k​w\\forall i,0\\leq\\eta\_\{i\}\\leq\\frac\{2\}\{kw\},∑i=1∞ηi=∞,∑i=1∞ηi2<∞\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}=\\infty,\\sum\_\{i=1\}^\{\\infty\}\\eta\_\{i\}^\{2\}<\\infty,

lim supi→∞𝔼⁡\[‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV\]≤ℰPDFM,lim infi→∞𝔼⁡\[Puθ′\(λ⁡\(i\)\)​\(𝒞\)\]≥b−ℰPDFM,\\limsup\_\{i\\to\\infty\}\\mathbb\{E\}\\left\[\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\right\]\\leq\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\\quad\\liminf\_\{i\\to\\infty\}\\mathbb\{E\}\\left\[P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\\right\]\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\(98\)ℰPDFM=emax\+2​emaxk​\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3,e⁡\(λ\):=ε\+μ⁡\(λ\)​\(A1​ν\+A2​ν2\)2​\(μ​\(λ\)−λ\),\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}=e\_\{\\max\}\+\\frac\{2e\_\{\\max\}\}\{k\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\},\\quad e\(\\lambda\):=\\sqrt\{\\frac\{\\varepsilon\+\\mu\(\\lambda\)\(A\_\{1\}\\nu\+A\_\{2\}\\nu^\{2\}\)\}\{2\(\\mu\(\\lambda\)\-\\lambda\)\}\},\(99\)in whichμ⁡\(λ\)=12​\(λ\+1\+\(λ\+1\)2−4​λ​\(1−Q⁡\(𝒞\)\)\)\\mu\(\\lambda\)=\\frac\{1\}\{2\}\\left\(\\lambda\+1\+\\sqrt\{\(\\lambda\+1\)^\{2\}\-4\\lambda\(1\-Q\(\\mathcal\{C\}\)\)\}\\right\)andemax=maxλ∈\[0,λmax\]⁡e⁡\(λ\)e\_\{\\max\}=\\max\_\{\\lambda\\in\[0,\\lambda\_\{\\max\}\]\}e\(\\lambda\)\.

###### Proof\.

By Lemma[F\.1](https://arxiv.org/html/2610.12699#A6.Thmtheorem1)and Lemma[D\.2](https://arxiv.org/html/2610.12699#A4.Thmtheorem2),

‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV\\displaystyle\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}≤‖Puθ′\(λ⁡\(i\)\)−P\(λ⁡\(i\)\)‖TV\+‖P\(λ⁡\(i\)\)−P\(λ∗\)‖TV\\displaystyle\\leq\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda\(i\)\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\+\\left\\\|\{P^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\(100\)≤e⁡\(λ⁡\(i\)\)\+1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​\|λ⁡\(i\)−λ∗\|\.\\displaystyle\\leq e\(\\lambda\(i\)\)\+\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\|\{\\lambda\(i\)\}\-\{\\lambda^\{\*\}\}\|\.Therefore,

𝔼⁡\[‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV\]\\displaystyle\\mathbb\{E\}\\left\[\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\right\]≤𝔼⁡\[e⁡\(λ⁡\(i\)\)\]\+1\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3​𝔼​\[\|λ⁡\(i\)−λ∗\|\]\\displaystyle\\leq\\mathbb\{E\}\[e\(\\lambda\(i\)\)\]\+\\frac\{1\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\\mathbb\{E\}\\left\[\|\{\\lambda\(i\)\}\-\{\\lambda^\{\*\}\}\|\\right\]\(101\)≤𝔼⁡\[e⁡\(λ⁡\(i\)\)\]\+𝔼⁡\[\|λ⁡\(i\)−λ∗\|2\]\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3\\displaystyle\\leq\\mathbb\{E\}\[e\(\\lambda\(i\)\)\]\+\\frac\{\\sqrt\{\\mathbb\{E\}\\left\[\|\{\\lambda\(i\)\}\-\{\\lambda^\{\*\}\}\|^\{2\}\\right\]\}\}\{\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}
By𝔼⁡\[e⁡\(λ⁡\(i\)\)\]≤emax\\mathbb\{E\}\[e\(\\lambda\(i\)\)\]\\leq e\_\{\\max\}and Lemma[G\.2](https://arxiv.org/html/2610.12699#A7.Thmtheorem2),

lim supi→∞𝔼⁡\[‖Puθ′\(λ⁡\(i\)\)−P\(λ∗\)‖TV\]≤ℰPDFM,ℰPDFM=emax\+2​emaxk​\(\(1−Q⁡\(𝒞\)\)1/3\+Q​\(𝒞\)1/3\)3\.\\limsup\_\{i\\to\\infty\}\\mathbb\{E\}\\left\[\\left\\\|\{P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\}\-P^\{\(\\lambda^\{\*\}\)\}\\right\\\|\_\{\\mathrm\{TV\}\}\\right\]\\leq\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\},\\quad\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}=e\_\{\\max\}\+\\frac\{2e\_\{\\max\}\}\{k\\left\(\(1\-Q\(\{\\mathcal\{C\}\}\)\)^\{1/3\}\+Q\(\{\\mathcal\{C\}\}\)^\{1/3\}\\right\)^\{3\}\}\.\(102\)and by Lemma[3\.1](https://arxiv.org/html/2610.12699#S3.Thmtheorem1)the expected constraint satisfaction rate satisfies

lim infi→∞𝔼⁡\[Puθ′\(λ⁡\(i\)\)​\(𝒞\)\]≥b−ℰPDFM\.\\liminf\_\{i\\to\\infty\}\\mathbb\{E\}\\left\[P\_\{u^\{\\prime\}\_\{\\theta\}\}^\{\(\\lambda\(i\)\)\}\(\\mathcal\{C\}\)\\right\]\\geq b\-\\mathcal\{E\}\_\{\\mathrm\{PDFM\}\}\.\(103\)
∎

## Appendix HExperiment Details

All experiments are run on an NVIDIA A100 GPU with8080GB of memory\.

We implement standard FM using independent random coupling and linear interpolants, following Algorithm 2 \(Simplified Conditional Flow Matching\) in[Tong et al\. \[2024\]](https://arxiv.org/html/2610.12699#bib.bib4)\. The network architecture for each experiment is specified in the corresponding subsection\. During FM training, the learning rate decays from10−410^\{\-4\}to10−510^\{\-5\}during training\. During subsequent primal dual training, we use the settings specified below\.

### H\.1Illustrative22\-D Constrained Generation \(Q⁡\(𝒞\)<bQ\(\\mathcal\{C\}\)<b\)

#### Network architectures:

The velocity fielduθ​\(xt,t\)u\_\{\\theta\}\(x\_\{t\},t\)is parameterized by a fully connected network taking\(xt,t\)∈ℝ3\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{3\}as input\. It contains five hidden layers of width6464with SELU activations and a two\-dimensional linear output representinguθ​\(xt,t\)∈ℝ2u\_\{\\theta\}\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{2\}\. The feasibility predictorfϕ​\(xt,t\)f\_\{\\phi\}\(x\_\{t\},t\)uses the same hidden architecture with a scalar output\. A sigmoid function is applied to the output logit to obtain the predicted probability of terminal constraint satisfaction\.

#### Hyperparameters:

learning rate for the velocity fieldη\(θ\)=1×10−5\\eta^\{\(\\theta\)\}=1\\times 10^\{\-5\}, learning rate for the classifierη\(ϕ\)=1×10−4\\eta^\{\(\\phi\)\}=1\\times 10^\{\-4\}, dual variable learning rateηi=500\\eta\_\{i\}=500\. Largeηi\\eta\_\{i\}is applied because a high constraint satisfaction target corresponds to a large optimal dual variable\. A larger update step therefore accelerates convergence\.

The visualizations of the clean distribution,QQ,P\(λ∗\)P^\{\(\\lambda^\{\*\}\)\}, the distributions induced by baselines \(1\), \(2\), and PDFM are shown in Fig\.[1](https://arxiv.org/html/2610.12699#A8.F1)\.

![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_clean.png)\(a\) Clean distribution![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_Q.png)\(b\) Training distributionQQQ⁡\(𝒞\)≈96%Q\(\\mathcal\{C\}\)\\approx 96\\%![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_Opt.png)\(c\) Optimal solutionP\(λ∗\)P^\{\(\\lambda^\{\*\}\)\},P\(λ∗\)​\(𝒞\)=99\.5%P^\{\(\\lambda^\{\*\}\)\}\(\\mathcal\{C\}\)=99\.5\\%![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_FM_Q.png)\(d\) FM trained onQQ,P⁡\(𝒞\)=96\.37%P\(\\mathcal\{C\}\)=96\.37\\%,SWD=0\.1384\\mathrm\{SWD\}=0\.1384![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_FM_trunc.png)\(e\) FM trained onQ′Q^\{\\prime\},P⁡\(𝒞\)=98\.74%P\(\\mathcal\{C\}\)=98\.74\\%,SWD=0\.1156\\mathrm\{SWD\}=0\.1156![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/2D/uniform2_PDFM.png)\(f\) PDFM trained onQQ\(b=99\.5%b=99\.5\\%\),P⁡\(𝒞\)=99\.51%P\(\\mathcal\{C\}\)=99\.51\\%,SWD=0\.1096\\mathrm\{SWD\}=0\.1096

Figure 1:Histogram plots of5000050000samples from the compared distributions\.

### H\.2Constrained Generation withℓ2\\ell\_\{2\}\-Ball Constraints \(Q⁡\(𝒞\)≥bQ\(\\mathcal\{C\}\)\\geq b\)

#### Network architectures:

For theℓ2\\ell\_\{2\}\-ball experiments, whered∈\{8,20\}d\\in\\\{8,20\\\}, the velocity network separately embedsxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}andt∈ℝt\\in\\mathbb\{R\}\. The time encoder contains two linear layers, while the state encoder contains three linear layers; both produce6464\-dimensional representations\. These representations are added and passed through a SELU activation, followed by a three\-layer prediction network with hidden width6464and SELU activations\. Its output isuθ​\(xt,t\)∈ℝdu\_\{\\theta\}\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{d\}\. The feasibility predictorfϕ​\(xt,t\)f\_\{\\phi\}\(x\_\{t\},t\)uses the same architecture with separate parameters and a scalar output logit\. A sigmoid function converts this logit into the predicted probability of terminal constraint satisfaction\.

#### Hyperparameters:

learning rate for the velocity fieldη\(θ\)=1×10−5\\eta^\{\(\\theta\)\}=1\\times 10^\{\-5\}, learning rate for the classifierη\(ϕ\)=1×10−4\\eta^\{\(\\phi\)\}=1\\times 10^\{\-4\}, dual variable learning rateηi=500\\eta\_\{i\}=500\. Largeηi\\eta\_\{i\}is applied because a high constraint satisfaction target corresponds to a large optimal dual variable\. A larger update step therefore accelerates convergence\.

Table 4:Results on88\-D and2020\-D constrained generation tasks with standard deviations\.P⁡\(𝒞\)P\(\\mathcal\{C\}\)denotes the mean constrained\-satisfaction rate of1000010000samples, averaged over100100generation trials\. SWD is computed with1000010000generated samples and samples fromQQ, averaged over100100generation trials\.FMMDMNAMMPDMPCFMFM\-REPDFM88\-Dℓ2\\ell\_\{2\}ballP⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)97\.633%97\.633\\%\[\-1pt\]±0\.143%\\pm\\,0\.143\\%100%100\\%\[\-1pt\]±0\\pm\\,099\.986%99\.986\\%\[\-1pt\]±0\.008%\\pm\\,0\.008\\%100%100\\%\[\-1pt\]±0\\pm\\,0100%100\\%\[\-1pt\]±0\\pm\\,099\.923%99\.923\\%\[\-1pt\]±0\.025%\\pm\\,0\.025\\%99\.906%99\.906\\%\[\-1pt\]±0\.026%\\pm\\,0\.026\\%SWD\(↓\)\(\\downarrow\)0\.01930\.0193\[\-1pt\]±0\.0029\\pm\\,0\.00290\.02170\.0217\[\-1pt\]±0\.0030\\pm\\,0\.00300\.02280\.0228\[\-1pt\]±0\.0037\\pm\\,0\.00370\.02000\.0200\[\-1pt\]±0\.0031\\pm\\,0\.00310\.02840\.0284\[\-1pt\]±0\.0042\\pm\\,0\.00420\.01940\.0194\[\-1pt\]±0\.0029\\pm\\,0\.00290\.01720\.0172\[\-1pt\]±0\.0028\\pm\\,0\.00282020\-Dℓ2\\ell\_\{2\}ballP⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)90\.918%90\.918\\%\[\-1pt\]±0\.318%\\pm\\,0\.318\\%100%100\\%\[\-1pt\]±0\\pm\\,099\.711%99\.711\\%\[\-1pt\]±0\.053%\\pm\\,0\.053\\%100%100\\%\[\-1pt\]±0\\pm\\,0100%100\\%\[\-1pt\]±0\\pm\\,099\.748%99\.748\\%\[\-1pt\]±0\.052%\\pm\\,0\.052\\%99\.755%99\.755\\%\[\-1pt\]±0\.0051%\\pm\\,0\.0051\\%SWD\(↓\)\(\\downarrow\)0\.00870\.0087\[\-1pt\]±0\.0008\\pm\\,0\.00080\.01440\.0144\[\-1pt\]±0\.0012\\pm\\,0\.00120\.02050\.0205\[\-1pt\]±0\.0008\\pm\\,0\.00080\.02570\.0257\[\-1pt\]±0\.0018\\pm\\,0\.00180\.01480\.0148\[\-1pt\]±0\.0009\\pm\\,0\.00090\.01320\.0132\[\-1pt\]±0\.0010\\pm\\,0\.00100\.01010\.0101\[\-1pt\]±0\.0007\\pm\\,0\.0007

### H\.3Unconditional PDE\-Constrained Generation of Spatiotemporal Dynamics

#### Dataset construction and residual computation:

We construct a dataset of Kuramoto–Sivashinsky dynamics by numerically solving the conservative PDE with fixed solver parameters and randomly sampled truncated\-Fourier initial conditions, using Exponax\[[Koehler et al\., 2024](https://arxiv.org/html/2610.12699#bib.bib17)\]\. From each long trajectory, we extract non\-overlapping space–time patchesx∈ℝd×dx\\in\\mathbb\{R\}^\{d\\times d\}, wherex\[:,i\]x\[:,i\]denotes the spatial state at time indexii\. LetF⁡\(⋅\)F\(\\cdot\)denote one application of the numerical stepper\. Letx^\\hat\{x\}denote the generated sample, and for eachi=0,…,d−2i=0,\\dots,d\-2, we regardx^\[:,i\]\\hat\{x\}\[:,i\]as the true current state, compute the next true statex\[:,i\+1\]=F\(x^\[:,i\]\)\{x\}\[:,\{i\+1\}\]=F\(\\hat\{x\}\[:,i\]\), and compare it with the next estimated statex^\[:,i\+1\]\\hat\{x\}\[:,i\+1\], yielding the residualri=x\[:,i\+1\]−x^\[:,i\+1\]r\_\{i\}=x\[:,i\+1\]\-\\hat\{x\}\[:,\{i\+1\}\]\. The resulting residual sequence\{ri\}i=0d−2\\\{r\_\{i\}\\\}\_\{i=0\}^\{d\-2\}defines the size\-normalized residualδLn:=\(1\(d−1\)​d​∑i=0d−2‖ri‖nn\)1/n\\delta\_\{L\_\{n\}\}:=\\left\(\\frac\{1\}\{\(d\-1\)d\}\\sum\_\{i=0\}^\{d\-2\}\\\|r\_\{i\}\\\|\_\{n\}^\{n\}\\right\)^\{1/n\}, with the usual extension forn=∞n=\\infty\. A generated sample is defined as satisfying the constraint ifδLn<τ\\delta\_\{L\_\{n\}\}<\\tau, in whichτ\\tauis a prescribed tolerance\. In this task, we setd=64,n=1d=64,n=1\.

#### Network architectures:

The velocity networkuθu\_\{\\theta\}takesxt∈ℝ1×64×64x\_\{t\}\\in\\mathbb\{R\}^\{1\\times 64\\times 64\}and the timet∈ℝt\\in\\mathbb\{R\}as inputs\. A128128\-dimensional sinusoidal time embedding is projected to dimension256256, whilextx\_\{t\}is projected to6464channels\. The network then applies five residual space\-time blocks, each combining5×15\\times 1spatial,1×51\\times 5physical\-time, and3×33\\times 3joint convolutions with group normalization, SiLU activations, and time\-conditioned transformations\. A final3×33\\times 3convolution producesuθ​\(xt,t\)∈ℝ1×64×64u\_\{\\theta\}\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{1\\times 64\\times 64\}\. The feasibility predictor uses the same backbone with separate parameters, followed by global average pooling, a scalar linear output, and a sigmoid function producingfϕ​\(xt,t\)∈\(0,1\)f\_\{\\phi\}\(x\_\{t\},t\)\\in\(0,1\)\.

#### Hyperparameters:

for both cases, learning rate for the velocity fieldη\(θ\)=1×10−5\\eta^\{\(\\theta\)\}=1\\times 10^\{\-5\}, learning rate for the classifierη\(ϕ\)=1×10−4\\eta^\{\(\\phi\)\}=1\\times 10^\{\-4\}, dual variable learning rateηi=2\\eta\_\{i\}=2\.

#### Significance for chaotic PDE emulation\.

The Kuramoto–Sivashinsky setting is representative of generative emulation tasks for chaotic systems, where samples are drawn from the emergent chaotic attractor\. High\-quality emulation requires both local dynamical accuracy and statistical fidelity to that attractor\. The two terms in the PDFM objective parallel this decomposition: the FM loss matches the attractor distribution, while the chance constraint enforces local consistency with the dynamics, providing a principled probabilistic alternative to the combination of next\-step prediction error and statistical\-fidelity losses common in deterministic emulators of chaotic systems\[[Jiang et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib20),[Melo et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib26)\]\.

The visualization of the generated space\-time patch samples in Case 1 \(τ=0\.033,b=98%\\tau=0\.033,b=98\\%\) is provided in Fig\.[2](https://arxiv.org/html/2610.12699#A9.F2)and Fig\.[3](https://arxiv.org/html/2610.12699#A9.F3)\. To quantify the discrepancy between generated samples and the simulator predictions, we extract the first column of each generated sample as the initial state and use the simulator to predict the subsequent6363states\. The comparisons between the generated samples and the simulator\-predicted truth, along with the histograms of the summary statistic, are provided in Fig\.[4](https://arxiv.org/html/2610.12699#A9.F4)and Fig\.[5](https://arxiv.org/html/2610.12699#A9.F5)\. The visualizations of the mean histogram over20002000samples are provided in Fig\.[6](https://arxiv.org/html/2610.12699#A9.F6)and Fig\.[7](https://arxiv.org/html/2610.12699#A9.F7)\.

Table 5:Comparison across the considered methods for Case 1 and Case 2, evaluated using20002000generated samples per method in each of1010trials\. Results are reported as the mean and standard deviation\.MethodCase 1\(Q⁡\(𝒞\)=1\)\(Q\(\\mathcal\{C\}\)=1\)Case 2\(Q⁡\(𝒞\)≈39\.47%\)\(Q\(\\mathcal\{C\}\)\\approx 39\.47\\%\)P⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)AvgδLn\\delta\_\{L\_\{n\}\}\(↓\)\(\\downarrow\)HE\(↓\)\(\\downarrow\)P⁡\(𝒞\)P\(\\mathcal\{C\}\)\(↑\)\(\\uparrow\)AvgδLn\\delta\_\{L\_\{n\}\}\(↓\)\(\\downarrow\)HEclean\\mathrm\{HE\}\_\{\\mathrm\{clean\}\}\(↓\)\(\\downarrow\)HEdeg\\mathrm\{HE\}\_\{\\mathrm\{deg\}\}\(↓\)\(\\downarrow\)FM \(QQ\)14\.19%14\.19\\%\[\-1pt\]±0\.732%\\pm\\,0\.732\\%0\.04090\.0409\[\-1pt\]±0\.0002\\pm\\,0\.0002\(0\.0087,0\.0453\)\(0\.0087,0\.0453\)\[\-1pt\]±\(0\.0002,0\.0005\)\\pm\\,\(0\.0002,0\.0005\)9\.82%9\.82\\%\[\-1pt\]±0\.767%\\pm\\,0\.767\\%0\.06220\.0622\[\-1pt\]±0\.0005\\pm\\,0\.0005\(0\.0094,0\.0465\)\(0\.0094,0\.0465\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)\(0\.0086,0\.0565\)\(0\.0086,0\.0565\)\[\-1pt\]±\(0\.0002,0\.0005\)\\pm\\,\(0\.0002,0\.0005\)FM \(Q′Q^\{\\prime\}\)N/AN/AN/A33\.24%33\.24\\%\[\-1pt\]±0\.470%\\pm\\,0\.470\\%0\.05390\.0539\[\-1pt\]±0\.0001\\pm\\,0\.0001\(0\.0135,0\.0477\)\(0\.0135,0\.0477\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)\(0\.0114,0\.0489\)\(0\.0114,0\.0489\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)PCFM6\.03%6\.03\\%\[\-1pt\]±0\.383%\\pm\\,0\.383\\%0\.04840\.0484\[\-1pt\]±0\.0001\\pm\\,0\.0001\(0\.0156,0\.1547\)\(0\.0156,0\.1547\)\[\-1pt\]±\(0\.0002,0\.0008\)\\pm\\,\(0\.0002,0\.0008\)0\.06%0\.06\\%\[\-1pt\]±0\.054%\\pm\\,0\.054\\%0\.10220\.1022\[\-1pt\]±0\.0010\\pm\\,0\.0010\(0\.0173,0\.0729\)\(0\.0173,0\.0729\)\[\-1pt\]±\(0\.0002,0\.0005\)\\pm\\,\(0\.0002,0\.0005\)\(0\.0186,0\.0823\)\(0\.0186,0\.0823\)\[\-1pt\]±\(0\.0002,0\.0005\)\\pm\\,\(0\.0002,0\.0005\)FM\-RE43\.17%43\.17\\%\[\-1pt\]±1\.496%\\pm\\,1\.496\\%0\.03390\.0339\[\-1pt\]±0\.0002\\pm\\,0\.0002\(0\.0117,0\.0329\)\(0\.0117,0\.0329\)\[\-1pt\]±\(0\.0002,0\.0003\)\\pm\\,\(0\.0002,0\.0003\)46\.33%46\.33\\%\[\-1pt\]±1\.007%\\pm\\,1\.007\\%0\.05130\.0513\[\-1pt\]±0\.0003\\pm\\,0\.0003\(0\.0118,0\.0558\)\(0\.0118,0\.0558\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)\(0\.0097,0\.0570\)\(0\.0097,0\.0570\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)PDFM97\.74%97\.74\\%\[\-1pt\]±0\.531%\\pm\\,0\.531\\%0\.02350\.0235\[\-1pt\]±0\.0001\\pm\\,0\.0001\(0\.0107,0\.0269\)\(0\.0107,0\.0269\)\[\-1pt\]±\(0\.0003,0\.0003\)\\pm\\,\(0\.0003,0\.0003\)90\.48%90\.48\\%\[\-1pt\]±0\.525%\\pm\\,0\.525\\%0\.04060\.0406\[\-1pt\]±0\.0002\\pm\\,0\.0002\(0\.0118,0\.0435\)\(0\.0118,0\.0435\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)\(0\.0103,0\.0457\)\(0\.0103,0\.0457\)\[\-1pt\]±\(0\.0002,0\.0004\)\\pm\\,\(0\.0002,0\.0004\)

### H\.4SOCOFing Fingerprint Generation with Ridge\-Connectivity Constraint

We consider the task of generating fingerprint images with a ridge\-connectivity constraint, using the Sokoto Coventry Fingerprint Dataset \(SOCOFing\)\[[Shehu et al\., 2018](https://arxiv.org/html/2610.12699#bib.bib19)\]for training\. SOCOFing is a public grayscale fingerprint image dataset containing real fingerprint samples and synthetically altered variants\. It includes fingerprints collected from multiple subjects and provides altered versions with varying levels of distortion\.

Collected fingerprint images and the altered variants can be corrupted by sensor noise, partial occlusion, pressure variation, or preprocessing artifacts, which may break continuous ridge structures into disconnected fragments\. We therefore consider a ridge\-connectivity constraint, which requires the main fingerprint ridges to remain sufficiently continuous; satisfying this constraint indicates that the image preserves coherent ridge structure\.

Concretely, we evaluate this constraint using a non\-differentiable image\-processing oracle\. For each grayscale fingerprint image, we first apply Otsu thresholding to separate ridge pixels from the background, treating ridges as the foreground\. We then skeletonize the foreground ridge map into a one\-pixel\-wide representation and count the number of connected components in the skeletonized image\. A sample is considered feasible if the number of connected ridge components is below the prescribed threshold5050\. We use the Real and Altered\-Easy partitions of the SOCOFing as the training set\. Among the2393123931training samples,1623516235of them satisfy this ridge\-connectivity constraint, indicatingQ⁡\(𝒞\)≈67\.84%Q\(\\mathcal\{C\}\)\\approx 67\.84\\%\. The target constraint satisfaction rate is set tob=99%b=99\\%\.

#### Network architectures:

The velocity network takesxt∈ℝ1×96×96x\_\{t\}\\in\\mathbb\{R\}^\{1\\times 96\\times 96\}andt∈ℝt\\in\\mathbb\{R\}as inputs\. It uses a four\-level U\-Net with channel dimensions6464,128128,256256, and512512, three downsampling and three upsampling layers, skip connections, and two residual blocks at the bottleneck\. A256256\-dimensional sinusoidal embedding ofttis projected and added to the features in each residual block\. A final3×33\\times 3convolution producesuθ​\(xt,t\)∈ℝ1×96×96u\_\{\\theta\}\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{1\\times 96\\times 96\}\. The feasibility predictor uses a three\-level encoder with channel dimensions6464,128128, and256256, followed by two residual blocks, global average pooling, and a sigmoid outputfϕ​\(xt,t\)∈\(0,1\)f\_\{\\phi\}\(x\_\{t\},t\)\\in\(0,1\)\.

#### Hyperparameters:

learning rate for the velocity fieldη\(θ\)=1×10−5\\eta^\{\(\\theta\)\}=1\\times 10^\{\-5\}, learning rate for the classifierη\(ϕ\)=1×10−4\\eta^\{\(\\phi\)\}=1\\times 10^\{\-4\}, dual variable learning rateηi=5\\eta\_\{i\}=5\.

For this oracle\-constrained task, only constraint\-aware methods compatible with membership oracle feedback are applicable\. Therefore, we compare PDFM against FM\-RE and the vanilla FM baseline\. The performances are evaluated in two aspects: constraint satisfaction rate and negative log\-likelihood \(NLL\)\. NLL is computed by treating the finite\-step sampler as a sequence of invertible transformations from a base noise distribution to the data space\. For each evaluation sample, the discrete updates are inverted to recover the corresponding latent noise, the log density of this latent under the base distribution is evaluated, and the log\-determinant correction associated with each discrete transformation is accumulated to obtain the sample log likelihood; the NLL is then the negative average log likelihood over the evaluation set\. Lower NLL indicates that the model assigns a higher probability density to the evaluated samples under this discrete likelihood\. In this setting, we estimate NLL using100100Euler discretization steps on two evaluation sets: the full training set, denoted by𝒟train\\mathcal\{D\_\{\\mathrm\{train\}\}\}, which includes both feasible and infeasible samples, and the feasible subset\{x∈Dtrain:x∈𝒞\}\\\{x\\in D\_\{\\mathrm\{train\}\}:x\\in\\mathcal\{C\}\\\}, obtained by discarding infeasible samples\. We report these asNLLtrain\\mathrm\{NLL\}\_\{\\mathrm\{train\}\}andNLLfeas\\mathrm\{NLL\}\_\{\\mathrm\{feas\}\}, respectively\.

The statistical results in Table[6](https://arxiv.org/html/2610.12699#A8.T6)show that while FM\-RE improves the feasible rate over FM, a degraded induced distribution is observed\. PDFM successfully achieves the target constraint satisfaction rate of99%99\\%, while maintaining NLLs comparable to those of FM\. The visualizations of the training samples and the generated samples are provided in Fig\.[8](https://arxiv.org/html/2610.12699#A9.F8)\. The generated samples from FM\-RE display a slightly perturbed background, indicating a degraded induced distribution\. PDFM and FM generate samples that are visually close to the training samples\. This further validates that PDFM is able to achieve the desired feasibility control while preserving distributional quality and visual fidelity\.

Table 6:Constraint satisfaction is evaluated using 1000 generated samples per trial\.NLLtrain\\mathrm\{NLL\}\_\{\\mathrm\{train\}\}andNLLfeas\\mathrm\{NLL\}\_\{\\mathrm\{feas\}\}are evaluated using 1000 samples randomly drawn from the full training set and its feasible subset, respectively, per trial\. Results are reported as the mean and standard deviation over 10 trials\.MethodP\(𝒞\)\(↑\)P\(\\mathcal\{C\}\)\(\\uparrow\)NLLtrain\(↓\)\\mathrm\{NLL\}\_\{\\mathrm\{train\}\}\(\\downarrow\)NLLfeas\(↓\)\\mathrm\{NLL\}\_\{\\mathrm\{feas\}\}\(\\downarrow\)FM72\.74%±1\.57%72\.74\\%\\pm 1\.57\\%−4\.0851±0\.0063\-4\.0851\\pm 0\.0063−4\.1525±0\.0061\-4\.1525\\pm 0\.0061FM\-RE88\.14%±1\.37%88\.14\\%\\pm 1\.37\\%−0\.9254±0\.0039\-0\.9254\\pm 0\.0039−0\.9368±0\.0057\-0\.9368\\pm 0\.0057PDFM99\.28%±1\.33%99\.28\\%\\pm 1\.33\\%−3\.8904±0\.0122\-3\.8904\\pm 0\.0122−4\.0768±0\.0130\-4\.0768\\pm 0\.0130

### H\.5QM933\-D Molecule Generation

We evaluate PDFM on33\-D molecule generation using the QM9 dataset\[[Ramakrishnan et al\., 2014](https://arxiv.org/html/2610.12699#bib.bib7)\], a standard benchmark of small organic molecules with atom types and33\-D coordinates\. This task is important for the discovery of new drug molecules\. The molecules are represented as\(x,h\)\(x,h\), in whichxxrepresents the33\-D atom coordinates andhhcontains atom features such as atom types and charges\. Here, the sample\-wise constraint is molecule stability: a generated molecule is considered feasible if its atoms satisfy valid valency rules after bond inference, so that the generated structure is chemically stable at the atom level\. In this dataset,95\.2%95\.2\\%of the samples are stable, indicatingQ⁡\(𝒞\)≈95\.2%Q\(\\mathcal\{C\}\)\\approx 95\.2\\%\. Note that the stability constraint is a membership oracle constraint, without differentiable distances to the feasible set or available projections\.

Existing works such as\[[Song et al\., 2023](https://arxiv.org/html/2610.12699#bib.bib32)\]introduce molecule\-specific architectural and training designs, including hybrid probability paths for coordinatesxxand atom featureshh, equivariant optimal transport for coordinates, Hungarian–Kabsch alignment, and information\-aligned paths for atom features, achieving superior stability rates\. In this section, we investigate whether PDFM can improve constraint satisfaction using only binary feasibility feedback, without relying on such domain\-specific knowledge\.

#### Network architectures\.

Following the EGNN backbone of[Satorras et al\. \[2021\]](https://arxiv.org/html/2610.12699#bib.bib33), each molecular statext∈ℝN×8x\_\{t\}\\in\\mathbb\{R\}^\{N\\times 8\}is represented as a graph with three coordinates and five atom features per node\. For both FM and PDFM, the velocity network uses four time\-conditioned EGNN blocks with hidden dimension6464, where pairwise relative coordinates and distances are used to update node features and coordinates\. A6464\-dimensional sinusoidal embedding ofttconditions each block, and the network outputsuθ​\(xt,t\)∈ℝN×8u\_\{\\theta\}\(x\_\{t\},t\)\\in\\mathbb\{R\}^\{N\\times 8\}\. The feasibility predictor uses a separate pairwise graph network with hidden dimension128128, a3232\-dimensional time embedding, and two interaction blocks\. Pairwise messages are constructed from node features, relative coordinates, and squared distances and are aggregated by averaging\. The pooled graph representation and summary statistics of pairwise distances are passed through an MLP and a sigmoid function to producefϕ​\(xt,t\)∈\(0,1\)f\_\{\\phi\}\(x\_\{t\},t\)\\in\(0,1\)\.

#### Hyperparameters:

learning rate for the velocity fieldη\(θ\)=1×10−5\\eta^\{\(\\theta\)\}=1\\times 10^\{\-5\}, learning rate for the classifierη\(ϕ\)=1×10−4\\eta^\{\(\\phi\)\}=1\\times 10^\{\-4\}, dual variable learning rateηi=10\\eta\_\{i\}=10\.

Following EDM\[[Hoogeboom et al\., 2022](https://arxiv.org/html/2610.12699#bib.bib34)\], we use NLL as a likelihood\-based measure of distributional alignment𝔼x∼Q​\[−log⁡pθ​\(x\)\]\\mathbb\{E\}\_\{x\\sim Q\}\[\-\\log p\_\{\\theta\}\(x\)\], together with molecule stability to evaluate sample\-wise chemical feasibility\. Lower NLL indicates that the model assigns a larger probability density to the observed data and therefore achieves a better distributional match\. In practice, NLL is approximated with a finite\-step Euler sampler by reversing the discrete updates to recover the latent noise, evaluating its prior probability, and accumulating each step’s density/volume\-change correction\. In this task, NLL is evaluated using100100discrete steps\.

The visualizations of the generated samples are provided in Fig\.[9](https://arxiv.org/html/2610.12699#A9.F9)and Fig\.[10](https://arxiv.org/html/2610.12699#A9.F10)\. Over1010trials, constraint satisfaction is evaluated using10001000generated samples per trial, while NLL is evaluated using100100randomly selected samples from the training set per trial\. FM achievesP⁡\(𝒞\)=\(32\.77±1\.22\)%P\(\\mathcal\{C\}\)=\(32\.77\\pm 1\.22\)\\%andNLL=−19\.74±0\.44\\mathrm\{NLL\}=\-19\.74\\pm 0\.44\. PDFM, trained with the targetb=60%b=60\\%, achievesP⁡\(𝒞\)=\(48\.75±1\.56\)%P\(\\mathcal\{C\}\)=\(48\.75\\pm 1\.56\)\\%andNLL=−20\.70±0\.25\\mathrm\{NLL\}=\-20\.70\\pm 0\.25\. Results are reported as the mean and standard deviation\. While PDFM is unable to achieve the target constraint satisfaction rate due to the design limitations, including the generic backbone and probability path without domain\-specific knowledge, it substantially improves constraint satisfaction over the FM baseline\. Moreover, the improvement is achieved without sacrificing likelihood quality: PDFM obtains a lower NLL, indicating higher distributional alignment with the training distribution\. This suggests that PDFM is able to improve constraint satisfaction while preserving, or even improving, distributional fidelity without task\-specific designs and with only membership oracle feedback\.

## Appendix IDiscussion

### I\.1Why Flow Matching?

The procedure of starting from[OPT](https://arxiv.org/html/2610.12699#S2.Ex1)then turning to the parameterizedL¯\\bar\{L\}and solvingL′\{L^\{\\prime\}\}practically can also be applied to other types of generative models, such as DDPM and score\-based models\. For example, in DDPM with a parameterized score denoted bysθs\_\{\\theta\}, one can generally replace Assumption[3\.2](https://arxiv.org/html/2610.12699#S3.Thmtheorem2)with a score estimation error in the form𝔼t,xt​\[‖sθ\(λ\)​\(xt,t\)−∇log⁡p\(λ\)​\(xt,t\)‖22\]≤ν2\\mathbb\{E\}\_\{t,x\_\{t\}\}\\left\[\\\|s^\{\{\(\\lambda\)\}\}\_\{\\theta\}\(x\_\{t\},t\)\-\\nabla\\log p^\{\(\\lambda\)\}\(x\_\{t\},t\)\\\|\_\{2\}^\{2\}\\right\]\\leq\\nu^\{2\}, leading to a similar conclusion of bounded TV as in Lemma[D\.1](https://arxiv.org/html/2610.12699#A4.Thmtheorem1)shown in[Li and Yan \[2025\]](https://arxiv.org/html/2610.12699#bib.bib1)\. As for practically computing the gradient forPsθ​\(𝒞\)P\_\{s\_\{\\theta\}\}\{\(\\mathcal\{C\}\)\}, one can apply FM\-RE’s\[[Huan et al\., 2026](https://arxiv.org/html/2610.12699#bib.bib16)\]optimization framework due to DDPM’s inherent stochasticity\.

However, DDPM’s fixed stochasticity, especially the large noise steps at the start of the sampling trajectory, together with hard binary feedback in the FM\-RE framework, make the optimization hard to perform in practice\. Empirical evidence shows that this approach is able to deal with simple constraints, e\.g\.,ℓ2\\ell\_\{2\}\-ball constraints in Sec\.[5\.2](https://arxiv.org/html/2610.12699#S5.SS2), but may induce degraded distribution when the complexity of the constraint increases\. In contrast, the FM framework provides more stable sampling trajectories, thereby being able to adapt to more complex constraints while preserving the distributional fidelity with the optimization strategy in Sec\.[4\.2](https://arxiv.org/html/2610.12699#S4.SS2)\.

### I\.2Implementation Tips and Relationship with FM\-RE

From an implementation perspective, a single PDFM iteration with fixedλ′=λ/w\\lambda^\{\\prime\}=\\lambda/wis closely related to FM\-RE, if the dual variable in PDFM is interpreted as the weight of the constraint\-aware term in FM\-RE\. Although the procedures for estimating∇θPuθ​\(𝒞\)\\nabla\_\{\\theta\}P\_\{u\_\{\\theta\}\}\(\\mathcal\{C\}\)differ, both methods share the same underlying idea: they provide feedback touθ​\(xt,t\)u\_\{\\theta\}\(x\_\{t\},t\)that is explicitly or implicitly derived from the terminal binary feasibility feedback\. Since FM\-RE requires sampling complete trajectories to obtain terminal oracle feedback, a natural question is whether PDFM also requires complete trajectory sampling during its updates\.

Only partially\. PDFM maintains bothuθu\_\{\\theta\}andfϕf\_\{\\phi\}during the primal update\. Trainingfϕf\_\{\\phi\}requires complete trajectory sampling, since the terminal oracle valuey=𝟏𝒞​\(x1\)y=\\mathbf\{1\}\_\{\\mathcal\{C\}\}\(x\_\{1\}\)serves as the training label for the intermediate predictionsfϕ​\(xt,t\)f\_\{\\phi\}\(x\_\{t\},t\)\. However, complete trajectory sampling is not strictly required to computegradθ\\mathrm\{grad\}\_\{\\theta\}in \([18](https://arxiv.org/html/2610.12699#S4.E18)\), because the feedback is provided directly by the fixed estimatorfϕf\_\{\\phi\}\.

One subtlety is that evaluating \([18](https://arxiv.org/html/2610.12699#S4.E18)\) still requires samples of the intermediatexi​Δ​tx\_\{i\\Delta t\}\. From an implementation perspective, these states can be approximated using random couplings and linear interpolants, i\.e\.,xi​Δ​t=\(1−i​Δ​t\)​x0\+i​Δ​t​x1x\_\{i\\Delta t\}=\(1\-i\\Delta t\)x\_\{0\}\+i\\Delta tx\_\{1\}, provided that the probability path does not deviate significantly, as in the PDE\-constrained and molecule generation tasks\.

Note thatfϕf\_\{\\phi\}, which is fundamentally trained as a classifier, is generally cheaper to train than the more complex velocity fielduθu\_\{\\theta\}and does not require highly accurate prediction to provide useful feedback\. Since PDFM does not require complete trajectory sampling for theθ\\theta\-update, it has a computational advantage over FM\-RE\. In single\-iteration evaluations fori\>0i\>0, PDFM training is approximately22–3×3\\timesfaster than FM\-RE; for example, the runtime is about1\.51\.5minutes vs\.44minutes for the88\-Dℓ2\\ell\_\{2\}ball constraint, and2525minutes vs\.7070minutes for Case 1 in Sec\.[5\.3](https://arxiv.org/html/2610.12699#S5.SS3)\. Users seeking faster training can repeat multipleθ\\theta\-updates between theϕ\\phi\-updates, provided that theseθ\\theta\-updates do not substantially change the constraint satisfaction behavior\.

### I\.3Limitations

The limitations of PDFM are summarized as follows:

- 1\.Sincefϕf\_\{\\phi\}is fundamentally trained as a classifier, it requires both positive and negative feedback\. Highly imbalanced feedback, especially when the feedback is overwhelmingly negative, meaning that most samples generated by the currentuθu\_\{\\theta\}violate the constraint, can preventfϕf\_\{\\phi\}from learning reliable decision boundaries and lead to inaccurate feedback\. Therefore, at the start of iterationi=1i=1in Alg\.[2](https://arxiv.org/html/2610.12699#alg2), PDFM requires the current velocity field to generate at least some feasible samples\.
- 2\.PDFM requires the feasible region to be sufficiently represented or reachable under the data and model distributions\. This means one cannot generally expect PDFM to learn a near\-optimal constrained distribution whenQQdeviates significantly from𝒞\\mathcal\{C\}\. In such cases, the FM objective can become severely misaligned with the constraint satisfaction objective, leading to a large optimization error in Assumption[4\.3](https://arxiv.org/html/2610.12699#S4.Thmtheorem3)\.

![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/FMsamples.png)Figure 2:Samples generated by FM in Case 1 \(τ=0\.033,b=98%\\tau=0\.033,b=98\\%\)\. The titles denote the residualδLn\\delta\_\{L\_\{n\}\}\. The constraint satisfaction rate is14\.19%14\.19\\%\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/PDFMsamples.png)Figure 3:Samples generated by PDFM in Case 1 \(τ=0\.033,b=98%\\tau=0\.033,b=98\\%\)\. The titles denote the residualδLn\\delta\_\{L\_\{n\}\}\. The constraint satisfaction rate is97\.74%97\.74\\%\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_fm1_trajectory.png)ttxxGenerated![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_fm1_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_fm1_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_pd2_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_pd2_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xPDFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_pd2_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}Truth![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth1_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth1_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth1_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth2_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth2_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/individual_truth2_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}

Figure 4:Single generated sample and its summary statistic in Case 1\. For each three\-image set, the first image shows sampled emulator dynamics, and the last two show histograms of summary statistic distributions\. First row: samples generated by the generative models\. Left: FM,δLn=0\.0421\>τ\\delta\_\{L\_\{n\}\}=0\.0421\>\\tau\. Right: PDFM,δLn=0\.0256<τ\\delta\_\{L\_\{n\}\}=0\.0256<\\tau\. Second row: simulator \(truth\) samples initialized from the same initial states in the first row\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_fm1_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_fm1_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_fm1_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_pd2_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_pd2_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xPDFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_pd2_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth1_trajectory.png)ttxxTruthGenerated![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth1_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth1_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth2_trajectory.png)ttxx![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth2_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_individual_truth2_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}

Figure 5:Single generated sample and its summary statistic in Case 2\. For each three\-image set, the first shows sampled emulator dynamics, and the last two show histograms of summary statistic distributions\. First row: samples generated by the generative models\. Left: FM,δLn=0\.0734\>τ\\delta\_\{L\_\{n\}\}=0\.0734\>\\tau\. Right: PDFM,δLn=0\.0354<τ\\delta\_\{L\_\{n\}\}=0\.0354<\\tau\. Second row: simulator \(truth\) samples initialized from the same initial states in the first row\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_truth_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xQQ∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_fm_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_pd_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xPDFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_truth_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_fm_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_pd_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}Figure 6:Summary statistic over20002000samples in Case 1\. First column: training samples\. Second column: FM’s generated samples\. Third column: PDFM’s generated samples\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_truth_ut_vs_ux.png)∂u/∂x\\partial u/\\partial xCleanQQFMPDFM∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_truth_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_fm_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_pd_ut_vs_ux.png)∂u/∂x\\partial u/\\partial x∂u/∂t\\partial u/\\partial t![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/batch_truth_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_fm_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/KSPDE/d_batch_pd_uxx_vs_ux.png)∂u/∂x\\partial u/\\partial x∂2u/∂x2\\partial^\{2\}u/\\partial x^\{2\}Figure 7:Summary statistic over20002000samples in Case 2\. First column: clean samples\. Second column: degraded samples used during training\. Third column: FM’s generated samples\. Fourth column: PDFM’s generated samples\.Train![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/fingerprint/train.png)FM![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/fingerprint/FM.png)FM\-RE![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/fingerprint/FMRE.png)PDFM![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/fingerprint/PDFM.png)Figure 8:Generated fingerprint samples from different methods\. The title of each sample reports its number of connected components, with infeasible samples highlighted in red\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/FM_mole.png)Figure 9:Visualization of the33\-D molecules generated by FM\. The titles indicate whether the generated sample satisfies the molecule\-stability constraint\.![Refer to caption](https://arxiv.org/html/2610.12699v1/figs/PDFM_mole.png)Figure 10:Visualization of the33\-D molecules generated by PDFM\. The titles indicate whether the generated sample satisfies the molecule\-stability constraint\.

## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The theoretical and empirical claims in the abstract match the content in the main text\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: Limitations are discussed in detail in Appendix[I](https://arxiv.org/html/2610.12699#A9)\.
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11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[Yes\]
14. Justification: The assumptions are provided in the main text, with detailed proofs provided in the appendix\.
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16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: The code is released\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[Yes\]
24. Justification: The PDFM for implementation is provided in Alg\.[2](https://arxiv.org/html/2610.12699#alg2), with more details discussed in Appendix[I](https://arxiv.org/html/2610.12699#A9)\. We will release the code if the paper is accepted\. - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
25. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: The settings are described in the main text as well as in the appendix\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: We report means and standard deviations over repeated trials in the results\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: The computation resource information is provided in Appendix[H](https://arxiv.org/html/2610.12699#A8)\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
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46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[Yes\]
49. Justification: The impact is discussed in the abstract, as well as in the contributions\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification:The paper poses no such risks\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: Related works are properly cited\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[N/A\]
64. Justification: Code will be released on GitHub once the paper is accepted\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: The paper does not involve crowdsourcing nor research with human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: The paper does not involve crowdsourcing nor research with human subjects\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: The core method development in this research does not involve LLMs as any important, original, or non\-standard components\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.