Conflict-Aware Additive Guidance for Flow Models under Compositional Rewards

# Conflict-Aware Additive Guidance for Flow Models under Compositional Rewards
Source: [https://arxiv.org/html/2605.20758](https://arxiv.org/html/2605.20758)
###### Abstract

Inference\-time guided sampling steers state\-of\-the\-art diffusion and flow models without fine\-tuning by interpreting the generation process as a controllable trajectory\. This provides a simple and flexible way to inject external constraints \(e\.g\., cost functions or pre\-trained verifiers\) for controlled generation\. However, existing methods often fail when composing multiple constraints simultaneously, which leads to deviations from the true data manifold\. In this work, we identify root causes of this off\-manifold drift and find that the approximation error scales severely with gradient misalignment\. Building on these findings, we propose Conflict\-Aware Additive Guidance \(gcarg^\{\\text\{car\}\}\), a lightweight and learnable method, which actively rectifies off\-manifold drift by dynamically detecting and resolving gradient conflicts\. We validategcarg^\{\\text\{car\}\}across diverse domains, ranging from synthetic datasets and image editing to generative decision\-making for planning and control\. Our results demonstrate thatgcarg^\{\\text\{car\}\}effectively rectifies off\-manifold drift, surpassing baselines in generation fidelity while using light compute\. Code is available at[github\.com/yuxuehui/CAR\-guidance](https://arxiv.org/html/2605.20758v1/github.com/yuxuehui/CAR-guidance)\.

inference\-time alignment, flow matching, generative model, controlled generation

## 1Introduction

![Refer to caption](https://arxiv.org/html/2605.20758v1/x1.png)

Figure 1:At inference time, our goal is to sample from the tilted targetp1′​\(x1\)∝p1base​\(x1\)​er​\(x1\)p^\{\\prime\}\_\{1\}\(x\_\{1\}\)\\propto p\_\{1\}^\{\\text\{base\}\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\. \(b\-c\) A single reward reweights the distribution towards specific attributes \(“red” or “dog”\)\. \(d\) Under compositional rewards \(“red” and “dog”\), the ideal samples lie at the intersection of high\-reward regions \(⋆\\star,![Refer to caption](https://arxiv.org/html/2605.20758v1/all-twemojis.pdf)\); however, existing methods often suffer from off\-manifold drift \(i\.e\., the distorted image,∙\\bullet,![Refer to caption](https://arxiv.org/html/2605.20758v1/all-twemojis.pdf)\)\.![Refer to caption](https://arxiv.org/html/2605.20758v1/x2.png)

Figure 2:\(a\) The base sampling trajectories\. \(b\) Guided sampling adds guidance to inference trajectories and with a fixed sourcepbase​\(x0\)p^\{\\text\{base\}\}\(x\_\{0\}\); this forces the trajectories to curve significantly to satisfy the constraint, resulting in unnecessarily long and high\-curvature paths\. \(c\) Gradient misalignment \(between→\\bm\{\\to\}and→\\bm\{\\to\}\) aggravates local curvature, yielding incorrect and unstable guidance that pushes the trajectory off\-manifold \(∙\\bullet\) into low\-density regions \(i\.e\., the “vanishing energy” traps visualised in Figure[4](https://arxiv.org/html/2605.20758#S5.F4)\)\. \(d\) Under gradient conflict,gcarg^\{\\text\{car\}\}rectifies the misaligned gradient via learned guidance \(→\\bm\{\\to\}\) that points directly to the targetx1x\_\{1\}, recovering straight and on\-manifold trajectories\.Continuous\-time flow models, such as Rectified Flow\(liu2023flow\), Flow Matching\(lipman2023flow;tong2023improving\), and Stochastic Interpolants\(albergo2023stochastic\), have emerged as a simple yet highly effective generative modeling paradigm\. Through data\-driven training at scale, these models acquire a robust generative prior capable of generalizing across a wide spectrum of applications\.

In this work, we study the problem of steering powerful generative priors to satisfy multiple, potentially competing objectives simultaneously, a setting commonly referred to as thecompositional reward problem\(du2024compositional\)\. This challenge is central to the real\-world deployment of large\-scale generative flow models, where inference\-time samples are required to satisfy a diverse and often heterogeneous set of runtime constraints\. For example, in generative decision\-making, generated trajectories must respect heterogeneous requirements, including safety constraints\(eiras2021twostage\), trajectory smoothness\(urain2023se3\), and dynamic consistency with learned world models\(du2025dynaguide\)\. Similarly, in text\-guided image manipulation, one often seeks to exploit large vision–language models such as CLIP\(radford2021clip\)to modify images according to natural language prompts\(yu2023freedom\)\.

Inference\-time alignment methods\(lipman2023flow\)address these challenges by encoding task requirements as reward functions and steering the sampling process directly at inference time, without retraining or fine\-tuning the underlying generative prior\. Among these, inference\-time guided sampling \(or guidance\) offers a lightweight and flexible interface for adapting pretrained generative models to complex and heterogeneous constraints by directly injecting reward signals during generation, which has been successfully applied to enforce a wide range of runtime constraints\(lipman2023flow;pokle2024training\)\. In particular, their compute\-efficient approximate forms even enable control over objectives unseen during training through heuristic guidance terms\(feng2025guidance\)\.

However, these approximate guidance methods often push samples into low\-density regions where the pretrained vector field is poorly calibrated, causing systematic deviations from the true data distribution\. This failure mode, commonly referred to asoff\-manifold drift, becomes particularly severe in compositional reward settings \(Figure[1](https://arxiv.org/html/2605.20758#S1.F1)\), where competing objectives induce conflicting gradients that collectively drive samples away from the data manifold\. While exact guidance methods\(feng2025guidance\)can mitigate this issue and achieve high\-fidelity alignment, they are typically computationally expensive and lack the flexibility required to efficiently adapt across diverse reward functions\.

In this paper, we develop a theoretical analysis that establishes an upper bound on the approximation error of guided sampling, decomposing it into three terms:*coupling shift*,*gradient misalignment*, and*localized approximation*error\. Our analysis reveals that, in compositional reward settings,approximation error grows sharply with gradient misalignment\(𝟏−𝐜𝐨𝐬⁡ϕ\)\\bm\{\(1\-\\cos\\phi\)\}and the number of reward functionsG\\bm\{G\}, whereϕ\\phidenotes the average angular divergence between guidance channels \(Figure[2](https://arxiv.org/html/2605.20758#S1.F2)\(c\); see Section[4](https://arxiv.org/html/2605.20758#S4)\)\.

Motivated by this insight, we proposeConflict\-AwaRe Additive Guidance \(CAR guidance, denotedgcarg^\{\\text\{car\}\}\), a lightweight guided sampling strategy that actively mitigates off\-manifold drift by detecting and correcting gradient misalignment\. CAR guidance introduces a conflict\-aware gating mechanism that selectively activates learnable corrections in regions of significant gradient conflict\.

We empirically evaluategcarg^\{\\text\{car\}\}across a broad range of domains, including 2D synthetic benchmarks, pixel\-space image editing, and generative decision\-making tasks spanning state\-based planning and 3D point\-cloud manipulation\. Across all settings,gcarg^\{\\text\{car\}\}consistently outperforms state\-of\-the\-art baselines in challenging compositional reward settings, achieving stronger alignment with on\-the\-fly constraints while preserving high sample fidelity\.gcarg^\{\\text\{car\}\}improves identity preservation by25\.4%25\.4\\%in image editing, and increases planning success rates by38\.75%38\.75\\%\. On robot manipulation tasks, where naïve inference\-time guidance is particularly prone to drifting off\-manifold and producing out\-of\-distribution trajectories,gcarg^\{\\text\{car\}\}cuts violation rates by78%78\\%and lifts success from9%9\\%to61%61\\%, enabling reactive obstacle avoidance\. Code, data, and pretrained checkpoints are available at[github\.com/yuxuehui/CAR\-guidance](https://arxiv.org/html/2605.20758v1/github.com/yuxuehui/CAR-guidance)\.

##### Conflict of Interest Disclosure\.

The authors declare no financial conflicts of interest related to this work\.

## 2Preliminaries

### 2\.1Flow matching and conditional flow matching

Flow Matching \(FM\)\(lipman2023flow\)is a simulation\-free method for training continuous normalizing flows, which learns a time\-dependent vector fieldvt​\(xt,t\)v\_\{t\}\(x\_\{t\},t\)to transport a source distributionp0p\_\{0\}to a target distributionp1p\_\{1\}\. A vector fieldvtv\_\{t\}is said to generate a probability density pathptp\_\{t\}if its flowϕt:\[0,1\]:ℝd→ℝd\\phi\_\{t\}:\[0,1\]:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}satisfies the continuity equation∂tpt\+∇⋅\(pt​vt\)=0\\partial\_\{t\}p\_\{t\}\+\\nabla\\cdot\(p\_\{t\}v\_\{t\}\)=0with boundary conditions matching the source and target distributions att=0t=0andt=1t=1, respectively\. Given a target probability density pathptp\_\{t\}and a corresponding target vector fieldvtv\_\{t\}, which generatesptp\_\{t\}, the FM objective is:

𝔼t∼𝒰​\(0,1\),xt∼pt​\[‖vθ​\(xt,t\)−vt​\(xt,t\)‖2\],\\mathbb\{E\}\_\{t\\sim\\mathcal\{U\}\(0,1\),\\,x\_\{t\}\\sim p\_\{t\}\}\\big\[\\\|v\_\{\\theta\}\(x\_\{t\},t\)\-v\_\{t\}\(x\_\{t\},t\)\\\|^\{2\}\\big\],\(1\)whereθ\\thetadenotes learnable parameters of the vector fieldvθv\_\{\\theta\}\.

To make training tractable, Conditional Flow Matching \(CFM\)\(lipman2023flow\)introduces a latent variablezzdistributed according to a coupling measureπ​\(z\)\\pi\(z\), and defines conditional probability pathspt​\(xt\|z\)p\_\{t\}\(x\_\{t\}\|z\)together with corresponding conditional vector fieldsut​\(xt,t\|z\)u\_\{t\}\(x\_\{t\},t\|z\)\. The CFM objective is given by:

𝔼t∼𝒰​\(0,1\)z∼π​\(z\)𝔼xt∼pt\(⋅\|z\)\[∥vθ\(xt,t\)−vt\(xt,t\|z\)∥2\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\mathcal\{U\}\(0,1\)\\\\ z\\sim\\pi\(z\)\\end\{subarray\}\}\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}\(\\cdot\|z\)\}\\big\[\\\|v\_\{\\theta\}\(x\_\{t\},t\)\-v\_\{t\}\(x\_\{t\},t\|z\)\\\|^\{2\}\\big\]\.\(2\)For arbitrary choices of the latent variablezzand coupling measureπ​\(z\)\\pi\(z\), minimizing the conditional flow matching loss in \([2](https://arxiv.org/html/2605.20758#S2.E2)\) is equivalent to minimizing the marginal flow matching objective in \([1](https://arxiv.org/html/2605.20758#S2.E1)\), and yields the optimal marginal vector fieldvt​\(xt,t\)=𝔼z∼π​\(z\|xt\)​\[vt​\(xt,t\|z\)\]v\_\{t\}\(x\_\{t\},t\)=\\mathbb\{E\}\_\{z\\sim\\pi\(z\|x\_\{t\}\)\}\[v\_\{t\}\(x\_\{t\},t\|z\)\]\(tong2024simulation\)\. In the general formulation, the latent variable is defined as the coupling pairz=\(x0,x1\)z=\(x\_\{0\},x\_\{1\}\), and the coupling measureπ​\(z\)=π​\(x0,x1\)\\pi\(z\)=\\pi\(x\_\{0\},x\_\{1\}\)represents the joint coupling between the source and target distributions\.

### 2\.2Guided sampling

At inference time, our goal is to alter the base vector fieldvt​\(xt,t\)v\_\{t\}\(x\_\{t\},t\), which generates the base target distributionp1​\(x1\)p\_\{1\}\(x\_\{1\}\), into a new vector fieldvt′​\(xt,t\)v\_\{t\}^\{\\prime\}\(x\_\{t\},t\)that generates samples from a reweighted target distributionp1′​\(x1\)∝p1​\(x1\)​er​\(x1\)p\_\{1\}^\{\\prime\}\(x\_\{1\}\)\\propto p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\. Here,r​\(x1\)=∑j=1Grj​\(x1\)r\(x\_\{1\}\)=\\sum\_\{j=1\}^\{G\}r\_\{j\}\(x\_\{1\}\)is a composition of measurable reward functionsrj:ℝd→ℝr\_\{j\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}to be maximized\.

Inference\-time guided sampling steers the generation process by injecting an additive termgt​\(xt,t\)g\_\{t\}\(x\_\{t\},t\)into the base vector field\(pokle2024training;feng2025guidance\):

vt′​\(xt,t\)=vt​\(xt,t\)\+gt​\(xt,t\),v\_\{t\}^\{\\prime\}\(x\_\{t\},t\)=v\_\{t\}\(x\_\{t\},t\)\+g\_\{t\}\(x\_\{t\},t\),\(3\)while still initializing trajectories from the fixed source distributionp0p\_\{0\}, as illustrated in Figure[2](https://arxiv.org/html/2605.20758#S1.F2)\(b,c\)\. Recall that the modified vector fieldvt′​\(xt,t\)v^\{\\prime\}\_\{t\}\(x\_\{t\},t\)preserves the pretrained prior; that is, the conditional vector fieldvt′​\(xt,t\|z\)=vt​\(xt,t\|z\)v^\{\\prime\}\_\{t\}\(x\_\{t\},t\|z\)=v\_\{t\}\(x\_\{t\},t\|z\)and the conditional probability pathpt′​\(xt\|z\)=pt​\(xt\|z\)p^\{\\prime\}\_\{t\}\(x\_\{t\}\|z\)=p\_\{t\}\(x\_\{t\}\|z\)remain invariant\. Consequently, guided sampling is formally equivalent to reweighting the coupling measureπ​\(z\)\\pi\(z\)over the latent coupling paths\. The modified marginal probability path and vector field are given by:

pt′​\(xt\)\\displaystyle p\_\{t\}^\{\\prime\}\(x\_\{t\}\)=∫pt​\(xt\|z\)​π′​\(z\)​𝑑z,\\displaystyle=\\int p\_\{t\}\(x\_\{t\}\|z\)\\,\\pi^\{\\prime\}\(z\)\\,dz,\(4\)vt′​\(xt,t\)\\displaystyle v\_\{t\}^\{\\prime\}\(x\_\{t\},t\)=∫vt​\(xt,t\|z\)​π′​\(z\|xt\)​𝑑z,\\displaystyle=\\int v\_\{t\}\(x\_\{t\},t\|z\)\\,\\pi^\{\\prime\}\(z\|x\_\{t\}\)\\,dz,\(5\)withπ′​\(z\)=1𝒵​𝒫​\(z\)​π​\(z\)​er​\(x1\)\\pi^\{\\prime\}\(z\)=\\frac\{1\}\{\\mathcal\{Z\}\}\\mathcal\{P\}\(z\)\\pi\(z\)e^\{r\(x\_\{1\}\)\}\. Here,𝒫​\(z\)≜π′​\(x0∣x1\)π​\(x0∣x1\)\\mathcal\{P\}\(z\)\\triangleq\\frac\{\\pi^\{\\prime\}\(x\_\{0\}\\mid x\_\{1\}\)\}\{\\pi\(x\_\{0\}\\mid x\_\{1\}\)\}is the coupling ratio, which captures the shift in the coupling measure: from a coupling perspective, constraining the target to the reweighted distributionp1′​\(x1\)p\_\{1\}^\{\\prime\}\(x\_\{1\}\)inherently changes the optimal coupling between source and target\. And the coupling shift term𝒫​\(z\)\\mathcal\{P\}\(z\)ensures that the sourcep0​\(x0\)p\_\{0\}\(x\_\{0\}\)remains unchanged under the guidance operation\.𝒵=∬π​\(x0,x1\)​er​\(x1\)​𝑑x0​𝑑x1\\mathcal\{Z\}=\\iint\\pi\(x\_\{0\},x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\\,dx\_\{0\}\\,dx\_\{1\}is the normalizing constant, andr​\(x1\)r\(x\_\{1\}\)is the reward function evaluated at the terminal state of the pathzz\.

By Equations \([3](https://arxiv.org/html/2605.20758#S2.E3)\) and \([5](https://arxiv.org/html/2605.20758#S2.E5)\),feng2025guidancegave a closed\-form guidance term:

gt​\(xt,t\)=∫\(𝒫​\(z\)​er​\(x1\)𝒵t​\(xt\)−1\)​vt​\(xt,t\|z\)​π​\(z\|xt\)​𝑑z\.\\displaystyle g\_\{t\}\(x\_\{t\},t\)=\\int\\left\(\\frac\{\\mathcal\{P\}\(z\)e^\{r\(x\_\{1\}\)\}\}\{\\mathcal\{Z\}\_\{t\}\(x\_\{t\}\)\}\-1\\right\)\\,v\_\{t\}\(x\_\{t\},t\|z\)\\,\\pi\(z\|x\_\{t\}\)\\,dz\.\(6\)where𝒵t​\(xt\)=∫𝒫​\(z\)​er​\(x1\)​π​\(z∣xt\)​𝑑z\\mathcal\{Z\}\_\{t\}\(x\_\{t\}\)=\\int\\mathcal\{P\}\(z\)e^\{r\(x\_\{1\}\)\}\\pi\(z\\mid x\_\{t\}\)dzis the normalizer\.

Existing guided sampling methods\(pokle2024training;yu2023freedom;Patel\_2025\_ICCV\)typically assume𝒫​\(z\)≈1\\mathcal\{P\}\(z\)\\approx 1and approximate the reward function via a first\-order Taylor expansionx^1​\(xt\)≜𝔼z∼π​\(z∣xt\)​\[x1\]\\hat\{x\}\_\{1\}\(x\_\{t\}\)\\triangleq\\mathbb\{E\}\_\{z\\sim\\pi\(z\\mid x\_\{t\}\)\}\[x\_\{1\}\]\. Then, the approximate guidance term:

gtapprox\(xt,t\)≈Covπ​\(z∣xt\)\(vt\(⋅∣z\),x1\)∇x^1r\(x^1\)\.\\displaystyle g^\{\\text\{approx\}\}\_\{t\}\(x\_\{t\},t\)\\approx\\text\{Cov\}\_\{\\pi\(z\\mid x\_\{t\}\)\}\\big\(v\_\{t\}\(\\cdot\\mid z\),x\_\{1\}\\big\)\\nabla\_\{\\hat\{x\}\_\{1\}\}r\(\\hat\{x\}\_\{1\}\)\.\(7\)In practice, this covariance matrix is often further simplified to a hyperparameter or a time\-dependent scalar value\.

## 3Related works

Inference\-time guided sampling methods aim to estimate the additive guidancegt​\(xt,t\)g\_\{t\}\(x\_\{t\},t\), which can be broadly categorized into approximate guidance and exact guidance\.

Approximate Guidance\.Lots of works use gradient∇xtr​\(x^1\)\\nabla\_\{x\_\{t\}\}r\(\\hat\{x\}\_\{1\}\)as defined in Equation \([7](https://arxiv.org/html/2605.20758#S2.E7)\) as guidance, and implicitly assume the coupling ratio𝒫​\(z\)≈1\\mathcal\{P\}\(z\)\\approx 1\. For example, in diffusion models, it is widely adopted by DPS\(chung2023diffusion\)and LGD\(song2023loss\)\. In flow models, FlowDPS\(pokle2024training\)applies this approximation to the OT\-ODE \(guiding trajectories via∇xtlog⁡p​\(y\|x^1\)\\nabla\_\{x\_\{t\}\}\\log p\(y\|\\hat\{x\}\_\{1\}\)\), and FlowChef\(Patel\_2025\_ICCV\)derives similar guidance under assumptions of a locally linear vector field and a constant Jacobian\. Both approaches converge to the form∇xtr​\(x^1\)\\nabla\_\{x\_\{t\}\}r\(\\hat\{x\}\_\{1\}\), which we refer to asgcov\-Gg^\{\\text\{cov\-G\}\}following the notation offeng2025guidance\. These approximate methods are simple, compute\-light, and flexible\. Particularly for rectified flows\(liu2023flow\), straight paths enable efficient terminal prediction via a single Euler step\. However, these methods often result in off\-manifold drift\.

Exact Guidance\.Exact guidance methods explicitly learn the ground\-truth guidance, falling into two sub\-categories\.*Training\-based*methods regress to the ground\-truth guidance \(Equation \([6](https://arxiv.org/html/2605.20758#S2.E6)\)\)\. For example, Guidance Matching \(GM\)\(feng2025guidance\)directly learns the guidance by regressing tovt​\(xt\|x1\)=\(x1−xt\)/\(1−t\)v\_\{t\}\(x\_\{t\}\|x\_\{1\}\)=\(x\_\{1\}\-x\_\{t\}\)/\(1\-t\), wherex1x\_\{1\}is sampled from the unnormalized target distributionp1′​\(x1\)p^\{\\prime\}\_\{1\}\(x\_\{1\}\)\. Although GM achieves high fidelity, it is often impractical: ground\-truth samplesx1x\_\{1\}are inaccessible in tasks such as image editing, and training a guidance network per reward function is computationally expensive and introduces additional confounding errors from a learned network\.*Sample\-based, training\-free*methods instead estimategt​\(xt\)g\_\{t\}\(x\_\{t\}\)via SDE or ODE sampling\(feng2025guidance;holderrieth2026glass\)\. The recent GLASS\-FKS\(holderrieth2026glass\), which steers GLASS flows via Feynman\-Kac sampling, improves sampling efficiency over prior particle methods, but still still inherits the high variance and heavy per\-sample compute cost characteristic of this family\.

Positioned between these two categories, we aim to improve compute\-light approximate guidance by adding a fraction of extra compute to correct the approximation error \(see Appendix[A](https://arxiv.org/html/2605.20758#A1)for further discussion\)\.

## 4Approximation errors of guided sampling

In this section, we analyze approximation errors within the framework of measure transport\.

Guided sampling targets a tilted terminal distributionp1⋆​\(x1\)∝p1​\(x1\)​er​\(x1\)p^\{\\star\}\_\{1\}\(x\_\{1\}\)\\propto p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\. Consistent with this objective, the guided marginal density at any intermediate timettcan be expressed aspt⋆​\(xt\)∝pt​\(xt\)​eV​\(xt,t\)p^\{\\star\}\_\{t\}\(x\_\{t\}\)\\propto p\_\{t\}\(x\_\{t\}\)e^\{V\(x\_\{t\},t\)\}, where the value functionV​\(xt,t\)V\(x\_\{t\},t\)is defined as the log\-expected exponentiated future reward:

V​\(xt,t\)\\displaystyle V\(x\_\{t\},t\)≜log⁡𝔼x1∼p\(⋅∣xt\)​\[er​\(x1\)\]\\displaystyle\\triangleq\\log\\mathbb\{E\}\_\{x\_\{1\}\\sim p\(\\cdot\\mid x\_\{t\}\)\}\\\!\\left\[e^\{r\(x\_\{1\}\)\}\\right\]\(8\)=log⁡𝔼z∼π\(⋅∣xt\)​\[eR​\(z\)\],\\displaystyle=\\log\\mathbb\{E\}\_\{z\\sim\\pi\(\\cdot\\mid x\_\{t\}\)\}\\left\[e^\{R\(z\)\}\\right\],\(9\)where the second equality holds due to the deterministic mapping from latent space to data space, i\.e\.,x1=Ψ1​\(z\)x\_\{1\}=\\Psi\_\{1\}\(z\)\(whereΨt\\Psi\_\{t\}is the flow map at timett\), inherent in flow models\. This enables evaluating the reward directly on the latent variablezzviaR​\(z\)≜r​\(Ψ1​\(z\)\)R\(z\)\\triangleq r\(\\Psi\_\{1\}\(z\)\)\.

The optimal coupling measure is defined as:

π⋆​\(z\)=1𝒵​𝒫​\(z\)​eR​\(z\)​π​\(z\),\\pi^\{\\star\}\(z\)=\\frac\{1\}\{\\mathcal\{Z\}\}\{\\color\[rgb\]\{0\.2421875,0\.30859375,0\.58203125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.2421875,0\.30859375,0\.58203125\}\\mathcal\{P\}\(z\)\}\{\\color\[rgb\]\{0\.40625,0\.140625,0\.53125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.40625,0\.140625,0\.53125\}e^\{R\(z\)\}\}\\pi\(z\),\(10\)whereπ​\(z\)\\pi\(z\)is the base prior,eR​\(z\)e^\{R\(z\)\}is the trajectory reward, and𝒫​\(z\)\\mathcal\{P\}\(z\)is the coupling shift ratio\. From exact to the approximate guidance, we follow the approximation process

π⋆​\(z\)→𝒫​\(z\)≈1Coupling\-InvariantπCI​\(z\)→V^​\(xt\)Localizeπapprox​\(z\),\\pi^\{\\star\}\(z\)\\xrightarrow\[\\mathcal\{P\}\(z\)\\approx 1\]\{\\text\{Coupling\-Invariant\}\}\\pi^\{\\mathrm\{CI\}\}\(z\)\\xrightarrow\[\\hat\{V\}\(x\_\{t\}\)\]\{\\text\{Localize\}\}\\pi^\{\\text\{approx\}\}\(z\),\(11\)with a fixed sourcep0p\_\{0\}\. We can then attribute the approximation error to two approximation steps: First, by defining the shift ratio as𝒫​\(z\)≜d​π⋆d​πCI​\(z\)\\mathcal\{P\}\(z\)\\triangleq\\frac\{d\\pi^\{\\star\}\}\{d\\pi^\{\\mathrm\{CI\}\}\}\(z\)and assuming𝒫​\(z\)≡1\\mathcal\{P\}\(z\)\\equiv 1, we haveπCI​\(z\)∝eR​\(z\)​π​\(z\)\\pi^\{\\mathrm\{CI\}\}\(z\)\\propto e^\{R\(z\)\}\\pi\(z\)which ignores the coupling shift in the optimal transport, i\.e\., optimal coupling is invariant\. Second, the intractable value gradient∇xtV​\(xt\)\\nabla\_\{x\_\{t\}\}V\(x\_\{t\}\)is locally approximated via a two\-step proxy: \(i\) In diffusion models, a common simplification is to move the expectation inside the log\-exponential via Jensen’s inequality, yielding∇log⁡𝔼z\|xt​\[eR​\(z\)\]≈∇𝔼z\|xt​\[R​\(z\)\]\\nabla\\log\\mathbb\{E\}\_\{z\|x\_\{t\}\}\[e^\{R\(z\)\}\]\\approx\\nabla\\mathbb\{E\}\_\{z\|x\_\{t\}\}\[R\(z\)\]\. \(ii\) We approximate the expected reward using a first\-order Taylor expansion around the mean,𝔼​\[R​\(z\)\]≈r​\(x^1\)\\mathbb\{E\}\[R\(z\)\]\\approx r\(\\hat\{x\}\_\{1\}\), withx^1=𝔼​\[x1\|xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\|x\_\{t\}\]\. Then, we have a surrogateV^​\(xt\)≜∑j=1Grj​\(x^1\)\\hat\{V\}\(x\_\{t\}\)\\triangleq\\sum\_\{j=1\}^\{G\}r\_\{j\}\(\\hat\{x\}\_\{1\}\), which drives the guidance through the aggregated gradients∑j=1G∇rj​\(x^1\)\\sum\_\{j=1\}^\{G\}\\nabla r\_\{j\}\(\\hat\{x\}\_\{1\}\)\.

To set the stage for our error analysis, we first introduce the gradient misalignment between rewards\.

###### Definition 4\.1\(Gradient Misalignment\)\.

Letgk​\(xt\)≜∇xtrk​\(x^1\)g\_\{k\}\(x\_\{t\}\)\\triangleq\\nabla\_\{x\_\{t\}\}r\_\{k\}\(\\hat\{x\}\_\{1\}\)denote the guidance contributed by thekk\-th reward, evaluated at the predicted terminal statex^1=𝔼​\[x1\|xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\|x\_\{t\}\]\. Letϕj​k\\phi\_\{jk\}be the angle betweengjg\_\{j\}andgkg\_\{k\}\. The gradients are defined as*misaligned*when they are not perfectly collinear, i\.e\.,1−cos⁡ϕj​k\>01\-\\cos\\phi\_\{jk\}\>0\.

For simplicity, we define the average pairwise cosine similarity ascos⁡ϕ:=2G​\(G−1\)​∑j<kcos⁡ϕj​k\\cos\\phi\\;:=\\;\\frac\{2\}\{G\(G\-1\)\}\\sum\_\{j<k\}\\cos\\phi\_\{jk\}\. Broadly, gradient misalignment is quantified by1−cos⁡ϕ\>01\-\\cos\\phi\>0\.

###### Theorem 4\.2\(Upper Bound of Approximation Error\)\.

The approximation errorℰ≜W22​\(p1⋆,p^1\)\\mathcal\{E\}\\triangleq W\_\{2\}^\{2\}\(p\_\{1\}^\{\\star\},\\hat\{p\}\_\{1\}\)between the exact and realized target distributions admits the three\-term decomposition:

ℰ≲\\displaystyle\\mathcal\{E\}\\;\\lesssim\\;CCI​∫01𝔼​\[𝔼πCI​\[\(𝒫​\(z\)−1\)2\]\]​𝑑t⏟\(A\) coupling shift error\\displaystyle\\underbrace\{\{\\color\[rgb\]\{0\.2421875,0\.30859375,0\.58203125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.2421875,0\.30859375,0\.58203125\}C\_\{\\mathrm\{CI\}\}\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{CI\}\}\}\\\!\\big\[\(\\mathcal\{P\}\(z\)\-1\)^\{2\}\\big\]\\right\]dt\}\}\_\{\\text\{\(A\) coupling shift error\}\}\+G​\(G−1\)​μ2​∫01𝔼​\[1−cos⁡ϕt​\(xt\)\]​𝑑t⏟\(B\) gradient misalignment error\\displaystyle\\;\\;\+\\;\\underbrace\{\{\\color\[rgb\]\{0\.40625,0\.140625,0\.53125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.40625,0\.140625,0\.53125\}G\(G\-1\)\\,\\mu^\{2\}\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[1\-\\cos\\phi\_\{t\}\(x\_\{t\}\)\\right\]dt\}\}\_\{\\text\{\(B\) gradient misalignment error\}\}\+G​∫01𝔼​\[\(λh​σ1​der​\(x^1\)\)2​\(C1\+C2\)\]​𝑑t⏟\(C\) localized approximation error\(feng2025guidance\)\\displaystyle\\;\\;\+\\;\\underbrace\{G\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\left\(\\frac\{\\lambda\_\{h\}\\,\\sigma\_\{1\}\\,d\}\{e^\{r\(\\hat\{x\}\_\{1\}\)\}\}\\right\)^\{\\\!2\}\\\!\(C\_\{1\}\+C\_\{2\}\)\\right\]dt\}\_\{\\text\{\(C\) localized approximation error \\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{feng2025guidance\}\{\\@@citephrase\{, \}\}\{\}\)\}\}\}\(12\)whereμ:=‖gjCI‖\\mu:=\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|is the per\-reward gradient norm,CCIC\_\{\\mathrm\{CI\}\}only depends on base velocity field,λh\\lambda\_\{h\}is the spectral norm of the Hessian ofere^\{r\},GGis the number of rewards,\(1−cos⁡ϕt\)\(1\-\\cos\\phi\_\{t\}\)is the gradient misalignment,σ1​\(t\)\\sigma\_\{1\}\(t\)measures posterior uncertainty, andC1,C2C\_\{1\},C\_\{2\}are constants depending on the base velocity field\. We provide an intuitive interpretation in Appendix[B](https://arxiv.org/html/2605.20758#A2)and a more detailed proof in Appendix[C](https://arxiv.org/html/2605.20758#A3)\.

Theorem[4\.2](https://arxiv.org/html/2605.20758#S4.Thmtheorem2)provides an error bound and offers a theoretical roadmap, identifying the key design space for optimizing guided sampling:

1. 1\.the error scales with the number of reward functionsGGand gradient misalignment\(1−cos⁡ϕ\)\(1\-\\cos\\phi\);
2. 2\.the error is small when the coupling shift is negligible \(𝒫​\(z\)≈1\\mathcal\{P\}\(z\)\\approx 1\), which is reasonable in many practical flow matching methods with dependent couplings, such as mini\-batch OT\-FM\(tong2023improving\), but can be problematic in OT\-FM\(onken2021ot\);
3. 3\.the error is small when the reward landscape is smooth, i\.e\., smallλh=‖∇2er‖2\\lambda\_\{h\}=\\\|\\nabla^\{2\}e^\{r\}\\\|\_\{2\};
4. 4\.the error is small when the predicted endpointx^1=𝔼​\[x1\|xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\|x\_\{t\}\]lies in a high\-reward region \(larger​\(x^1\)r\(\\hat\{x\}\_\{1\}\)\);
5. 5\.the error decreases asσ1\\sigma\_\{1\}shrinks \(e\.g\.,t→1t\\\!\\to\\\!1\)\.

## 5Conflict\-aware additive guidance

![Refer to caption](https://arxiv.org/html/2605.20758v1/x3.png)Figure 3:\(a\) When gradients are misaligned \(largeϕ\\phi\), the approximate guidancegapproxg^\{\\text\{approx\}\}\(→\\bm\{\\to\}\), the vector sum of→\\bm\{\\to\}and→\\bm\{\\to\}, points off\-manifold; the conflict\-aware weightwt≈1w\_\{t\}\\approx 1, sogcar≈gψ​\(xt,t\)g^\{\\text\{car\}\}\\approx g\_\{\\psi\}\(x\_\{t\},t\)\(→\\bm\{\\to\}\) corrects the trajectory toward the true targetx1x\_\{1\}\(⋆\\star\)\. \(b\) When gradients align \(ϕ≈0\\phi\\approx 0\),gapproxg^\{\\text\{approx\}\}is already accurate;wt≈0w\_\{t\}\\approx 0, sogcar≈gapproxg^\{\\text\{car\}\}\\approx g^\{\\text\{approx\}\}\(→\\bm\{\\to\}\)\.We improve the compute\-light approximate guidance by adding extra compute to correct the approximation error:

###### Definition 5\.1\(Conflict\-Aware Additive Guidance\)\.

Letℳ=\(vt,p0,p1\)\\mathcal\{M\}=\(v\_\{t\},p\_\{0\},p\_\{1\}\)be a base flow model\. Given a target distributionp1′​\(x1\)∝p1​\(x1\)​er​\(x1\)p^\{\\prime\}\_\{1\}\(x\_\{1\}\)\\propto p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}with rewardsr​\(x1\)=∑j=1Grj​\(x1\)r\(x\_\{1\}\)=\\sum\_\{j=1\}^\{G\}r\_\{j\}\(x\_\{1\}\), the conflict\-aware additive guidance \(gcarg^\{\\text\{car\}\}\) transforms the base velocityvtv\_\{t\}into a guided velocityvt′v^\{\\prime\}\_\{t\}via:

vt′​\(xt,t\)\\displaystyle v^\{\\prime\}\_\{t\}\(x\_\{t\},t\)=vtbase​\(xt,t\)\+gcar​\(xt,t\),\\displaystyle\\;=\\;v^\{\\text\{base\}\}\_\{t\}\(x\_\{t\},t\)\\;\+\\;g^\{\\text\{car\}\}\(x\_\{t\},t\),\(13\)gcar​\(xt,t\)\\displaystyle g^\{\\text\{car\}\}\(x\_\{t\},t\)=\(1−wt\)​gapprox\+wt​gψ​\(xt,t\),\\displaystyle\\;=\\;\(1\-w\_\{t\}\)\\,g^\{\\text\{approx\}\}\\;\+\\;w\_\{t\}\\,g\_\{\\psi\}\(x\_\{t\},t\),\(14\)where the guidance is a composition of a fast approximationgapproxg^\{\\text\{approx\}\}and a learnable guidancegψg\_\{\\psi\}, which is controlled by a conflict\-aware weight

wraw​\(xt\)=1−2G​\(G−1\)​∑j<k⟨gj,gk⟩‖gj‖​‖gk‖\+ε,w\_\{\\text\{raw\}\}\(x\_\{t\}\)=1\-\\frac\{2\}\{G\(G\-1\)\}\\sum\_\{j<k\}\\frac\{\\langle g\_\{j\},g\_\{k\}\\rangle\}\{\\\|g\_\{j\}\\\|\\\|g\_\{k\}\\\|\+\\varepsilon\},\(15\)The raw conflict scorewraw​\(xt\)∈\[0,2\]w\_\{\\mathrm\{raw\}\}\(x\_\{t\}\)\\in\[0,2\]is then mapped tow​\(xt\)∈\(0,1\)w\(x\_\{t\}\)\\in\(0,1\)\. Theε\\varepsilonensures numerical stability\. Finally,gψg\_\{\\psi\}is trained to approximate the exact guidance by data \(see Section[5\.1](https://arxiv.org/html/2605.20758#S5.SS1)for details\)\.

As illustrated in Figure[3](https://arxiv.org/html/2605.20758#S5.F3), in regions of gradient misalignment, ourgcarg^\{\\text\{car\}\}leverages a learned lightweight guidancegψg\_\{\\psi\}to correct the systematic errors inherent in approximate methods, thereby effectively rectifying off\-manifold drift\. This design positionsgcarg^\{\\text\{car\}\}effectively between approximate and exact approaches, delivering near\-exact performance with significantly lighter compute compared to exact guidance\.

### 5\.1Value gradient

To bypass the intractability of the closed\-form target in Eq\. \([6](https://arxiv.org/html/2605.20758#S2.E6)\), we seek to learn the exact guidancegψg\_\{\\psi\}directly from data\. By viewing the flow generation process as deterministic ODE dynamics, we evaluateV​\(xt,t\)V\(x\_\{t\},t\)as the cumulative return of a trajectory induced by the current policyvt′v^\{\\prime\}\_\{t\}defined in Eq\. \([13](https://arxiv.org/html/2605.20758#S5.E13)\)\. Mathematically,V​\(xt,t\)V\(x\_\{t\},t\)serves as the fixed point of the Bellman backup operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}\. Details can be found in Appendix[D](https://arxiv.org/html/2605.20758#A4)\.

###### Proposition 5\.2\(Fitted Value Evaluation\)\.

Letℱ\\mathcal\{F\}denote the function class \(e\.g\., neural networks\) used to approximate the value function\. We collect a dataset of transitions𝒟=\{\(x,t,r,x′,t′\)\}\\mathcal\{D\}=\\\{\(x,t,r,x^\{\\prime\},t^\{\\prime\}\)\\\}generated under the current guided dynamics, wheret′=t\+Δ​tt^\{\\prime\}=t\+\\Delta tandrris the reward\. The value function can be estimated empirically by minimizing a least\-squares Bellman residual:

V^k\+1=arg⁡minV∈ℱ⁡𝔼𝒟​\[\(r\+γ​V^k​\(x′,t′\)−V​\(x,t\)\)2\]\.\\hat\{V\}\_\{k\+1\}=\\arg\\min\_\{V\\in\\mathcal\{F\}\}\\;\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\left\[\\big\(r\+\\gamma\\hat\{V\}\_\{k\}\(x^\{\\prime\},t^\{\\prime\}\)\-V\(x,t\)\\big\)^\{2\}\\right\]\.\(16\)subject to the boundary conditionV^k​\(x,1\)≡r​\(x\)\\hat\{V\}\_\{k\}\(x,1\)\\equiv r\(x\)for terminal states\. Here,γ∈\(0,1\]\\gamma\\in\(0,1\]is the discount factor, andV^k\\hat\{V\}\_\{k\}is the target from the previous iteration111For pure terminal optimization,r=0r=0andγ=1\\gamma=1\.\. Equation \([16](https://arxiv.org/html/2605.20758#S5.E16)\) empirically approximates the Bellman backup operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}using finite data and a function classℱ\\mathcal\{F\}\. Upon convergence, the learnable guidance is the gradient of the estimated value:

g​\(xt,t\)≜∇xV^​\(xt,t\),g\(x\_\{t\},t\)\\;\\triangleq\\;\\nabla\_\{x\}\\hat\{V\}\(x\_\{t\},t\),\(17\)which is a Markovian surrogate for the exact guidance\.

Fitted Value Evaluation can diverge due to the deadly triad, the instability arising from the interplay of function approximation, off\-policy data, and bootstrapping \(Proposition[D\.2](https://arxiv.org/html/2605.20758#A4.Thmtheorem2), Appendix[D](https://arxiv.org/html/2605.20758#A4)\)\. Leveraging the deterministic dynamics and terminal\-only rewards of flow matching, we propose Terminal Value Regression \(TVR\)\. By regressing directly against the terminal rewardr​\(x1\)r\(x\_\{1\}\), TVR eliminates the need for bootstrapping, effectively breaking the deadly triad and ensuring stable convergence\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x4.png)Figure 4:Visualization results on the synthetic dataset under\[1,0\]\[1,0\]constraints\. \(c–e\)gcov\-Gg^\{\\text\{cov\-G\}\}shows significant off\-manifold drift due to an “energy trap” \(highlighted by the red circle\), where conflicting gradients lead to erratic sampling trajectories\. \(f–h\) Ourgcarg^\{\\text\{car\}\}restores the accurate reward landscape, thereby rectifying the off\-manifold drift\. An intuitive interpretation of the “energy trap” caused by gradient misalignment can be found in Appendix[B](https://arxiv.org/html/2605.20758#A2)\.###### Proposition 5\.3\(Terminal Value Regression\)\.

Letℱ\\mathcal\{F\}denote a function class used to approximate the value function\. We collect a dataset of terminal rollouts𝒟=\{\(xt,t,x1\)\}\\mathcal\{D\}=\\\{\(x\_\{t\},t,x\_\{1\}\)\\\}, wherex1x\_\{1\}is the terminal state reached fromxtx\_\{t\}by integrating the current guided dynamics\. The value function is estimated by minimizing the following regression objective:

V^=arg⁡minV∈ℱ⁡𝔼\(xt,t,x1\)∼𝒟​\[\(r​\(x1\)−V​\(xt,t\)\)2\]\.\\hat\{V\}=\\arg\\min\_\{V\\in\\mathcal\{F\}\}\\;\\mathbb\{E\}\_\{\(x\_\{t\},t,x\_\{1\}\)\\sim\\mathcal\{D\}\}\\Big\[\\big\(r\(x\_\{1\}\)\-V\(x\_\{t\},t\)\\big\)^\{2\}\\Big\]\.\(18\)Unlike the bootstrapped target in Eq\. \([16](https://arxiv.org/html/2605.20758#S5.E16)\), the terminal rewardr​\(x1\)r\(x\_\{1\}\)serves as a stable, unbiased regression target, which is enabled by the deterministic nature of the flow\.

We parameterize the scalar value functionV​\(xt,t\)V\(x\_\{t\},t\), then define the guidance asgψ​\(xt,t\)≜∇xtVψ​\(xt,t\)g\_\{\\psi\}\(x\_\{t\},t\)\\triangleq\\nabla\_\{x\_\{t\}\}V\_\{\\psi\}\(x\_\{t\},t\)\. Building on Proposition[5\.3](https://arxiv.org/html/2605.20758#S5.Thmtheorem3), we optimizeψ\\psiby minimizing a masked regression loss against the terminal reward:

ℒ​\(ψ\)=𝔼\(xt,t,x1\)∼𝒟​\[𝕀t⋅\(r​\(x1\)−Vψ​\(xt,t\)\)2\],\\mathcal\{L\}\(\\psi\)=\\mathbb\{E\}\_\{\(x\_\{t\},t,x\_\{1\}\)\\sim\\mathcal\{D\}\}\\left\[\\mathbb\{I\}\_\{t\}\\cdot\\big\(r\(x\_\{1\}\)\-V\_\{\\psi\}\(x\_\{t\},t\)\\big\)^\{2\}\\right\],\(19\)wherex1x\_\{1\}is the terminal state obtained from online rollouts\. We allocate compute budget efficiently by using𝕀t≜𝟙​\(w​\(xt\)\>τ\)\\mathbb\{I\}\_\{t\}\\triangleq\\mathds\{1\}\(w\(x\_\{t\}\)\>\\tau\), which ensures the guidancegψg\_\{\\psi\}is updated solely in regions showing high gradient conflict\. While directly parameterizing the vector field∇V​\(xt,t\)\\nabla V\(x\_\{t\},t\)is common in diffusion models\(song2021train\), unconstrained neural vector fields are not guaranteed to be conservative \(i\.e\., curl\-free\)\(balcerak2025energy\)\. See Appendix[E\.1](https://arxiv.org/html/2605.20758#A5.SS1)for further discussion\.

## 6Experiments

Our experiments are designed to answer two core questions: \(1\) Cangcarg^\{\\text\{car\}\}effectively rectify off\-manifold drift under compositional reward settings? \(2\) Is thegcarg^\{\\text\{car\}\}compute light?

Table 1:Comparison with baselines on the Synthetic dataset\. Each number is evaluated over10​k10\\text\{k\}generated samples\. Here\[⋅,⋅\]\[\\,\\cdot\\,,\\,\\cdot\\,\]denotes the target labels of two classifiers, where\[1,0\]\[1,0\]represents the gradient conflict scenario\. For GLASS\-FKS, we reportK=16K\{=\}16; Forgcarg^\{\\text\{car\}\},τ=0\.50\\tau=0\.50\. Full results are in Appendix[E\.4\.3](https://arxiv.org/html/2605.20758#A5.SS4.SSS3)\.Bold textis the best performance\.Table 2:Comparison with baselines on Maze2D\. Compositional reward settings: \(1\) static obstacle: two random static obstacles; \(2\) static goal: two random goals; \(3\) dynamic obstacle: two random dynamic agents; \(4\) hybrid composition: one static and one dynamic obstacles\. Metrics include Inference Time \(ms/sample\), Safe \(collision\-free rate %\), Violation \(mean constraint violations \#\), Success \(success rate %\), and Steps \(\#\)\. Results are averaged over 100 samples with conflict thresholdτ=0\.20\\tau=0\.20\. Note that we do not use inpainting, which allows us to better observe the capability of inference\-time alignment methods in preserving the base model prior\. Furthermore, comparing the success rates reveals that directly applying GLASS\-FKS, MPPI,gcov\-Gg^\{\\text\{cov\-G\}\}, and their PCGrad orgcarg^\{\\text\{car\}\}corrections as inference\-time techniques to the base flow to satisfy constraints compromises prior preservation\. Thegcarg^\{\\text\{car\}\}method requires an online training period of10\.2±0\.110\.2\\pm 0\.1min \(mean±\\pmstd across 5 random seeds\) for 4 training steps prior to being used for inference\.\(1\) static obstacles\(2\) static goal\(3\) dynamic obstacles\(4\) hybrid compositionMethodTime↓\\downarrowSafety↑\\uparrowViol\.↓\\downarrowSucc\.↑\\uparrowSteps↓\\downarrowSafety↑\\uparrowViol\.↓\\downarrowSucc\.↑\\uparrowSteps↓\\downarrowSafety↑\\uparrowViol\.↓\\downarrowSucc\.↑\\uparrowSteps↓\\downarrowSafety↑\\uparrowViol\.↓\\downarrowSucc\.↑\\uparrowSteps↓\\downarrowGLASS\-FKS532\.7780\.3100\-560\.6100\-710\.4100\-680\.5100\-±\\pm53\.2±\\pm3\.2±\\pm0\.1±\\pm0\.0±\\pm4\.5±\\pm0\.2±\\pm0\.0±\\pm3\.8±\\pm0\.1±\\pm0\.0±\\pm4\.1±\\pm0\.2±\\pm0\.0MPPI242\.41000\.04130621\.56930580\.56330470\.46930±\\pm12\.5±\\pm0\.0±\\pm0\.0±\\pm2\.4±\\pm3\.5±\\pm0\.3±\\pm3\.1±\\pm2\.8±\\pm0\.2±\\pm2\.9±\\pm3\.1±\\pm0\.2±\\pm3\.4MPPI \+gcarg^\{\\text\{car\}\}\(ours\)265\.81000\.098↑5730100↑380\.0↓1\.596↑273096↑380\.1↓0\.494↑313095↑480\.2↓0\.292↑2330±\\pm14\.2±\\pm0\.0±\\pm0\.0±\\pm1\.2±\\pm0\.0±\\pm0\.0±\\pm1\.5±\\pm1\.4±\\pm0\.0±\\pm1\.8±\\pm1\.6±\\pm0\.1±\\pm2\.1gcov\-Gg^\{\\text\{cov\-G\}\}150\.0390\.316\-411\.723\-390\.942\-341\.112\-±\\pm5\.4±\\pm2\.1±\\pm0\.1±\\pm1\.4±\\pm2\.4±\\pm0\.3±\\pm1\.8±\\pm2\.2±\\pm0\.2±\\pm2\.5±\\pm1\.9±\\pm0\.2±\\pm1\.1PCGrad175\.244↑50\.2↓0\.127↑11\-47↑61\.5↓0\.232↑9\-41↑20\.7↓0\.246↑4\-35↑10\.9↓0\.218↑6\-±\\pm18\.0±\\pm2\.3±\\pm0\.1±\\pm1\.7±\\pm2\.6±\\pm0\.3±\\pm2\.1±\\pm2\.1±\\pm0\.1±\\pm2\.8±\\pm1\.8±\\pm0\.2±\\pm1\.5gcarg^\{\\text\{car\}\}\(ours\)168\.474↑350\.0↓0\.379↑63463↑220\.8↓0\.972↑49449↑100\.2↓0\.761↑19443↑90\.3↓0\.836↑244±\\pm8\.6±\\pm1\.5±\\pm0\.0±\\pm1\.8±\\pm2\.2±\\pm0\.1±\\pm1\.9±\\pm1\.7±\\pm0\.1±\\pm2\.0±\\pm1\.8±\\pm0\.1±\\pm1\.6
Note:Bold textindicates the best performance\. Rows withgray backgroundsindicate methods that use ourgcarg^\{\\text\{car\}\}for conflict correction\.Purple superscriptsshow the performance change ofgcarg^\{\\text\{car\}\}overgcov\-Gg^\{\\text\{cov\-G\}\}, andteal superscriptsshow the change of PCGrad overgcov\-Gg^\{\\text\{cov\-G\}\}, where↑\\uparrowdenotes improvement and↓\\downarrowdenotes degradation\. For all metrics, we report the mean \(top row\) and standard deviation across 5 random seeds \(bottom row\)\.

### 6\.1Experimental setup

Baselines\.Our baselines span the full spectrum of inference\-time guidance methods, letting us probe whether each exhibits off\-manifold drift: \(1\) approximate guidance \(𝒈cov\-G\\bm\{g^\{\\textbf\{cov\-G\}\}\}\), which is compute\-light but susceptible to off\-manifold drift; and \(2\) exact guidance, including the sample\-basedGLASS\-FKS\(holderrieth2026glass\)and the training\-basedGM\(feng2025guidance\)\. Since GM requires ground\-truth samplesx1x\_\{1\}that satisfy all constraints, its use is limited to synthetic datasets where such samples are accessible by construction\. Task\-specific SOTA baselines are also included, detailed in the following sections\. Ourgcarg^\{\\text\{car\}\}corrects off\-manifold drift by adding only a fraction of extra compute\. We includePCGrad\(yu2020pcgrad\), a gradient conflict resolution method from multi\-objective optimisation, to show that our conflict\-aware mechanism rectifies off\-manifold drift more effectively than projection\-based deconfliction\. See details and additional results in Appendix[E](https://arxiv.org/html/2605.20758#A5)\.

### 6\.2Synthetic dataset

Tasks\.We begin with a 2\-dimensional Mixture of Gaussians \(see Appendix[E\.4](https://arxiv.org/html/2605.20758#A5.SS4)for details\), where the ground\-truth densityptp\_\{t\}is known, allowing for precise quantitative evaluation\. We train a flow matching model\(lipman2024flowmatchingguidecode\)to transport a standard Gaussian source to a Mixture of Gaussians target\. We steer the generation using two pre\-trained classifiers that impose differing label constraints on the target samples\. These constraints are specifically configured to induce severe gradient conflicts, thereby creating a “stress test” to benchmark the robustness of different guidance methods against off\-manifold drift\.

Evaluation metrics\.We present intuitive visualizations, and also designed four quantitative metrics: \(i\) posterior coverage \(higher is better on data manifold\); \(ii\) inference time \(lower is better speed\); \(iii\) data usage \(lower indicates better training data efficiency\); \(iv\) constraint satisfaction \(higher indicates better recovery of ground\-truth posterior\)\.

Results\.We first visualize the failure modes of the standard approximation guidancegcov\-Gg^\{\\text\{cov\-G\}\}in Figure[4](https://arxiv.org/html/2605.20758#S5.F4)\. Notably,gcov\-Gg^\{\\text\{cov\-G\}\}generates off\-manifold samples \(highlighted by the red circle in Figure[4](https://arxiv.org/html/2605.20758#S5.F4)c\)\. By analyzing the underlying energy landscape \(Figure[4](https://arxiv.org/html/2605.20758#S5.F4)d\), we observe an “energy trap”, a spurious artifact arising from gradient conflict\. This trap captures the sampling process \(Figure[4](https://arxiv.org/html/2605.20758#S5.F4)e\), forcing trajectories to ultimately diverge from the data manifold\. Ourgcarg^\{\\text\{car\}\}rectifies the vanishing energy by learning a residual guidance within these conflict regions, thereby eliminating off\-manifold drift\.

Quantitative results in Table[1](https://arxiv.org/html/2605.20758#S6.T1)show, under the conflict constraint\[1,0\]\[1,0\],gcov\-Gg^\{\\text\{cov\-G\}\}drifts off\-manifold for nearly 30% of samples \(only 71\.70% PC\), and PCGrad provides only marginal correction \(75\.20% PC,↑\\uparrow3\.5\), confirming that generic projection\-based deconfliction is insufficient at inference time\. Ourgcarg^\{\\text\{car\}\}reduces off\-manifold drift to6\.2%6\.2\\%\(93\.80% PC\), incurring only a small compute overhead overgcov\-Gg^\{\\text\{cov\-G\}\}\(1\.65 vs\. 0\.37 ms/sample\)\. The exact baselines come with their own trade\-offs: GM requires roughly20×20\\timesmore training data thangcarg^\{\\text\{car\}\}and shows higher variance, while GLASS\-FKS avoids off\-manifold drift but suffers from high computational cost and is sensitive to the particle count\.

### 6\.3Generative decision\-making as planners

Tasks\.We conduct experiments on generative decision making tasks where generative models have been used as planners\. We focus on the Maze2D\(luo2024potential\), involving a point\-robot with states∈ℝ4s\\in\\mathbb\{R\}^\{4\}\(position and velocity\) and actiona∈ℝ2a\\in\\mathbb\{R\}^\{2\}\(force\)\. We aim to steer a base planner pre\-trained on expert demonstrations, where the maze layout, start and goal are randomly generated\. Followingluo2024potential, we collect a diverse set of collision\-free demonstrations viaBIT∗\\text\{BIT\}^\{\*\}to train the base CFM model\(tong2023improving\)\. The model serves as a learned prior, generating trajectoriesx=\(s0:H−1,a0:H−1\)∈ℝ6​Hx=\(s\_\{0:H\-1\},a\_\{0:H\-1\}\)\\in\\mathbb\{R\}^\{6H\}conditioned on the maze layout, start, and goal\. Ther​\(x1\)r\(x\_\{1\}\)includes: \(i\) static obstacle avoidance for unseen environments; \(ii\) goal reaching \(e\.g\., object grasping\); \(iii\) dynamic collision avoidance against agents with random linear trajectories\(romer2025diffusion;bouvier2025ddat\)\.

Table 3:Comparison on the ManiSkill2 StackCube task\. \(1\) static obstacles: two random static obstacles; \(2\) hybrid composition: two random static obstacles and trajectory smoothness\. Full results, including PickCube, are in Table[8](https://arxiv.org/html/2605.20758#A5.T8)\.Note:Bold textindicates the best performance\. Rows withgray backgroundsindicate methods that utilize ourgcarg^\{\\text\{car\}\}for conflict correction\.Purple superscriptsshow the performance change ofgcarg^\{\\text\{car\}\}overgcov\-Gg^\{\\text\{cov\-G\}\}, andteal superscriptsshow the change of PCGrad overgcov\-Gg^\{\\text\{cov\-G\}\}, where↑\\uparrowdenotes improvement and↓\\downarrowdenotes degradation\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x5.png)Figure 5:Visualization on ManiSkill2 StackCube withτ=0\.20\\tau=0\.20\. OOD: the trajectory leaves the data manifold, producing physically incoherent motions \(e\.g\., erratic spinning or tangled paths\); Fail:the trajectory stays on the manifold but fails the task \(e\.g\., does not reach the goal\)\.Baselines and metrics\.Besides the baselines introduced in Section[6\.1](https://arxiv.org/html/2605.20758#S6.SS1), we include MPPI\(williams2017mppi\), a classical optimization\-based planner effective for constrained trajectory problems, and itsgcarg^\{\\text\{car\}\}\-augmented variant MPPI\+gcarg^\{\\text\{car\}\}\. We report five metrics: \(i\) safety rate: the percentage of trajectories that are collision\-free with respect to the static maze layout, which reflects the prior preservation\. \(ii\) violations: average number of inference\-time constraint violations \(e\.g\., collisions with new obstacles or missed goals\) per trajectory\. \(iii\) success rate: the percentage of trajectories that successfully reach the task goals\. \(iv\) steps: the number of iterations required to train the learnable guidance \(forgcarg^\{\\text\{car\}\}\) or optimize the action sequence via importance sampling \(for MPPI\)\. \(v\) time: wall\-clock training and inference time per sample\.

Results\.Under the four compositional reward settings in Table[2](https://arxiv.org/html/2605.20758#S6.T2),gcov\-Gg^\{\\text\{cov\-G\}\}generates hallucinated paths, trajectories that jump across obstacles or have sharp kinks \(see Figure[16](https://arxiv.org/html/2605.20758#A5.F16)for visualisations\)\. Ourgcarg^\{\\text\{car\}\}rectifies these failures, with average gains of19\.0%19\.0\\%safety,38\.75%38\.75\\%success, and0\.680\.68fewer violations per trajectory, at only44training iterations; PCGrad provides only marginal correction\. GLASS\-FKS, while strong overall, reaches100%100\\%success at the cost of non\-zero violations and∼3\.2×\\sim 3\.2\\timesslower inference \(533533vs\.168168ms\)\. MPPI alone is a strong planning method, collision\-free on static obstacles \(100100safety,0\.00\.0violations\), but its success drops to6363–69%69\\%in dynamic and hybrid settings\. MPPI\+gcarg^\{\\text\{car\}\}outperforms vanilla MPPI and achieves the best performance, lifting average safety to97\.75%97\.75\\%and success to95%95\\%, including the previously hard static\-goal setting \(100100safety,9696success\)\.

We further evaluate robustness by increasing the number of clustered static obstacles from22to66\(Figure[15](https://arxiv.org/html/2605.20758#A5.F15), Appendix[E\.5](https://arxiv.org/html/2605.20758#A5.SS5)\):gcov\-Gg^\{\\text\{cov\-G\}\}’s success rate collapses to12%12\\%, while our method maintains34%34\\%in the most complex setting\.

### 6\.4Generative Decision\-Making as Policies

Tasks\.We evaluate our method on high\-dimensional manipulation tasks \(PickCube and StackCube\) in ManiSkill2\(gu2023maniskill2\), where the policy predicts action chunks of horizonTTfrom 3D point cloud observations\. We aim to test the capability to reactively steer a general\-purpose base policy to satisfy on\-the\-fly requirements\. The action is defined as𝐚t=\[Δ​𝐩,Δ​𝐫,g\]∈ℝ7\\mathbf\{a\}\_\{t\}=\\big\[\\Delta\\mathbf\{p\},\\Delta\\mathbf\{r\},g\\big\]\\in\\mathbb\{R\}^\{7\}, representing delta translation, rotation, and gripper state\. We train a base CFM model solely on unconstrained demonstrations\. Reward functionsr​\(x\)r\(x\)include: \(i\) static obstacles; \(ii\) trajectory smoothness costs\.

Evaluation metrics\.We evaluate violations, success rate, training steps and time, following the definitions in Section[6\.3](https://arxiv.org/html/2605.20758#S6.SS3)\.

Results\.The base CFM model achieves a100%100\\%success rate on both training and test sets \(Appendix[E\.6](https://arxiv.org/html/2605.20758#A5.SS6)\)\. Naïve inference\-time guidance tends to drift off\-manifold and produce out\-of\-distribution behavior:gcov\-Gg^\{\\text\{cov\-G\}\}generates hallucinated paths and fails to complete the tasks, and PCGrad even drives success to0%0\\%on both StackCube settings; onlygcarg^\{\\text\{car\}\}rectifies these failures \(Figures[5](https://arxiv.org/html/2605.20758#S6.F5)and[19](https://arxiv.org/html/2605.20758#A5.F19)\)\. Quantitatively \(Table[3](https://arxiv.org/html/2605.20758#S6.T3)\), under hybrid constraints,gcarg^\{\\text\{car\}\}reduces the baseline’s violation rate from1\.81\.8to0\.40\.4and boosts the success rate from9%9\\%to61%61\\%, with only∼10%\{\\sim\}10\\%inference overhead overgcov\-Gg^\{\\text\{cov\-G\}\}\(203203vs\.185185ms\)\. GLASS\-FKS struggles on StackCube \(2424–32%32\\%\), which requires stably and precisely placing one cube onto another, due to its high sampling variance\.

### 6\.5Text\-guided Image Manipulation

Table 4:Comparison with baselines on CelebA\-HQ\. We evaluate composed prompts including*sad \+ angry*,*sad \+ happy*, and*sad \+ curly hair*\. We report results withτ=0\.20\\tau=0\.20\. Full results are in Table[10](https://arxiv.org/html/2605.20758#A5.T10)\.Note:Bold textindicates the best performance\. Rows withgray backgroundsindicate methods that use ourgcarg^\{\\text\{car\}\}for conflict correction\.Purple superscriptsshow the performance change ofgcarg^\{\\text\{car\}\}overgcov\-Gg^\{\\text\{cov\-G\}\}, andteal superscriptsshow the change of PCGrad overgcov\-Gg^\{\\text\{cov\-G\}\}, where↑\\uparrowdenotes improvement and↓\\downarrowdenotes degradation\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x6.png)Figure 6:Visualization of text\-guided generated faces\.Tasks\.To evaluate the scalability ofgcarg^\{\\text\{car\}\}in high\-dimensional pixel spaces, we conduct text\-guided image manipulation on CelebA\-HQ\. Followingliu2023flowgrad, we use a pre\-trained Rectified Flow model as the generative prior and steer it toward compositional text guidance \{*sad \+ angry*,*sad \+ happy*,*sad \+ curly hair*\}, which target different facial expressions or traits\. Following the same setup asliu2023flowgrad, given an imagex1x\_\{1\}, the reward for alignment with the text prompt is evaluated by the CLIP model\(radford2021clip\), which is used to score the similarity between arbitrary image\-text pairs\. Experiment details are in Appendix[E\.7](https://arxiv.org/html/2605.20758#A5.SS7)\.

Baselines and metrics\.Besides the baselines from Section[6\.1](https://arxiv.org/html/2605.20758#S6.SS1), we additionally compare against the SOTA image editing method FlowGrad\(liu2023flowgrad\)\. We report metrics in three groups: \(i\)*Text\-image alignment*\(higher is better\): CLIP\(radford2021clip\), plus two stronger perceptual measures, BLIP\-ITM\(li2022blip\)\(a strict binary image\-text matcher\) and VQAScore\(vqascore\)\(compositional reasoning via LLaVA\-1\.5\); \(ii\)*Image quality*: LPIPS \(lower is better preservation of the reference\), ID similarity \(higher is better identity preservation\), and CLIP\-IQA\(wang2022exploring\)\(higher is fewer visual artifacts\); \(iii\)*Efficiency*: training time and per\-sample inference time\.

Results\.Visual comparisons \(Figure[6](https://arxiv.org/html/2605.20758#S6.F6)\) and quantitative metrics \(Table[4](https://arxiv.org/html/2605.20758#S6.T4)\) reveal thatgcov\-Gg^\{\\text\{cov\-G\}\}is prone to off\-manifold drift and produces hallucinated generations, with CLIP\-IQA score only0\.5350\.535\. PCGrad cannot recover from off\-manifold drift and struggles to balance multiple constraints, leaving some targets unfulfilled \(e\.g\., failing to generate an “angry” expression, with BLIP\-ITMP1=0\.650P\_\{1\}\{=\}0\.650vs\.P2=0\.337P\_\{2\}\{=\}0\.337; see Table[10](https://arxiv.org/html/2605.20758#A5.T10)\)\. FlowGrad similarly lacks the ability to balance multiple constraints\. GLASS\-FKS fails to preserve the reference image due to its high sampling variance, with the worst LPIPS \(0\.3130\.313\) and ID \(0\.3290\.329\)\. Ourgcarg^\{\\text\{car\}\}finds a sweet spot between text\-image alignment and image quality, achieving the highest ID \(0\.6810\.681\), CLIP\-IQA \(0\.5430\.543\), and alignment scores \(CLIP0\.2910\.291, BLIP\-ITM0\.5970\.597, VQAScore0\.7550\.755\); it effectively rectifies the off\-manifold drift observed ingcov\-Gg^\{\\text\{cov\-G\}\}and generates faces with minimal distortion\.

## 7Conclusions, Limitations, and Future Work

In this paper, we proposed Conflict\-Aware Additive Guidance \(gcarg^\{\\text\{car\}\}\), a lightweight guided sampling method that incorporates a conflict\-aware gating mechanism to actively detect and rectify trajectory deviations\. Experimental results showed that flow models equipped withgcarg^\{\\text\{car\}\}achieved state\-of\-the\-art steerability at inference time across diverse domains, including text\-guided image editing, robotic planning, and manipulation\.

A remaining challenge lies in convergence under complex reward landscapes\. In high\-dimensional tasks like text\-guided image editing, the CLIP reward signal is non\-smooth, producing adversarial artifacts that maximize the score without semantic improvement, makinggψg\_\{\\psi\}hard to train stably\. Lighter alternatives togψg\_\{\\psi\}, richer reward compositions, and more structured latent representations\(yu2024skillaware;dunion2023cmid\)replacing the CLIP reward to yield smoother landscapes are all promising directions for future work\.

## Acknowledgment

This research is supported by the RIE2025 Industry Alignment Fund – Industry Collaboration Projects \(IAF\-ICP\) \(Grant No\. I2501E0041\), administered by A\*STAR, as well as supported by Schaeffler \(Singapore\) PTE\. LTD\. and NTU Singapore through Schaeffler\-NTU Corporate Lab: Intelligent Mechatronics Hub\.

## Impact Statement

This work improves the reliability of inference\-time guidance for generative models under multiple, potentially conflicting objectives\. By identifying gradient misalignment as a key source of off\-manifold drift and proposing a lightweight correction mechanism, our method enables more faithful and stable generation without retraining large pretrained models\. These advances support safer and more robust deployment of generative models in applications such as planning, control, and interactive content generation, where adherence to heterogeneous constraints is critical\.

## References

## Appendix AExtended related works

![Refer to caption](https://arxiv.org/html/2605.20758v1/x7.png)Figure 7:Inference\-time guidance methods arranged by computational cost\. Approximate guidance methods \(e\.g\.,gcov\-Gg^\{\\text\{cov\-G\}\}\) are lightweight but accumulate local approximation error, leading to off\-manifold drift\. Exact guidance methods \(e\.g\., Guidance Matching, sample\-based guidance\) are exact but require substantially more compute\. This work \(gcarg^\{\\text\{car\}\}\), which sits in the middle, aims to improve the compute\-light approximate guidance by adding extra compute\.### A\.1Inference\-time alignment

Inference\-time alignment methods for flow matching models refer to steering the generated samples toward some constraints, e\.g\., sampling from a distribution weighted with some objective function\(lu2023contrastive\)or conditioned on class labels\(song2021scorebased\), and all constraints can be framed as reward functionsr:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}\. Approaches to inference\-time reward alignment for flow models can be divided into three broad paradigms:

##### Inference\-time guidance\.

Inference\-time guidance addresses the reward\-tilted sampling problem by adding a guidance vector fieldgt​\(xt\)g\_\{t\}\(x\_\{t\}\)to the pretrained velocityvθ​\(xt,t\)v\_\{\\theta\}\(x\_\{t\},t\)during ODE integration, leaving the pretrained model unchanged\. A unified theoretical framework for this family was established byfeng2025guidance, who derive the exact guidance vector field for general flow matching:

gt​\(xt\)=𝔼z∼p​\(z\|xt\)​\[\(er​\(x1\)Zt​\(xt\)−1\)​vt\|z​\(xt\|z\)\],g\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{z\\sim p\(z\|x\_\{t\}\)\}\\\!\\left\[\\\!\\left\(\\tfrac\{e^\{r\(x\_\{1\}\)\}\}\{Z\_\{t\}\(x\_\{t\}\)\}\-1\\right\)v\_\{t\|z\}\(x\_\{t\}\|z\)\\right\],whereZt​\(xt\)=𝔼z∼p​\(z\|xt\)​\[er​\(x1\)\]Z\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{z\\sim p\(z\|x\_\{t\}\)\}\[e^\{r\(x\_\{1\}\)\}\]is an intractable normalising constant\. Methods in this family differ in how they approximategtg\_\{t\}\.

*Approximate guidance*replaces the intractable posterior average with a point estimate via Tweedie’s formula, adapting well\-studied diffusion guidance methods including DPS\(chung2023diffusion\),Π\\PiGDM\(song2023pseudoinverse\), and LGD\(song2023loss\)to the flow matching setting;feng2025guidanceunify these under the flow\-matching extensiongcov\-Gg^\{\\text\{cov\-G\}\}\. These methods are computationally lightweight but incur an approximation error, as we show in Section[4](https://arxiv.org/html/2605.20758#S4); see alsofeng2025guidance\.

*Exact guidance*methods avoid this bias at the cost of additional computation\. On the training\-free side, Monte Carlo guidance \(gMCg^\{\\text\{MC\}\},feng2025guidance\) estimatesgtg\_\{t\}by drawingNNsamples from the priorp​\(z\)p\(z\)and self\-normalising; it is asymptotically exact but suffers from high variance, especially in high\-dimensional spaces\. GLASS\-FKS\(holderrieth2026glass\), which steers GLASS flows via Feynman\-Kac sampling, improves sampling efficiency but still inherits the high variance\. On the training\-based side, Guidance Matching\(feng2025guidance\)learns a networkgψg\_\{\\psi\}to directly approximategtg\_\{t\}via tractable surrogate losses, which however require ground\-truth samples sastify all constraints\.

##### Optimization\-based controlled generation\.

A second paradigm frames controlled generation as a direct optimization problem: given a differentiable objective \(cost or reward\), one searches for an initial noise, latent trajectory, or auxiliary variable that, after running the generative ODE/SDE, produces a sample of high reward\. Representative methods*differentiate through*the entire sampling ODE to back\-propagate reward gradients into the input space, including D\-Flow\(benhamu2024dflow\), FlowGrad\(liu2023flowgrad\), and source\-guided flow matching\(wang2025source\)\. These methods pursue a fundamentally different objective from the guidance framework we adopt: rather than sampling from the reward\-tilted distributionp1′​\(x1\)∝p1base​\(x1\)​er​\(x1\)p^\{\\prime\}\_\{1\}\(x\_\{1\}\)\\propto p^\{\\text\{base\}\}\_\{1\}\(x\_\{1\}\)\\,e^\{r\(x\_\{1\}\)\}, they solve an optimization problem\.

##### Reward fine\-tuning\.

A third paradigm modifies the pretrained model weights to maximize the reward, based on GRPO\(liu2025flowgrpo\), stochastic optimal control\(domingo2024adjoint\), DPO\(wallace2024diffusion\), or other reinforcement learning approaches\. They differ in how this optimization problem is solved; for example, VGG\-Flow\(liu2025value\)fine\-tunes the velocity field via a reward\-importance\-weighted flow matching loss, whereas Adjoint Matching\(domingo2024adjoint\)back\-propagates through the entire ODE trajectory using the continuous adjoint equations\. Many fine\-tuning methods require DDPM/SDE sampling for exploration during training\(liu2025flowgrpo;domingo2024adjoint\), which is significantly less efficient than ODE sampling and couples the method to a specific reward at training time; adapting to a new reward requires retraining from scratch\. We instead focus on exploring how to best leverage the pretrained flow model at inference time, without any fine\-tuning\.

### A\.2Value gradient guidance

A related line of work defines the guidance signal as the gradient of a learned value functiong​\(xt,t\)≜∇xtV​\(xt,t\)g\(x\_\{t\},t\)\\triangleq\\nabla\_\{x\_\{t\}\}V\(x\_\{t\},t\), withV​\(xt,t\)≈𝔼​\[r​\(x1\)∣xt\]V\(x\_\{t\},t\)\\approx\\mathbb\{E\}\[r\(x\_\{1\}\)\\mid x\_\{t\}\]\. VGG\-Flow\(liu2025value\)instantiates this idea by co\-training a value\-gradient network with the fine\-tuned velocity via an HJB consistency loss\. As a fine\-tuning method, however, VGG\-Flow couples the model to a single fixed reward at training time and targets a different problem from the one we tackle\. In contrast,gcarg^\{\\text\{car\}\}focuses on off\-manifold drift at inference time, redirecting trajectories back onto the data manifold via value\-gradient guidance without modifying pretrained weights, and can be applied on top of any approximate guidance\.

## Appendix BGeometric interpretation: “energy trap” under gradient misalignment

Before the formal analysis in Appendix[C](https://arxiv.org/html/2605.20758#A3), we provide a geometric interpretation of why gradient misalignment creates an “energy trap” in the guidance field\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x8.png)Figure 8:Spurious local minimum from gradient misalignment\.\(a, b\)Individual energy landscapes for two multi\-modal reward functions,Ej=−rjE\_\{j\}=\-r\_\{j\}, each with two global minima \(stars\)\. One mode at\(8,−8\)\(8,\-8\)is shared, i\.e\., thex1⋆x\_\{1\}^\{\\star\}\.\(c\)The compositional energy landscapeE=E1\+E2=−\(r1\+r2\)E=E\_\{1\}\+E\_\{2\}=\-\(r\_\{1\}\+r\_\{2\}\)has a*spurious local minimum*x†x^\{\\dagger\}\(top\):x†≠x1⋆x^\{\\dagger\}\\neq x\_\{1\}^\{\\star\}, andx†x^\{\\dagger\}maximizes neither any individual rewardrjr\_\{j\}nor their sum\.\(d\)The spurious local minimum coincides with the region of maximum gradient conflict \(dark red, where∇E1≈−∇E2\\nabla E\_\{1\}\\approx\-\\nabla E\_\{2\}\), where energy dissipation traps nearby trajectories rather than steering them tox1⋆x\_\{1\}^\{\\star\}\.##### Global optimum and spurious local minimum\.

Consider a compositional reward problem withGGreward functions\{rj\}j=1G\\\{r\_\{j\}\\\}\_\{j=1\}^\{G\}\. Theglobal optimumis expected to maximize all rewards:

x1⋆=arg⁡maxx1​∑j=1Grj​\(x1\)\.x\_\{1\}^\{\\star\}\\;=\\;\\arg\\max\_\{x\_\{1\}\}\\sum\_\{j=1\}^\{G\}r\_\{j\}\(x\_\{1\}\)\.\(20\)
From the energy\-guided sampling perspective, the compositional energy landscape on the predicted terminal statex^1=𝔼​\[x1∣xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\\mid x\_\{t\}\]isE​\(x^1\)≜−∑j=1Grj​\(x^1\)E\(\\hat\{x\}\_\{1\}\)\\triangleq\-\\sum\_\{j=1\}^\{G\}r\_\{j\}\(\\hat\{x\}\_\{1\}\), with per\-reward guidancegj​\(xt\)≜∇xtrj​\(x^1\)g\_\{j\}\(x\_\{t\}\)\\triangleq\\nabla\_\{x\_\{t\}\}r\_\{j\}\(\\hat\{x\}\_\{1\}\)and compositional guidancegt​\(xt\)≜∑j=1Ggj​\(xt\)g\_\{t\}\(x\_\{t\}\)\\triangleq\\sum\_\{j=1\}^\{G\}g\_\{j\}\(x\_\{t\}\)\. The guided trajectory evolves asx˙t=vtbase​\(xt\)\+gt​\(xt\)\\dot\{x\}\_\{t\}=v\_\{t\}^\{\\text\{base\}\}\(x\_\{t\}\)\+g\_\{t\}\(x\_\{t\}\), and the sampler’s terminal states lie in the set of stable equilibria ofEE,

𝒮≜\{x:∇E​\(x\)=0,Hess​E​\(x\)≻0\}\.\\mathcal\{S\}\\;\\triangleq\\;\\Big\\\{x:\\nabla E\(x\)=0,\\;\\;\\mathrm\{Hess\}\\,E\(x\)\\succ 0\\Big\\\}\.\(21\)By construction,x1⋆∈𝒮x\_\{1\}^\{\\star\}\\in\\mathcal\{S\}: the global optimum is a stable equilibrium\. In general, however,\{x1⋆\}⊊𝒮\\\{x\_\{1\}^\{\\star\}\\\}\\subsetneq\\mathcal\{S\}, formally:

###### Definition B\.1\(Spurious local minimum\)\.

A pointx†x^\{\\dagger\}is a*spurious local minimum*of the compositional energyEEif it is a stable equilibrium that is*not*the global optimum:

x†∈𝒮∖\{x1⋆\},i\.e\.,∇E​\(x†\)=0,Hess​E​\(x†\)≻0,and​x†≠x1⋆\.x^\{\\dagger\}\\;\\in\\;\\mathcal\{S\}\\setminus\\\{x\_\{1\}^\{\\star\}\\\},\\quad\\text\{i\.e\.,\}\\quad\\nabla E\(x^\{\\dagger\}\)=0,\\;\\;\\mathrm\{Hess\}\\,E\(x^\{\\dagger\}\)\\succ 0,\\;\\;\\text\{and\}\\;\\;x^\{\\dagger\}\\neq x\_\{1\}^\{\\star\}\.\(22\)By construction, the compositional guidance vanishes atx†x^\{\\dagger\}\(∑j=1Ggj​\(x†\)=0\\sum\_\{j=1\}^\{G\}g\_\{j\}\(x^\{\\dagger\}\)=0\), butx†x^\{\\dagger\}maximizes neither any individual rewardrjr\_\{j\}nor their sum; the vanishing arises through*destructive interference between non\-zero reward gradients*\(yu2020pcgrad\)rather than through reward maximization\.222From a probabilistic perspective, the log\-density landscape∑jrj\\sum\_\{j\}r\_\{j\}corresponds to a Product of Experts \(PoE\) formulation\. A well\-documented theoretical pathology of PoE and energy\-based models is their propensity to generate*spurious modes*— unintended attractors that emerge between the unaligned peaks of the constituent distributions\. In the optimization literature, these are formally referred to as*spurious local minima*\.

As the trajectory approaches the basin of anyx†x^\{\\dagger\}, it drifts off\-manifold\.

##### Energy dissipation under gradient misalignment\.

To characterize the mechanism that drives trajectories off\-manifold, we quantify the effective driving force of the compositional guidance via its squared norm\. Expanding‖gt​\(xt\)‖2\\\|g\_\{t\}\(x\_\{t\}\)\\\|^\{2\}at any statextx\_\{t\}:

‖∑j=1Ggj​\(xt\)‖2=∑j‖gj​\(xt\)‖2⏟self\-energy\+2​∑j<k‖gj​\(xt\)‖​‖gk​\(xt\)‖​cos⁡ϕj​k​\(xt\)⏟cross\-energy,\\Big\\\|\\sum\_\{j=1\}^\{G\}g\_\{j\}\(x\_\{t\}\)\\Big\\\|^\{2\}\\;=\\;\\underbrace\{\\sum\_\{j\}\\\|g\_\{j\}\(x\_\{t\}\)\\\|^\{2\}\}\_\{\\text\{self\-energy\}\}\\;\+\\;\\underbrace\{2\\sum\_\{j<k\}\\\|g\_\{j\}\(x\_\{t\}\)\\\|\\,\\\|g\_\{k\}\(x\_\{t\}\)\\\|\\,\\cos\\phi\_\{jk\}\(x\_\{t\}\)\}\_\{\\text\{cross\-energy\}\},\(23\)whereϕj​k​\(xt\)\\phi\_\{jk\}\(x\_\{t\}\)denotes the angle betweengj​\(xt\)g\_\{j\}\(x\_\{t\}\)andgk​\(xt\)g\_\{k\}\(x\_\{t\}\)\. By the triangle inequality, the maximum compositional guidance is\(∑j‖gj​\(xt\)‖\)2\\big\(\\sum\_\{j\}\\\|g\_\{j\}\(x\_\{t\}\)\\\|\\big\)^\{2\}, realized if and only if all gradients are perfectly collinear \(cos⁡ϕj​k​\(xt\)=1\\cos\\phi\_\{jk\}\(x\_\{t\}\)=1for allj<kj<k\)\. We define the deficit between this collinear capacity and the realized compositional guidance as the energy dissipation:

###### Definition B\.2\(Energy dissipation under gradient misalignment\)\.

For any statextx\_\{t\}, the*energy dissipation*of the compositional guidance is

Δ​E​\(xt\)≜\(∑j‖gj​\(xt\)‖\)2−‖∑jgj​\(xt\)‖2=2​∑j<k‖gj​\(xt\)‖​‖gk​\(xt\)‖​\(1−cos⁡ϕj​k​\(xt\)\)≥0\.\\Delta E\(x\_\{t\}\)\\;\\triangleq\\;\\Big\(\\sum\_\{j\}\\\|g\_\{j\}\(x\_\{t\}\)\\\|\\Big\)^\{2\}\\;\-\\;\\Big\\\|\\sum\_\{j\}g\_\{j\}\(x\_\{t\}\)\\Big\\\|^\{2\}\\;=\\;2\\sum\_\{j<k\}\\\|g\_\{j\}\(x\_\{t\}\)\\\|\\,\\\|g\_\{k\}\(x\_\{t\}\)\\\|\\,\\big\(1\-\\cos\\phi\_\{jk\}\(x\_\{t\}\)\\big\)\\;\\geq\\;0\.\(24\)

Δ​E​\(xt\)≥0\\Delta E\(x\_\{t\}\)\\geq 0, with equality if and only if all reward gradients are perfectly aligned atxtx\_\{t\}\(cos⁡ϕj​k​\(xt\)≡1\\cos\\phi\_\{jk\}\(x\_\{t\}\)\\equiv 1\)\. As pairwise gradient misalignment grows, the compositional guidance energy structurally dissipates: the trajectory loses its driving force and becomes trapped at a spurious local minimum of the energy landscape \(as visualised in Figure[9](https://arxiv.org/html/2605.20758#A2.F9)\(d\)\)\.

At the terminal time step, the trajectory enters the basin of a spurious local minimumx†∈𝒮∖\{x1⋆\}x^\{\\dagger\}\\in\\mathcal\{S\}\\setminus\\\{x\_\{1\}^\{\\star\}\\\}, where∑jgj​\(x†\)=0\\sum\_\{j\}g\_\{j\}\(x^\{\\dagger\}\)=0and the dissipated energy isΔ​E​\(x†\)=2​∑j<k‖gj​\(x†\)‖​‖gk​\(x†\)‖​\(1−cos⁡ϕj​k​\(x†\)\)\\Delta E\(x^\{\\dagger\}\)=2\\sum\_\{j<k\}\\\|g\_\{j\}\(x^\{\\dagger\}\)\\\|\\,\\\|g\_\{k\}\(x^\{\\dagger\}\)\\\|\\,\\big\(1\-\\cos\\phi\_\{jk\}\(x^\{\\dagger\}\)\\big\)\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x9.png)Figure 9:Energy dissipation under gradient misalignment\.\(a\)Whenϕj​k=0∘\\phi\_\{jk\}=0^\{\\circ\}, reward gradients are perfectly collinear,Δ​E​\(xt\)=0\\Delta E\(x\_\{t\}\)=0, and no correction is needed\.\(b\)When0∘<ϕj​k<90∘0^\{\\circ\}<\\phi\_\{jk\}<90^\{\\circ\}, gradients are misaligned but remain in the same half\-space\. PCGrad detects no conflict \(cos⁡ϕj​k\>0\\cos\\phi\_\{jk\}\>0\) and takes no action, yetΔ​E​\(xt\)\>0\\Delta E\(x\_\{t\}\)\>0; ourgcarg^\{\\text\{car\}\}identifies this misalignment and corrects it\.\(c\)Whenϕj​k\>90∘\\phi\_\{jk\}\>90^\{\\circ\}, gradients undergo destructive interference\. PCGrad intervenes via projection, whereasgcarg^\{\\text\{car\}\}corrects the trajectory via learned residual guidance\.\(d\)Energy dissipationΔ​E∝‖gj‖​‖gk‖​\(1−cos⁡ϕj​k\)\\Delta E\\propto\\\|g\_\{j\}\\\|\\\|g\_\{k\}\\\|\(1\-\\cos\\phi\_\{jk\}\)scales with the gradient angle\.
##### PCGrad and its structural limitations\.

PCGrad\(yu2020pcgrad\)is a widely adopted gradient\-conflict resolution method for multi\-objective optimisation, but it addresses only destructive interference \(cos⁡ϕj​k<0\\cos\\phi\_\{jk\}<0\) and resolves it via gradient surgery, projecting the conflicting component of each gradient onto the normal plane of the other\. In standard multi\-task optimisation, with smooth losses and thousands of accumulated optimiser steps, a transiently weakened gradient update is easily recovered in subsequent iterations, and the composite gradient reliably descends the loss landscape\.

In inference\-time guided sampling, however, this logic breaks down\. The guidance signalgt​\(xt\)g\_\{t\}\(x\_\{t\}\)must steer the trajectory towardx1x\_\{1\}at every timestep, and there are no future updates at a fixed state to recover the dissipated guidance energy\. Whenever pairwise misalignment1−cos⁡ϕj​k\>01\-\\cos\\phi\_\{jk\}\>0, trajectory loses driving force and drifts toward a spurious equilibriumx†x^\{\\dagger\}\.

## Appendix CGuided sampling and approximation error

In this section, we analyze approximation errors utilizing the optimal coupling formulation and provide a detailed proof of the approximation error bound\.

### C\.1Optimal coupling and triad decomposition

Optimal guided sampling modifies the generation process by reweighting the latent couplingπ​\(z\)\\pi\(z\)to target a tilted distributionp1⋆​\(x1\)∝p1​\(x1\)​er​\(x1\)p\_\{1\}^\{\\star\}\(x\_\{1\}\)\\propto p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\. In the context of the transport problem, a change in the target marginalp1⋆​\(x1\)p\_\{1\}^\{\\star\}\(x\_\{1\}\)implies a modification of the optimal transport plan\. To quantify this discrepancy, we consider the measuresπ\\piandπ⋆\\pi^\{\\star\}on the latent space\. Assuming absolute continuity of the tilted couplingπ⋆\(⋅∣x1\)\\pi^\{\\star\}\(\\cdot\\mid x\_\{1\}\)with respect to the base couplingπ\(⋅∣x1\)\\pi\(\\cdot\\mid x\_\{1\}\)\(i\.e\.,π⋆≪π\\pi^\{\\star\}\\ll\\pi\), the Radon\-Nikodym derivative exists and we term this derivative the coupling shift:

𝒫​\(z\)≜dπ⋆\(⋅∣x1\)dπ\(⋅∣x1\)​\(z\)\.\\mathcal\{P\}\(z\)\\triangleq\\frac\{d\\pi^\{\\star\}\(\\cdot\\mid x\_\{1\}\)\}\{d\\pi\(\\cdot\\mid x\_\{1\}\)\}\(z\)\.\(25\)Formally, conditioned on a fixed terminal statex1x\_\{1\},𝒫​\(z\)\\mathcal\{P\}\(z\)measures the relative density shift between the conditional distribution of the true optimal transport planπ⋆\(⋅∣x1\)\\pi^\{\\star\}\(\\cdot\\mid x\_\{1\}\)and the pre\-trained base couplingπ\(⋅∣x1\)\\pi\(\\cdot\\mid x\_\{1\}\)\. If we were to simply perform Bayesian reweighting \(as in standard classifier guidance\),𝒫​\(z\)≡1\\mathcal\{P\}\(z\)\\equiv 1\. However, enforcing optimality in the transport cost introduces a shift𝒫​\(z\)≠1\\mathcal\{P\}\(z\)\\neq 1\.

Recall the definition of the conditional probabilityπ​\(z\)=π​\(x0∣x1\)​p1​\(x1\)\\pi\(z\)=\\pi\(x\_\{0\}\\mid x\_\{1\}\)p\_\{1\}\(x\_\{1\}\)\. Substituting the target marginalp1⋆​\(x1\)∝p1​\(x1\)​er​\(x1\)p\_\{1\}^\{\\star\}\(x\_\{1\}\)\\propto p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}and the coupling shift𝒫​\(z\)=π⋆​\(x0∣x1\)π​\(x0∣x1\)\\mathcal\{P\}\(z\)=\\frac\{\\pi^\{\\star\}\(x\_\{0\}\\mid x\_\{1\}\)\}\{\\pi\(x\_\{0\}\\mid x\_\{1\}\)\}, we derive the triad decomposition \(Equation \([10](https://arxiv.org/html/2605.20758#S4.E10)\) in main text\) as follows:

π⋆​\(z\)\\displaystyle\\pi^\{\\star\}\(z\)=π⋆​\(x0∣x1\)​p1⋆​\(x1\)\\displaystyle=\\pi^\{\\star\}\(x\_\{0\}\\mid x\_\{1\}\)\\,p\_\{1\}^\{\\star\}\(x\_\{1\}\)=\[𝒫​\(z\)⋅π​\(x0∣x1\)\]⋅\[1𝒵⋆​p1​\(x1\)​er​\(x1\)\]\\displaystyle=\\left\[\\mathcal\{P\}\(z\)\\cdot\\pi\(x\_\{0\}\\mid x\_\{1\}\)\\right\]\\cdot\\left\[\\frac\{1\}\{\\mathcal\{Z\}^\{\\star\}\}p\_\{1\}\(x\_\{1\}\)e^\{r\(x\_\{1\}\)\}\\right\]=1𝒵⋆⋅𝒫​\(z\)⏟Coupling⋅eR​\(z\)⏟Reward⋅π​\(z\)⏟Base Prior,\\displaystyle=\\frac\{1\}\{\\mathcal\{Z\}^\{\\star\}\}\\cdot\\underbrace\{\\mathcal\{P\}\(z\)\}\_\{\\text\{\{\\color\[rgb\]\{0\.2421875,0\.30859375,0\.58203125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.2421875,0\.30859375,0\.58203125\}Coupling\}\}\}\\cdot\\underbrace\{e^\{R\(z\)\}\}\_\{\\text\{\{\\color\[rgb\]\{0\.40625,0\.140625,0\.53125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.40625,0\.140625,0\.53125\}Reward\}\}\}\\cdot\\underbrace\{\\pi\(z\)\}\_\{\\text\{\{\\color\[rgb\]\{0\.08203125,0\.390625,0\.203125\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.08203125,0\.390625,0\.203125\}Base Prior\}\}\},\(26\)whereR​\(z\)≜r​\(Ψ1​\(z\)\)R\(z\)\\triangleq r\(\\Psi\_\{1\}\(z\)\)denotes the trajectory\-level reward, andπ​\(z\)=π​\(x0∣x1\)​p1​\(x1\)\\pi\(z\)=\\pi\(x\_\{0\}\\mid x\_\{1\}\)p\_\{1\}\(x\_\{1\}\)\. Note that, in a guided transport problem, the source distribution must remain anchored to the pre\-defined prior \(e\.g\., standard Gaussian\) to ensure tractable inference; Thus,𝒫​\(z\)\\mathcal\{P\}\(z\)acts as a structural correction term: it represents the necessary re\-organization of the transport plan, specifically the shift in the conditionalπ​\(x0∣x1\)\\pi\(x\_\{0\}\\mid x\_\{1\}\), necessary to satisfy the new target boundaryp1⋆p\_\{1\}^\{\\star\}while simultaneously preserving the fixed source marginalp0p\_\{0\}\.

##### Two\-stage approximation\.

As outlined in the main text, practical guided sampling simplifies Equation \([26](https://arxiv.org/html/2605.20758#A3.E26)\) via two approximations\.

π⋆→𝒫​\(z\)≈1Coupling\-InvariantπCI→V^​\(xt\)Local Approx\.πapprox\.\\pi^\{\\star\}\\xrightarrow\[\\mathcal\{P\}\(z\)\\approx 1\]\{\\text\{Coupling\-Invariant\}\}\\pi^\{\\mathrm\{CI\}\}\\xrightarrow\[\\hat\{V\}\(x\_\{t\}\)\]\{\\text\{Local Approx\.\}\}\\pi^\{\\text\{approx\}\}\.First, we have theCoupling\-Invariant Approximation \(CIA\), which assumes the conditional transportπ​\(x0\|x1\)\\pi\(x\_\{0\}\|x\_\{1\}\)remains unchanged, i\.e\.,𝒫​\(z\)≡1\\mathcal\{P\}\(z\)\\equiv 1:

πCI​\(z\)=eR​\(z\)𝒵CI​π​\(z\),𝒵CI=𝔼π​\(z\)​\[eR​\(z\)\]\.\\pi^\{\\mathrm\{CI\}\}\(z\)=\\frac\{e^\{R\(z\)\}\}\{\\mathcal\{Z\}^\{\\mathrm\{CI\}\}\}\\,\\pi\(z\),\\qquad\\mathcal\{Z\}^\{\\mathrm\{CI\}\}=\\mathbb\{E\}\_\{\\pi\(z\)\}\[e^\{R\(z\)\}\]\.\(27\)The coupling shift𝒫​\(z\)\\mathcal\{P\}\(z\)is required to anchor the transport to the fixed priorp0p\_\{0\}\. By assuming𝒫​\(z\)≡1\\mathcal\{P\}\(z\)\\equiv 1, the Coupling\-Invariant Approximation theoretically shifts the optimal source to a reweighted densityp0bias​\(x0\)∝p0​\(x0\)​𝔼​\[er​\(x1\)\|x0\]p\_\{0\}^\{\\text\{bias\}\}\(x\_\{0\}\)\\propto p\_\{0\}\(x\_\{0\}\)\\mathbb\{E\}\[e^\{r\(x\_\{1\}\)\}\|x\_\{0\}\]\. Since inference restricts sampling top0p\_\{0\}rather thanp0biasp\_\{0\}^\{\\text\{bias\}\}, a boundary mismatch arises\. This discrepancy leads to off\-manifold drift and error accumulation\.

Second, to make the guidance realizable at any time steptt, we apply aLocalized Approximation\. We approximate the trajectory rewardV​\(z\)V\(z\)using a first\-order Taylor expansion around the expected future statex^1=𝔼​\[x1\|xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\|x\_\{t\}\]:V^​\(z\)≈V​\(xt\)\+∇V​\(xt\)⊤​\(x1−x^1\)\\hat\{V\}\(z\)\\approx V\(x\_\{t\}\)\+\\nabla V\(x\_\{t\}\)^\{\\top\}\(x\_\{1\}\-\\hat\{x\}\_\{1\}\)\. Since the constant terms cancel out during normalization, this results in the realizable coupling measure driven by the gradient:

πapprox​\(z\)=eV^​\(z\)𝒵^​π​\(z\),𝒵^=𝔼π​\(z\)​\[eV^​\(z\)\]\.\\pi^\{\\text\{approx\}\}\(z\)=\\frac\{e^\{\\hat\{V\}\(z\)\}\}\{\\hat\{\\mathcal\{Z\}\}\}\\,\\pi\(z\),\\qquad\\hat\{\\mathcal\{Z\}\}=\\mathbb\{E\}\_\{\\pi\(z\)\}\[e^\{\\hat\{V\}\(z\)\}\]\.\(28\)The effective guidance is thus determined solely by the gradient direction∇V​\(xt\)\\nabla V\(x\_\{t\}\)\.333Formally, letCt≜V​\(xt\)−∇V​\(xt\)⊤​x^1C\_\{t\}\\triangleq V\(x\_\{t\}\)\-\\nabla V\(x\_\{t\}\)^\{\\top\}\\hat\{x\}\_\{1\}denote the terms constant with respect tozz\. The normalization implies:πapprox​\(z∣xt\)=eCt\+∇V​\(xt\)⊤​x1​π​\(z∣xt\)∫eCt\+∇V​\(xt\)⊤​x1​π​\(z∣xt\)​𝑑z=eCt​e∇V​\(xt\)⊤​x1​π​\(z∣xt\)eCt​∫e∇V​\(xt\)⊤​x1​π​\(z∣xt\)​𝑑z=eCteCt​e∇V​\(xt\)⊤​x1𝒵^​π​\(z∣xt\)\.\\pi^\{\\text\{approx\}\}\(z\\mid x\_\{t\}\)=\\frac\{e^\{C\_\{t\}\+\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}\}\\pi\(z\\mid x\_\{t\}\)\}\{\\int e^\{C\_\{t\}\+\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}\}\\pi\(z\\mid x\_\{t\}\)\\,dz\}=\\frac\{e^\{C\_\{t\}\}e^\{\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}\}\\pi\(z\\mid x\_\{t\}\)\}\{e^\{C\_\{t\}\}\\int e^\{\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}\}\\pi\(z\\mid x\_\{t\}\)\\,dz\}=\\frac\{\\cancel\{e^\{C\_\{t\}\}\}\}\{\\cancel\{e^\{C\_\{t\}\}\}\}\\frac\{e^\{\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}\}\}\{\\hat\{\\mathcal\{Z\}\}\}\\pi\(z\\mid x\_\{t\}\)\.This derivation shows that the guidance is driven purely by the gradient component∇V​\(xt\)⊤​x1\\nabla V\(x\_\{t\}\)^\{\\top\}x\_\{1\}, rendering the absolute magnitude of the valueV​\(xt\)V\(x\_\{t\}\)irrelevant\.

Equation \([26](https://arxiv.org/html/2605.20758#A3.E26)\) explicitly demonstrates that the optimal coupling is governed by a triad of factors: the structural coupling shift \(𝒫​\(z\)\\mathcal\{P\}\(z\)\), the trajectory reward \(eR​\(z\)e^\{R\(z\)\}\), and the base prior \(π​\(z\)\\pi\(z\)\)\. We formally derive the upper bound of the approximation error shortly\. Before doing so, we revisit the two\-stage approximation to clarify how it specifically targets these components: the coupling shift \(𝒫\\mathcal\{P\}\) is neglected via the Coupling\-Invariant assumption, and the trajectory reward \(eR​\(z\)e^\{R\(z\)\}\) is estimated via the Localized Approximation\.

### C\.2Approximation error via the Benamou–Brenier theorem

Flow models construct probabilistic transport plans that move mass from a source measurep0p\_\{0\}to a target measurep1p\_\{1\}, and the squared Wasserstein distanceW22​\(p0,p1\)W\_\{2\}^\{2\}\(p\_\{0\},p\_\{1\}\)quantifies the minimal kinetic energy required for this transport\. By the Benamou–Brenier theorem:

W22​\(p0,p1⋆\)=inf\(pt,vt\):p0→vtp1⋆∫01𝔼xt∼pt​\[‖vt​\(xt\)‖2\]​𝑑t=∫01𝔼​\[‖vtbase\+gt⋆‖2\]​𝑑t,W\_\{2\}^\{2\}\(p\_\{0\},p\_\{1\}^\{\\star\}\)=\\inf\_\{\(p\_\{t\},\\,v\_\{t\}\):\\,p\_\{0\}\\xrightarrow\{v\_\{t\}\}p\_\{1\}^\{\\star\}\}\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}\}\\\!\\big\[\\\|v\_\{t\}\(x\_\{t\}\)\\\|^\{2\}\\big\]\\,dt=\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\\|v\_\{t\}^\{\\mathrm\{base\}\}\+g\_\{t\}^\{\\star\}\\\|^\{2\}\\right\]dt,\(29\)wheregt⋆g\_\{t\}^\{\\star\}is the optimal guidance field that steers mass toward the reward\-tilted targetp1⋆p\_\{1\}^\{\\star\}\. Under the two\-stage approximation \(CIA \+ Localized Approximation\) for compositional rewardsR=∑jrjR=\\sum\_\{j\}r\_\{j\},gt⋆g\_\{t\}^\{\\star\}is replaced by the realized guidancegtapprox=∑jgjapproxg\_\{t\}^\{\\mathrm\{approx\}\}=\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{approx\}\}, where eachgjapprox=∇xtrj​\(x^1\)g\_\{j\}^\{\\mathrm\{approx\}\}=\\nabla\_\{x\_\{t\}\}r\_\{j\}\(\\hat\{x\}\_\{1\}\)withx^1=𝔼​\[x1∣xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\\mid x\_\{t\}\]\. The realized fieldv^t=vtbase\+gtapprox\\hat\{v\}\_\{t\}=v\_\{t\}^\{\\mathrm\{base\}\}\+g\_\{t\}^\{\\mathrm\{approx\}\}therefore only transportsp0p\_\{0\}top^1≠p1⋆\\hat\{p\}\_\{1\}\\neq p\_\{1\}^\{\\star\}\.

We quantify the resulting approximation errorℰ≜W22​\(p^1,p1⋆\)\\mathcal\{E\}\\triangleq W\_\{2\}^\{2\}\(\\hat\{p\}\_\{1\},p\_\{1\}^\{\\star\}\)using the stability of the continuity equation\(villani2009optimal\), which bounds the terminal distributional discrepancy by the time\-integrated squared velocity field difference along the optimal pathpt⋆p\_\{t\}^\{\\star\}:

ℰ≜W22​\(p^1,p1⋆\)≤∫01𝔼xt∼pt⋆​\[‖vt⋆​\(xt\)−v^t​\(xt\)‖2\]​𝑑t=∫01𝔼xt∼pt⋆​\[‖gt⋆−gtapprox‖2\]​𝑑t\.\\mathcal\{E\}\\triangleq W\_\{2\}^\{2\}\(\\hat\{p\}\_\{1\},p\_\{1\}^\{\\star\}\)\\;\\leq\\;\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\left\[\\\|v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\hat\{v\}\_\{t\}\(x\_\{t\}\)\\\|^\{2\}\\right\]dt=\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\left\[\\\|g\_\{t\}^\{\\star\}\-g\_\{t\}^\{\\mathrm\{approx\}\}\\\|^\{2\}\\right\]dt\.\(30\)
Since compositional guided sampling sums per\-reward gradients directly,gtapprox=∑jgjapproxg\_\{t\}^\{\\mathrm\{approx\}\}=\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{approx\}\}, the∑jgjCI\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\(distinct fromgtCIg\_\{t\}^\{\\mathrm\{CI\}\}forRR\) sits naturally betweengtCIg\_\{t\}^\{\\mathrm\{CI\}\}andgtapproxg\_\{t\}^\{\\mathrm\{approx\}\}\. Young’s inequality‖a\+b‖2≤2​‖a‖2\+2​‖b‖2\\\|a\+b\\\|^\{2\}\\leq 2\\\|a\\\|^\{2\}\+2\\\|b\\\|^\{2\}along the chain

gt⋆→gtCI→∑jgjCI→gtapprox,g\_\{t\}^\{\\star\}\\;\\to\\;g\_\{t\}^\{\\mathrm\{CI\}\}\\;\\to\\;\\textstyle\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\;\\to\\;g\_\{t\}^\{\\mathrm\{approx\}\},yields three terms: the CIA step \(first arrow\), the Localized step \(third\), and an analytical decomposition \(middle\) specific to the compositional setting:

ℰ\\displaystyle\\mathcal\{E\}≤∫01𝔼xt∼pt⋆​\[‖gt⋆−gtapprox‖2\]​𝑑t\\displaystyle\\leq\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\left\[\\\|g\_\{t\}^\{\\star\}\-g\_\{t\}^\{\\mathrm\{approx\}\}\\\|^\{2\}\\right\]dt≤∫01𝔼xt∼pt⋆​\[2​‖gt⋆−gtCI‖2\+2​‖gtCI−∑jgjCI‖2\+2​‖∑jgjCI−gtapprox‖2\]​𝑑t\\displaystyle\\leq\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\Bigg\[2\\\|g\_\{t\}^\{\\star\}\-g\_\{t\}^\{\\mathrm\{CI\}\}\\\|^\{2\}\+2\\left\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\-\\textstyle\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\right\\\|^\{2\}\+2\\left\\\|\\textstyle\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\-g\_\{t\}^\{\\mathrm\{approx\}\}\\right\\\|^\{2\}\\Bigg\]dt≲CCI​∫01𝔼​\[𝔼πCI​\[\(𝒫​\(z\)−1\)2\]\]​𝑑t⏟\(A\) Coupling shift error\+𝒦deficit⏟\(B\) gradient misalignment error\+G​∫01𝔼​\[\(λh​σ1​der​\(x^1\)\)2​\(C1\+C2\)\]​𝑑t⏟\(C\) Localized approximation error\(feng2025guidance\), scaled by​G\.\\displaystyle\\lesssim\\underbrace\{C\_\{\\mathrm\{CI\}\}\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{CI\}\}\}\\\!\\big\[\(\\mathcal\{P\}\(z\)\-1\)^\{2\}\\big\]\\right\]dt\}\_\{\\text\{\(A\) Coupling shift error\}\}\+\\underbrace\{\\mathcal\{K\}\_\{\\mathrm\{deficit\}\}\}\_\{\\text\{\(B\) gradient misalignment error\}\}\+\\underbrace\{G\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\left\(\\frac\{\\lambda\_\{h\}\\sigma\_\{1\}d\}\{e^\{r\(\\hat\{x\}\_\{1\}\)\}\}\\right\)^\{\\\!2\}\(C\_\{1\}\+C\_\{2\}\)\\right\]dt\}\_\{\\text\{\(C\) Localized approximation error~\\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{feng2025guidance\}\{\\@@citephrase\{, \}\}\{\}\)\},\\ scaled by \}G\}\.\(31\)Each term corresponds to one step in the approximation chain\. We detail each component below\.

##### \(A\) Coupling shift error\.

Term \(A\) arises from replacinggt⋆g\_\{t\}^\{\\star\}withgtCIg\_\{t\}^\{\\mathrm\{CI\}\}under the Coupling\-Invariant Approximation,𝒫​\(z\):=dπ⋆\(⋅\|x1\)dπ\(⋅\|x1\)​\(z\)≈1\\mathcal\{P\}\(z\):=\\frac\{d\\pi^\{\\star\}\(\\cdot\|x\_\{1\}\)\}\{d\\pi\(\\cdot\|x\_\{1\}\)\}\(z\)\\approx 1, i\.e\., assuming the conditional transport plan requires no reorganisation when the target shifts fromp1p\_\{1\}top1⋆p\_\{1\}^\{\\star\}\. By Cauchy–Schwarz:

‖gt⋆​\(xt\)−gtCI​\(xt\)‖22\\displaystyle\\\|g\_\{t\}^\{\\star\}\(x\_\{t\}\)\-g\_\{t\}^\{\\mathrm\{CI\}\}\(x\_\{t\}\)\\\|\_\{2\}^\{2\}=∥𝔼z∼πCI\(⋅\|xt\)\[\(𝒫\(z\)−1\)vt\|z\(xt\|z\)\]∥22\\displaystyle=\\left\\\|\\mathbb\{E\}\_\{z\\sim\\pi^\{\\mathrm\{CI\}\}\(\\cdot\|x\_\{t\}\)\}\\big\[\(\\mathcal\{P\}\(z\)\-1\)\\,v\_\{t\|z\}\(x\_\{t\}\|z\)\\big\]\\right\\\|\_\{2\}^\{2\}≤𝔼z∼πCI\(⋅\|xt\)\[\(𝒫\(z\)−1\)2\]⋅𝔼z∼πCI\(⋅\|xt\)\[∥vt\|z\(xt\|z\)∥22\]\.\\displaystyle\\leq\\mathbb\{E\}\_\{z\\sim\\pi^\{\\mathrm\{CI\}\}\(\\cdot\|x\_\{t\}\)\}\\big\[\(\\mathcal\{P\}\(z\)\-1\)^\{2\}\\big\]\\cdot\\mathbb\{E\}\_\{z\\sim\\pi^\{\\mathrm\{CI\}\}\(\\cdot\|x\_\{t\}\)\}\\big\[\\\|v\_\{t\|z\}\(x\_\{t\}\|z\)\\\|\_\{2\}^\{2\}\\big\]\.\(32\)Term \(A\) is small when the coupling shift is negligible \(𝒫​\(z\)≈1\\mathcal\{P\}\(z\)\\approx 1\), which holds for flow matching methods with dependent couplings such as mini\-batch OT\-FM\(tong2023improving\), but not for vanilla OT\-FM\(onken2021ot\)\.

##### \(B\) Gradient misalignment error\.

We analyze Term \(B\) in two cases: \(B1\) WhenG=1G=1orcos⁡ϕj​k=1\\cos\\phi\_\{jk\}=1for all pairs, Term \(B\)=0=0\. \(B2\) WhenG\>1G\>1andcos⁡ϕj​k<1\\cos\\phi\_\{jk\}<1for some pair, Term \(B\)\>0\>0\.

In case \(B2\), we further quantify Term \(B\) by showing how the pointwise energy dissipation accumulates along the sampling trajectory, ultimately trapping it at a spurious local minimum at the terminal time step \(Definition[B\.1](https://arxiv.org/html/2605.20758#A2.Thmtheorem1)\)\.

###### Proposition C\.1\(Trajectory\-level energy dissipation\)\.

ForG≥2G\\geq 2, letcos⁡ϕt​\(xt\):=2G​\(G−1\)​∑j<kcos⁡ϕj​k​\(xt\)\\cos\\phi\_\{t\}\(x\_\{t\}\):=\\frac\{2\}\{G\(G\-1\)\}\\sum\_\{j<k\}\\cos\\phi\_\{jk\}\(x\_\{t\}\)denote the average pairwise cosine similarity at statextx\_\{t\}\. Under the assumption‖gjCI‖≈μ\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|\\approx\\mufor alljj, the trajectory\-level energy deficit is

𝒦deficit\\displaystyle\\mathcal\{K\}\_\{\\mathrm\{deficit\}\}≜∫01𝔼xt∼pt⋆​\[Δ​E​\(xt\)\]​𝑑t=G​\(G−1\)​μ2​∫01𝔼xt∼pt⋆​\[1−cos⁡ϕt​\(xt\)\]​𝑑t≥0\.\\displaystyle\\;\\triangleq\\;\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\left\[\\Delta E\(x\_\{t\}\)\\right\]dt\\;=\\;G\(G\-1\)\\,\\mu^\{2\}\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}^\{\\star\}\}\\\!\\left\[1\-\\cos\\phi\_\{t\}\(x\_\{t\}\)\\right\]dt\\;\\geq\\;0\.\(33\)𝒦deficit=0\\mathcal\{K\}\_\{\\mathrm\{deficit\}\}=0iffcos⁡ϕt≡1\\cos\\phi\_\{t\}\\equiv 1almost everywhere, recovering case \(B1\)\. As conflict grows \(cos⁡ϕt→−1\\cos\\phi\_\{t\}\\to\-1\),𝒦deficit\\mathcal\{K\}\_\{\\mathrm\{deficit\}\}increases linearly in\(1−cos⁡ϕt\)\(1\-\\cos\\phi\_\{t\}\)and quadratically inGGthroughG​\(G−1\)​μ2G\(G\-1\)\\mu^\{2\}\. This dissipated energy is structurally unavoidable under additive guidance and must be explicitly supplied by a corrective fieldgψg\_\{\\psi\}to escape the basins of spurious local minima \(Definition[B\.1](https://arxiv.org/html/2605.20758#A2.Thmtheorem1)\)\.

Note that the squared discrepancy‖gtCI−∑jgjCI‖2\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\-\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}arises solely from gradient misalignment444Expanding directly:‖gtCI−∑jgjCI‖2=‖gtCI‖2−2​⟨gtCI,∑jgjCI⟩\+‖∑jgjCI‖2\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\-\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}=\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\\\|^\{2\}\-2\\langle g\_\{t\}^\{\\mathrm\{CI\}\},\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\rangle\+\\\|\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}\. Under the structural assumptions \(i\)‖gtCI‖2=\(∑j‖gjCI‖\)2\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\\\|^\{2\}=\(\\sum\_\{j\}\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|\)^\{2\}\(aligned\-ideal magnitude\) and \(ii\)⟨gtCI,∑jgjCI⟩=‖∑jgjCI‖2\\langle g\_\{t\}^\{\\mathrm\{CI\}\},\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\rangle=\\\|\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}\(full projection onto the sum direction\), the self\-terms∑j‖gjCI‖2\\sum\_\{j\}\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}cancel, leaving‖gtCI−∑jgjCI‖2=2​∑j<k‖gjCI‖​‖gkCI‖​\(1−cos⁡ϕj​k\)=Δ​E​\(xt\)\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\-\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\\|^\{2\}=2\\sum\_\{j<k\}\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|\\\|g\_\{k\}^\{\\mathrm\{CI\}\}\\\|\(1\-\\cos\\phi\_\{jk\}\)=\\Delta E\(x\_\{t\}\)\.\. Hence, Term \(B\) in Eq\. \([31](https://arxiv.org/html/2605.20758#A3.E31)\) is bounded by the energy deficit:

2​∫01𝔼​\[‖gtCI−∑jgjCI‖2\]​𝑑t≲𝒦deficit,2\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\left\\\|g\_\{t\}^\{\\mathrm\{CI\}\}\-\\textstyle\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\\right\\\|^\{2\}\\right\]dt\\;\\lesssim\\;\\mathcal\{K\}\_\{\\mathrm\{deficit\}\},\(34\)where absolute constants are absorbed into the≲\\lesssimsymbol\. In case \(B1\),𝒦deficit=0\\mathcal\{K\}\_\{\\mathrm\{deficit\}\}=0\.

##### \(C\) Localized approximation error\.

The third term arises from replacing eachgjCIg\_\{j\}^\{\\mathrm\{CI\}\}with its first\-order Taylor estimategjapproxg\_\{j\}^\{\\mathrm\{approx\}\}aroundx^1\\hat\{x\}\_\{1\}\. This linearization error has been analysed in detail byfeng2025guidancefor the single\-reward setting\. For the compositional setting, Cauchy–Schwarz across theGGrewards yields

‖∑jgjCI−gtapprox‖2≤G​∑j‖δ​gj‖2≲G​\(λh​σ1​der​\(x^1\)\)2​\(C1\+C2\),\\left\\\|\\sum\_\{j\}g\_\{j\}^\{\\mathrm\{CI\}\}\-g\_\{t\}^\{\\mathrm\{approx\}\}\\right\\\|^\{2\}\\;\\leq\\;G\\sum\_\{j\}\\\|\\delta g\_\{j\}\\\|^\{2\}\\;\\lesssim\\;G\\left\(\\frac\{\\lambda\_\{h\}\\,\\sigma\_\{1\}\\,d\}\{e^\{r\(\\hat\{x\}\_\{1\}\)\}\}\\right\)^\{\\\!2\}\(C\_\{1\}\+C\_\{2\}\),\(35\)i\.e\.,feng2025guidance’s per\-reward bound scaled linearly byGG\. Term \(C\) decreases when the reward is smooth \(smallλh\\lambda\_\{h\}\), neart→1t\\to 1\(smallσ1\\sigma\_\{1\}\), orx^1\\hat\{x\}\_\{1\}lies in a high\-reward region \(largeer​\(x^1\)e^\{r\(\\hat\{x\}\_\{1\}\)\}\)\.

##### Put it together\.

###### Theorem C\.2\(Upper bound of Approximation Error in Compositional Reward Setting\)\.

Letvt⋆v\_\{t\}^\{\\star\}be the exact guided velocity field andv^t\\hat\{v\}\_\{t\}be the realized field under the Coupling\-Invariant Approximation and Localized Approximation\. The total approximation errorℰ≜W22​\(p^1,p1⋆\)\\mathcal\{E\}\\triangleq W\_\{2\}^\{2\}\(\\hat\{p\}\_\{1\},p\_\{1\}^\{\\star\}\)satisfies:

ℰ≲\\displaystyle\\mathcal\{E\}\\;\\lesssim\\;CCI​∫01𝔼​\[𝔼πCI​\[\(𝒫​\(z\)−1\)2\]\]​𝑑t⏟\(A\) coupling shift error\\displaystyle\\underbrace\{C\_\{\\mathrm\{CI\}\}\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{CI\}\}\}\\\!\\big\[\(\\mathcal\{P\}\(z\)\-1\)^\{2\}\\big\]\\right\]dt\}\_\{\\text\{\(A\) coupling shift error\}\}\+\\displaystyle\+\\;G​\(G−1\)​μ2​∫01𝔼​\[1−cos⁡ϕt​\(xt\)\]​𝑑t⏟\(B\) gradient misalignment error \(Proposition[C\.1](https://arxiv.org/html/2605.20758#A3.Thmtheorem1)\)\\displaystyle\\underbrace\{G\(G\-1\)\\mu^\{2\}\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[1\-\\cos\\phi\_\{t\}\(x\_\{t\}\)\\right\]dt\}\_\{\\text\{\(B\) gradient misalignment error \(Proposition~\\ref\{prop:energy\_deficit\}\)\}\}\+\\displaystyle\+\\;G​∫01𝔼​\[\(λh​σ1​der​\(x^1\)\)2​\(C1\+C2\)\]​𝑑t⏟\(C\) localized approximation error\(feng2025guidance\),\\displaystyle\\underbrace\{G\\\!\\int\_\{0\}^\{1\}\\\!\\mathbb\{E\}\\\!\\left\[\\left\(\\frac\{\\lambda\_\{h\}\\,\\sigma\_\{1\}\\,d\}\{e^\{r\(\\hat\{x\}\_\{1\}\)\}\}\\right\)^\{\\\!2\}\(C\_\{1\}\+C\_\{2\}\)\\right\]dt\}\_\{\\text\{\(C\) localized approximation error \\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{feng2025guidance\}\{\\@@citephrase\{, \}\}\{\}\)\}\}\},\(36\)whereCCI:=supt∈\[0,1\]𝔼πCI​\[‖vt∣z‖22\]C\_\{\\mathrm\{CI\}\}:=\\sup\_\{t\\in\[0,1\]\}\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{CI\}\}\}\[\\\|v\_\{t\\mid z\}\\\|\_\{2\}^\{2\}\]only depends on the base velocity field,cos⁡ϕt​\(xt\):=2G​\(G−1\)​∑j<kcos⁡ϕj​k\\cos\\phi\_\{t\}\(x\_\{t\}\):=\\frac\{2\}\{G\(G\-1\)\}\\sum\_\{j<k\}\\cos\\phi\_\{jk\}is the average pairwise cosine similarity at\(xt,t\)\(x\_\{t\},t\),μ:=‖gjCI‖\\mu:=\\\|g\_\{j\}^\{\\mathrm\{CI\}\}\\\|is the per\-reward CI guidance magnitude\. Followingfeng2025guidance,λh\\lambda\_\{h\}is the spectral norm of the Hessian ofere^\{r\},σ1\\sigma\_\{1\}is the spectral norm of the conditional covarianceΣ1\|t\\Sigma\_\{1\|t\}, andC1,C2C\_\{1\},C\_\{2\}are constants depend on base flow\. The error bound provides four key insights into the realized guidanceg^t\\hat\{g\}\_\{t\}:

1. 1\.The error is small when thereward landscape is smooth, i\.e\., smallλh=‖∇2er‖2\\lambda\_\{h\}=\\\|\\nabla^\{2\}e^\{r\}\\\|\_\{2\}\. A flat reward landscape without sharp peaks or valleys implies less aggressive curvature, thereby minimizing the linearization error in Term \(C\)\.
2. 2\.The error is small whenσ1\\sigma\_\{1\}is small, i\.e\., the conditional covarianceΣ1\|t\\Sigma\_\{1\|t\}has small spectral norm, meaning that at the current statextx\_\{t\}the uncertainty about the terminal pointx1x\_\{1\}is low\. This is the case when the flow timet→1t\\to 1\(andσt→0\\sigma\_\{t\}\\to 0\), wherextx\_\{t\}reliably predictsx1x\_\{1\}\.
3. 3\.Themagnitude ofer​\(x^1\)e^\{r\(\\hat\{x\}\_\{1\}\)\}reflects how well the predicted endpointx^1=𝔼​\[x1\|xt\]\\hat\{x\}\_\{1\}=\\mathbb\{E\}\[x\_\{1\}\|x\_\{t\}\]matches the reward objective\. Ifx^1\\hat\{x\}\_\{1\}lies inside the region whererris large, the approximate guidance is more accurate, as the optimization is conducted locally and the gradient reflects the landscape well\. Ifer​\(x^1\)e^\{r\(\\hat\{x\}\_\{1\}\)\}is small, the gradient explores the sample space almost randomly, producing larger approximation error\.
4. 4\.The error scales withthe number of reward functionsGGandgradient misalignment\(1−cos⁡ϕ\)\(1\-\\cos\\phi\)\.

## Appendix DGuided sampling through the lens of fitted value evaluation

Unlike diffusion models, Flow models are governed by deterministic ODE processes\. By leveraging this deterministic coupling and applying Jensen’s inequality \(or assuming the reward variance over the posterior is small\), we approximate the soft value function with the expected return:V​\(xt,t\)≈𝔼z∼π​\(z∣xt\)​\[r​\(x1\)\]V\(x\_\{t\},t\)\\approx\\mathbb\{E\}\_\{z\\sim\\pi\(z\\mid x\_\{t\}\)\}\[r\(x\_\{1\}\)\]\. This simplification avoids the computational instability of the log\-sum\-exp operation while preserving the guidance direction\.

However, a central challenge remains: the optimal guidance depends on the future endpointx1∼p​\(x1∣xt\)x\_\{1\}\\sim p\(x\_\{1\}\\mid x\_\{t\}\), making analytical evaluation computationally prohibitive\. We address this by introducing a value \(reward\-to\-go\) functionV​\(x,t\)V\(x,t\)to summarize the expected terminal reward, modeled via the Bellman backup operator induced by the guided velocity field\.

###### Proposition D\.1\(Fitted Value Evaluation\)\.

Letℱ\\mathcal\{F\}denote the function class \(e\.g\., neural networks\) used to approximate the value function\. We collect a dataset of transitions𝒟=\{\(x,t,r,x′,t′\)\}\\mathcal\{D\}=\\\{\(x,t,r,x^\{\\prime\},t^\{\\prime\}\)\\\}generated under the current guided dynamics, wheret′=t\+Δ​tt^\{\\prime\}=t\+\\Delta tandrris the reward\. The value function can be estimated empirically by minimizing a least\-squares Bellman residual:

V^k\+1=arg⁡minV∈ℱ⁡𝔼𝒟​\[\(r\+γ​V^k​\(x′,t′\)−V​\(x,t\)\)2\]\.\\hat\{V\}\_\{k\+1\}=\\arg\\min\_\{V\\in\\mathcal\{F\}\}\\;\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\left\[\\big\(r\+\\gamma\\hat\{V\}\_\{k\}\(x^\{\\prime\},t^\{\\prime\}\)\-V\(x,t\)\\big\)^\{2\}\\right\]\.\(37\)subject to the boundary conditionV^k​\(x,1\)≡r​\(x\)\\hat\{V\}\_\{k\}\(x,1\)\\equiv r\(x\)for terminal states\. Here,γ∈\(0,1\]\\gamma\\in\(0,1\]is the discount factor, andV^k\\hat\{V\}\_\{k\}is the target from the previous iteration555For pure terminal optimization,r=0r=0andγ=1\\gamma=1\.\. Equation \([37](https://arxiv.org/html/2605.20758#A4.E37)\) empirically approximates the Bellman backup operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}using finite data and a function classℱ\\mathcal\{F\}\. Upon convergence, the learnable guidance is derived as the gradient of the estimated value:

g​\(xt,t\)≜∇xV^​\(xt,t\),g\(x\_\{t\},t\)\\;\\triangleq\\;\\nabla\_\{x\}\\hat\{V\}\(x\_\{t\},t\),\(38\)which provides a Markovian surrogate for the computationally expensive exact guidance \(Equation \([6](https://arxiv.org/html/2605.20758#S2.E6)\)\)\.

Notice that the definitiong​\(xt,t\)≜∇xV^​\(xt,t\)g\(x\_\{t\},t\)\\triangleq\\nabla\_\{x\}\\hat\{V\}\(x\_\{t\},t\)has also appeared in prior work\(liu2025value\), where it is motivated from an optimal control perspective in the context of controlled generation via differentiating through the ODE sampling process\. Recall that our goal here is to estimate the Bellman backup operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}\(i\.e\., the generative dynamic from an intermediate statextx\_\{t\}to the terminal statex1x\_\{1\}\)\. So we do not rely ong⋆g^\{\\star\}to quantify the optimality of the guidance term; instead, we use it purely as a tractable surrogate for the dependence of guidance on future states\. Below, we provide a simple proof to justify this construction\.

###### Proof\.

We view the generative dynamics as a Markov process whose policy is given by the guided velocityv′​\(x,t\)=vbase​\(x,t\)\+g​\(x,t\)\.v^\{\\prime\}\(x,t\)=v^\{\\mathrm\{base\}\}\(x,t\)\+g\(x,t\)\.Under this policy, we define a value \(reward\-to\-go\) function that summarizes the expected future reward induced by the guided dynamics\. Specifically, the value function is required to satisfy Bellman consistency

V\(x,t\)=𝔼\[r\+γV\(x′,t′\)\|x\],x′∼𝒯v′\(⋅∣x\),V\(x,t\)=\\mathbb\{E\}\\\!\\left\[r\+\\gamma\\,V\(x^\{\\prime\},t^\{\\prime\}\)\\;\\middle\|\\;x\\right\],\\qquad x^\{\\prime\}\\sim\\mathcal\{T\}^\{v^\{\\prime\}\}\(\\cdot\\mid x\),\(39\)where𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}denotes the transition operator induced by the guided velocity field \(stepping fromtttot′t^\{\\prime\}\) andγ∈\(0,1\]\\gamma\\in\(0,1\]is a discount factor\. In the generative setting considered here, the reward is sparse:r=0r=0for allt∈\[0,1\)t\\in\[0,1\), and reward is accrued only at the terminal statex1x\_\{1\}\.

The value functionVv′V^\{v^\{\\prime\}\}is thus characterized as a fixed point of the Bellman operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}, i\.e\.,Vv′=𝒯v′​Vv′V^\{v^\{\\prime\}\}=\\mathcal\{T\}^\{v^\{\\prime\}\}V^\{v^\{\\prime\}\}\. One possible approach is to compute the Bellman backup operator𝒯v′\\mathcal\{T\}^\{v^\{\\prime\}\}by exhaustive bootstrapping\. However, in high\-dimensional state spaces, it is infeasible to enumerate or traverse all states\. We therefore approximate the Bellman operator by data and function approximation, implemented via Fitted Value Evaluation \(FVE\)\.

Letℱ\\mathcal\{F\}denote a function class used to approximate the value function, and let𝒟=\{\(x\(i\),t\(i\),r\(i\),x′⁣\(i\),t′⁣\(i\)\)\}i=1N\\mathcal\{D\}=\\\{\(x^\{\(i\)\},t^\{\(i\)\},r^\{\(i\)\},x^\{\\prime\(i\)\},t^\{\\prime\(i\)\}\)\\\}\_\{i=1\}^\{N\}be a dataset of one\-step transitions collected from rollouts under the guided dynamics\. We define the empirical Bellman backup

𝒯^v′​V​\(x,t\)≜r\+γ​V​\(x′,t′\),\(x,t,r,x′,t′\)∼𝒟\.\\widehat\{\\mathcal\{T\}\}^\{v^\{\\prime\}\}V\(x,t\)\\;\\triangleq\\;r\+\\gamma V\(x^\{\\prime\},t^\{\\prime\}\),\\qquad\(x,t,r,x^\{\\prime\},t^\{\\prime\}\)\\sim\\mathcal\{D\}\.\(40\)Using this empirical operator, we perform fitted value evaluation \(FVE\) by iteratively projecting the Bellman backup ontoℱ\\mathcal\{F\}:

Vk\+1=arg⁡minV∈ℱ⁡𝔼𝒟​\[\(r\+γ​Vk​\(x′,t′\)−V​\(x,t\)\)2\]\.V\_\{k\+1\}=\\arg\\min\_\{V\\in\\mathcal\{F\}\}\\;\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\Big\[\\big\(r\+\\gamma V\_\{k\}\(x^\{\\prime\},t^\{\\prime\}\)\-V\(x,t\)\\big\)^\{2\}\\Big\]\.\(41\)This procedure yields an empirical approximationVVthat is Bellman\-consistent in expectation with respect to the guided dynamics\.

Whenℱ\\mathcal\{F\}is large \(or infinite\) andVVis parameterized asVθ∈ℱV\_\{\\theta\}\\in\\mathcal\{F\}, Equation \([41](https://arxiv.org/html/2605.20758#A4.E41)\) is typically solved by stochastic optimization\. In particular, treating the bootstrap targety=r\+γ​Vθk​\(x′,t′\)y=r\+\\gamma V\_\{\\theta\_\{k\}\}\(x^\{\\prime\},t^\{\\prime\}\)as fixed, we minimize the squared regression error via a semi\-gradient update:

θ←θ−α∇θ\(Vθ\(x,t\)−\[r\+γVθk\(x′,t′\)\]\)2,\\theta\\leftarrow\\theta\-\\alpha\\,\\nabla\_\{\\theta\}\\Big\(V\_\{\\theta\}\(x,t\)\-\\big\[r\+\\gamma V\_\{\\theta\_\{k\}\}\(x^\{\\prime\},t^\{\\prime\}\)\\big\]\\Big\)^\{2\},\(42\)whereα\>0\\alpha\>0is the learning rate\. Repeating Equation \([41](https://arxiv.org/html/2605.20758#A4.E41)\) \(or its stochastic variant Equation \([42](https://arxiv.org/html/2605.20758#A4.E42)\)\) yields a Bellman\-consistent value approximation for the guided dynamics\.

The Bellman\-consistent value functionV​\(x,t\)V\(x,t\)summarizes the expected terminal reward attainable from the current state under the guided dynamics\. Indeed,∇xV​\(x,t\)\\nabla\_\{x\}V\(x,t\)points in the direction of steepest increase of the expected future reward, and therefore represents the locally optimal infinitesimal adjustment to the dynamics\. This observation provides a principled bridge between value estimation and guidance construction: rather than explicitly conditioning on future endpointsx1x\_\{1\}, guidance can be implemented as a local ascent direction induced by the value function gradient\.

Once a Bellman\-consistent value functionV^\\hat\{V\}is obtained, we define the guidance vector field as

g​\(x,t\)≜∇xV^​\(x,t\),g\(x,t\)\\;\\triangleq\\;\\nabla\_\{x\}\\hat\{V\}\(x,t\),\(43\)which induces a local ascent direction in state space that maximally increases the expected terminal reward\. This construction yields a Markovian and tractable surrogate for the otherwise future\-dependent guidance implied by exact importance weighting\. ∎

This transformation effectively converts the intractable integral in Term \(C\) \(localized approximation error\) of Theorem[4\.2](https://arxiv.org/html/2605.20758#S4.Thmtheorem2)into a differentiable, Markovian vector field\. However, Fitted Value Evaluation \(FVE\) can diverge even when theoretical conditions are met\.

###### Proposition D\.2\(Divergence of Fitted Value Evaluation\)\.

Fitted value evaluation \(FVE\) can diverge even when all of the following conditions hold:

1. 1\.The dataset is infinite, i\.e\.,\|𝒟\|=∞\|\\mathcal\{D\}\|=\\infty;
2. 2\.The Bellman residual minimization is solved exactly at each iteration;
3. 3\.The function classℱ\\mathcal\{F\}is simple enough to be estimated, e\.g\., a one\-dimensional linear function classfθ​\(x\)=θ⊤​ϕ​\(x\)f\_\{\\theta\}\(x\)=\\theta^\{\\top\}\\phi\(x\);
4. 4\.The realizability assumption holds, i\.e\., the true value function satisfiesV∈ℱV\\in\\mathcal\{F\}\.

This phenomenon is commonly referred to as the*deadly triad*in empirical deep reinforcement learning, which arises from the interaction of function approximation, off\-policy data, and bootstrapping\. In the flow matching setting, however, the dynamics are deterministic and rewards are sparse \(evaluated at terminal statex1x\_\{1\}\)\. We exploit this property to propose Terminal Value Regression, a method that directly fits the terminal reward\. By removing the need for bootstrapping, this approach effectively breaks the deadly triad and ensures stable convergence\.

###### Proposition D\.3\(Terminal Value Regression\)\.

Letℱ\\mathcal\{F\}denote a function class used to approximate the value function\. We collect a dataset of terminal rollouts𝒟=\{\(xt,t,x1\)\}\\mathcal\{D\}=\\\{\(x\_\{t\},t,x\_\{1\}\)\\\}, wherex1x\_\{1\}is the terminal state reached fromxtx\_\{t\}by integrating the current guided dynamics\. The value function is estimated by minimizing the following regression objective:

V^=arg⁡minV∈ℱ⁡𝔼\(xt,t,x1\)∼𝒟​\[\(r​\(x1\)−V​\(xt,t\)\)2\]\.\\hat\{V\}=\\arg\\min\_\{V\\in\\mathcal\{F\}\}\\;\\mathbb\{E\}\_\{\(x\_\{t\},t,x\_\{1\}\)\\sim\\mathcal\{D\}\}\\Big\[\\big\(r\(x\_\{1\}\)\-V\(x\_\{t\},t\)\\big\)^\{2\}\\Big\]\.\(44\)Unlike the bootstrapped target in Equation \([37](https://arxiv.org/html/2605.20758#A4.E37)\), the terminal rewardr​\(x1\)r\(x\_\{1\}\)serves as a stable, unbiased regression target, which is enabled by the deterministic nature of the flow\.

Unlike fitted value evaluation, Equation \([44](https://arxiv.org/html/2605.20758#A4.E44)\) does not rely on bootstrapping and therefore avoids the instability associated with the deadly triad\. Since the flow matching dynamics are deterministic, the terminal rewardr​\(x1\)r\(x\_\{1\}\)serves as an unbiased Monte Carlo target for value estimation, yielding a stable procedure tailored to flow matching models\.

## Appendix EExperimental details

### E\.1Parameterization: value function vs\. vector field

While directly parameterizing the vector field is common in diffusion models\(song2021train\), unconstrained neural vector fields are not guaranteed to be conservative \(i\.e\., curl\-free\)\(balcerak2025energy\)\. Therefore, in Equation \([19](https://arxiv.org/html/2605.20758#S5.E19)\), we explicitly parameterize the scalar value functionV​\(xt,t\)V\(x\_\{t\},t\)and derive the guidance via automatic differentiation∇xtVψ​\(xt\)\\nabla\_\{x\_\{t\}\}V\_\{\\psi\}\(x\_\{t\}\), ensuring that the learned guidance corresponds to the gradient of a valid scalar reward landscape\. Crucially, as shown in Figure[10](https://arxiv.org/html/2605.20758#A5.F10)\(c\-e\), simply parameterizing∇V​\(xt,t\)\\nabla V\(x\_\{t\},t\)fails to rectify the off\-manifold drift\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x10.png)Figure 10:Comparative empirical results on parameterization strategies\.We compare two architectures for learning the residual guidance: \(Left: c–e\) Directly parameterizing the unconstrained vector field \(denoted as∇V\\nabla V\)\. As shown in \(d\), this lack of structural constraint leads to a non\-conservative field with a distorted, incoherent energy landscape, causing the “energy trap” and off\-manifold drift in \(e\)\. \(Right: f–h\) Explicitly parameterizing the scalar value functionVV\. By taking the gradient of a learned scalarVV, we enforce the field to be curl\-free by construction\. This results in the smooth, globally consistent energy landscape in \(g\), effectively rectifying the drift as shown in \(h\)\.Finally, to stabilize optimization when backpropagating through the parameterized value functionV​\(xt,t\)V\(x\_\{t\},t\), we apply gradient clipping to the derived gradients∇Vψ​\(xt,t\)\\nabla V\_\{\\psi\}\(x\_\{t\},t\)\. This prevents exploding gradients, particularly in regions where the learned energy surface becomes steep or singular\.

### E\.2Ablation: hard gate𝕀t\\mathbb\{I\}\_\{t\}and conflict thresholdτ\\tau

Theτ\\taucontrols the hard gate𝕀t\\mathbb\{I\}\_\{t\}in the training loss:

ℒ​\(ψ\)=𝔼\(xt,t,x1\)∼𝒟​\[𝕀t⋅\(r​\(x1\)−Vψ​\(xt,t\)\)2\]\\mathcal\{L\}\(\\psi\)=\\mathbb\{E\}\_\{\(x\_\{t\},t,x\_\{1\}\)\\sim\\mathcal\{D\}\}\\left\[\\mathbb\{I\}\_\{t\}\\cdot\\bigl\(r\(x\_\{1\}\)\-V\_\{\\psi\}\(x\_\{t\},t\)\\bigr\)^\{2\}\\right\]and serves two purposes: \(1\) reducing unnecessary computation by skipping low\-conflict regions, and \(2\) preservinggapproxg^\{\\text\{approx\}\}in those regions, where the approximate guidance is already accurate and adding a learned correction would introduce spurious perturbations\. Ifτ\\tauis too small, neither purpose is met, as the gate activates almost everywhere\. Conversely, too large aτ\\tauskips too many training steps, leavinggψg\_\{\\psi\}under\-trained\.

We suggest that the thresholdτ\\taucan be tuned according to the specific domain, and we empirically find thatτ=0\.2\\tau=0\.2is a robust sweet spot for most of our evaluated tasks \(Maze2D, CelebA\-HQ image editing, and ManiSkill2\)\. For the synthetic benchmark, we useτ=0\.5\\tau=0\.5, which works better under its different conflict distribution\. Therefore, we report experimental results usingτ=0\.2\\tau=0\.2for real\-world domains andτ=0\.5\\tau=0\.5for the synthetic benchmark\. We show the ablation results of the conflict thresholdτ\\tauin Figure[11](https://arxiv.org/html/2605.20758#A5.F11)\. A threshold that is too small \(e\.g\.,τ=0\.0\\tau=0\.0\) introduces spurious guidance in non\-conflict regions \(where approximation guidancegapproxg^\{\\text\{approx\}\}is already good enough\), degrading performance \(e\.g\., in the synthetic experiment, CS drops to68\.4%68\.4\\%vs\.∼94%\{\\sim\}94\\%forτ∈\[0\.2,0\.5\]\\tau\\in\[0\.2,0\.5\]\)\. Conversely, excessively high thresholds \(e\.g\.,τ=0\.8\\tau=0\.8\) skip too many updates, leavinggψg\_\{\\psi\}under\-trained\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x11.png)Figure 11:Ablation results on the conflict thresholdτ\\tau\.
### E\.3Ablation: learned correctiongψg\_\{\\psi\}and conflict\-aware weightwtw\_\{t\}

Our methodgcarg^\{\\text\{car\}\}integrates a learned correctiongψ​\(xt,t\)g\_\{\\psi\}\(x\_\{t\},t\)and a conflict\-aware weightwtw\_\{t\}into the guided velocity field:

vt′​\(xt,t\)\\displaystyle v^\{\\prime\}\_\{t\}\(x\_\{t\},t\)=vtbase​\(xt,t\)\+gcar​\(xt,t\),\\displaystyle\\;=\\;v^\{\\text\{base\}\}\_\{t\}\(x\_\{t\},t\)\\;\+\\;g^\{\\text\{car\}\}\(x\_\{t\},t\),gcar​\(xt,t\)\\displaystyle g^\{\\text\{car\}\}\(x\_\{t\},t\)=\(1−wt\)​gapprox\+wt​gψ​\(xt,t\)\.\\displaystyle\\;=\\;\(1\-w\_\{t\}\)\\,g^\{\\text\{approx\}\}\\;\+\\;w\_\{t\}\\,g\_\{\\psi\}\(x\_\{t\},t\)\.To understand the contribution of each component, we ablategψg\_\{\\psi\}andwtw\_\{t\}independently\. Table[5](https://arxiv.org/html/2605.20758#A5.T5)summarizes the three ablation studies, and Figure[12](https://arxiv.org/html/2605.20758#A5.F12)reports the quantitative results on the synthetic benchmark\.

Figure[12](https://arxiv.org/html/2605.20758#A5.F12)reports results across three domains\. Addinggψg\_\{\\psi\}without the gate togcov\-Gg^\{\\text\{cov\-G\}\}yields modest gains \(synthetic CS\+0\.5\+0\.5pp, CelebA\-HQ LPIPS−0\.014\-0\.014, Maze2D success\+5\+5\), confirming that the learned correction provides a useful residual signal to maximize rewards\. Constraining the correctiongψg\_\{\\psi\}to conflict regions via addingwtw\_\{t\}gives much larger improvements \(synthetic CS\+9\.8\+9\.8pp, PC\+3\.5\+3\.5pp; CelebA\-HQ LPIPS−0\.021\-0\.021, CLIP\+0\.011\+0\.011; Maze2D safety\+15\+15, success\+9\+9\), demonstrating that the conflict\-aware weight is the more critical component\. Training loss curves in Figure[12](https://arxiv.org/html/2605.20758#A5.F12)\(c,f,i\) confirm stable convergence across all settings\.

Table 5:Ablation study design for learned correctiongψg\_\{\\psi\}and conflict\-aware weightwtw\_\{t\}\. All configurations share the same pretrained base velocity fieldvbasev^\{\\text\{base\}\}and same approximation guidancegapproxg^\{\\text\{approx\}\}\(i\.e\.,gcov\-Gg^\{\\text\{cov\-G\}\}\)\.![Refer to caption](https://arxiv.org/html/2605.20758v1/fig/app_component_ablation.png)Figure 12:Component ablation on the synthetic benchmark\. \(a\) Mode Coverage \(CS\) and \(b\) Prior Preservation \(PC\)\. The baselinegcov\-Gg^\{\\text\{cov\-G\}\}suffers from severe gradient conflicts\. Applying the learned correction without the conflict gate \(gapprox\+gψg^\{\\text\{approx\}\}\+g\_\{\\psi\}\) improves PC but hurts CS due to spurious updates in low\-conflict regions\. Our full methodgcarg^\{\\text\{car\}\}leverages the gatewtw\_\{t\}to restrict corrections strictly to high\-conflict states, achieving optimal performance in both metrics\.
### E\.4Synthetic dataset

We consider a 2\-dimensional Mixture of Gaussians toy example \(see Figure[13](https://arxiv.org/html/2605.20758#A5.F13)\), where the ground\-truth densityptp\_\{t\}is known analytically, allowing for precise quantitative evaluation\. The source distributionπ0\\pi\_\{0\}is a standard Gaussianπ0​\(x\)=𝒩​\(x∣μ0,Σ0\)\\pi\_\{0\}\(x\)=\\mathcal\{N\}\(x\\mid\\mu\_\{0\},\\Sigma\_\{0\}\), whereμ0=\[0\.0,0\.0\]\\mu\_\{0\}=\[0\.0,0\.0\]andΣ0=I\\Sigma\_\{0\}=I\. The target distributionπ1\\pi\_\{1\}is a Mixture of Gaussians consisting ofK=3K=3modes, i\.e\.,π1​\(x\)=13​∑k=13𝒩​\(x∣μk,Σ1\)\\pi\_\{1\}\(x\)=\\frac\{1\}\{3\}\\sum\_\{k=1\}^\{3\}\\mathcal\{N\}\(x\\mid\\mu\_\{k\},\\Sigma\_\{1\}\), where each component shares the covarianceΣ1=I\\Sigma\_\{1\}=I\. We use a fixed configuration with centers located atμ1=\[8\.0,8\.0\]\\mu\_\{1\}=\[8\.0,8\.0\],μ2=\[8\.0,−8\.0\]\\mu\_\{2\}=\[8\.0,\-8\.0\], andμ3=\[0\.0,10\.0\]\\mu\_\{3\}=\[0\.0,10\.0\], corresponding to the base posterior visualized in Figure[13](https://arxiv.org/html/2605.20758#A5.F13)\(b\)\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x12.png)Figure 13:Visualization of synthetic experiments\.\(a\) The sampling dynamics of the base Rectified Flow model att=1t=1\. \(b\) The base posterior distributionpbase​\(x1\)p^\{\\text\{base\}\}\(x\_\{1\}\)consisting of three Gaussian modes\. \(c\)–\(e\) Ground\-truth posteriors under different classifier constraints \(c=\[0,0\]c=\[0,0\],\[1,0\]\[1,0\], and\[1,1\]\[1,1\]\), estimated via rejection sampling with 10k samples\.#### E\.4\.1Inference\-time Constraints

To evaluate the system under conflicting guidance, we employ two pre\-trained binary classifiers,𝒞1\\mathcal\{C\}\_\{1\}and𝒞2\\mathcal\{C\}\_\{2\}, which act as independent reward signals\. Each classifier assigns a labely∈\{0,1\}y\\in\\\{0,1\\\}to the generated samples\. The classifier labels for the three target modes are designed to create varying degrees of gradient alignment\. We visualize the ground\-truth posteriors under these different compositional rewards in Figure[13](https://arxiv.org/html/2605.20758#A5.F13)\(c–e\), generated via rejection sampling\. Specifically, the constraintc=\[1,0\]c=\[1,0\]\(shown in Figure[13](https://arxiv.org/html/2605.20758#A5.F13)\(d\)\) represents a scenario with significant gradient conflict \(or misalignment\), serving as a primary stress test for off\-manifold drift\.

#### E\.4\.2Evaluation metrics

\(1\) Posterior Coverage \(PC\) \(↑\\uparrow\)\.Fraction of generated samples residing within the2​σ2\\sigmaboundary of ground\-truth target mixture components, measured by anisotropic Mahalanobis distance:

PC=1N​∑i=1N𝕀​\[mink∈𝒦target⁡d​\(xi,μk\)≤2\],\\text\{PC\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbb\{I\}\\left\[\\min\_\{k\\in\\mathcal\{K\}\_\{\\text\{target\}\}\}d\(x\_\{i\},\\mu\_\{k\}\)\\leq 2\\right\],\(45\)where𝒦target\\mathcal\{K\}\_\{\\text\{target\}\}is the set of cluster indices satisfying the target labels\. Unlike soft classifier probabilities \(CS\), PC is a strict geometric oracle: a sample is valid only if it physically resides within the correct high\-density mode\.A drop in PC indicates off\-manifold drift or biased sampling\.

\(2\) Constraint satisfaction \(CS\) \(↑\\uparrow\)\.The average probability assigned to the target labelyyby the guidance classifiers\.

CS=1N​∑i=1Npϕ​\(y\|xi\)\.\\text\{CS\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}p\_\{\\phi\}\(y\|x\_\{i\}\)\.\(46\)High CS indicates the guidance successfully optimizes the reward, potentially including adversarial examples that satisfy the classifier but fail PC\.

\(3\) Inference Time \(↓\\downarrow\)\.Wall\-clock time per generated sample, aggregating: \(i\) trajectory generation \(if applicable\), \(ii\) learnable guidance training, and \(iii\) forward ODE solving\. Note that GM collects data offline \(highly parallelized\), whereasgcarg^\{\\text\{car\}\}relies on much slower online rollouts\.

\(4\) Data Usage \(↓\\downarrow\)denotes the total number of training trajectories \(fromx0x\_\{0\}tox1x\_\{1\}\) required to learn the learnable guidance; lower is more data\-efficient\.

Table 6:Quantitative comparison on the synthetic benchmark\. \(1\) Posterior Coverage \(PC\): lower values indicate off\-manifold drift or biased sampling\. \(2\) Constraint Satisfaction \(CS\): average classifier probability for the target labels\. \(3\) Time \(↓\\downarrow\): wall\-clock time per sample \(ms\)\. \(4\) Data Usage \(×103\\times 10^\{3\},↓\\downarrow\): total training samples consumed, reported in units of10310^\{3\}\. All results use conflict thresholdτ=0\.50\\tau\{=\}0\.50;ϵ\\epsilondenotes the early\-stopping threshold\. Each number is evaluated with10​k10\\text\{k\}generated samples\. Data Usage ofgcarg^\{\\text\{car\}\}reports the total number of training samples consumed before the fraction of conflict samples drops belowϵ\\epsilon\(see early\-stopping criterion in Appendix[E\.4\.3](https://arxiv.org/html/2605.20758#A5.SS4.SSS3)\)\.Note:Boldindicates best performance\. The\[1,0\]\[1,0\]column highlights the gradient conflict scenario\. Mean±\\pmstdare reported over 5 random seeds\.

#### E\.4\.3Experimental Results

We present comprehensive quantitative results in Table[6](https://arxiv.org/html/2605.20758#A5.T6), evaluated with10​k10\\text\{k\}generated samples per setting\. The evaluation covers all valid constraint configurations: \(i\) Single Guidance:\[0,∅\],\[1,∅\],\[∅,0\]\[0,\\varnothing\],\[1,\\varnothing\],\[\\varnothing,0\], and\[∅,1\]\[\\varnothing,1\]; \(ii\) Composed Guidance:\[0,0\]\[0,0\],\[1,0\]\[1,0\], and\[1,1\]\[1,1\]\. Note that no data samples satisfy\[0,1\]\[0,1\]\.

##### gcarg^\{\\text\{car\}\}resolves off\-manifold drift efficiently\.

Under single guidance, all methods achieve competitive Posterior Coverage \(PC,\>85%\{\>\}85\\%\) and Constraint Satisfaction \(CS,\>99%\{\>\}99\\%\)\. However, in the compositional reward setting—especially the\[1,0\]\[1,0\]gradient conflict scenario in[Table6](https://arxiv.org/html/2605.20758#A5.T6)—significant performance gaps emerge\. Withϵ=0\.05\\epsilon\{=\}0\.05,gcarg^\{\\text\{car\}\}achieves93\.80%±0\.0593\.80\\%\\pm 0\.05PC and a perfect100\.00%±0\.00100\.00\\%\\pm 0\.00CS on the\[1,0\]\[1,0\], outperforming GLASS\-FKS \(K=16K\{=\}16\) by3\.03\.0PC points while operating at a70×70\{\\times\}lower inference cost \(4\.20±0\.054\.20\\pm 0\.05vs\.≈296\\approx 296ms/sample\)\. Employing a tighter threshold \(ϵ=0\.00\\epsilon\{=\}0\.00\) further maximizes composed guidance fidelity \(averaging95\.10%±0\.0395\.10\\%\\pm 0\.03PC\) at the expense of maximum training data usage, whereasϵ=0\.10\\epsilon\{=\}0\.10provides highly competitive composed performance \(88\.80%±0\.0488\.80\\%\\pm 0\.04PC average\) with near\-zero transition data requirements\.

Key observations are:

- •gcov\-Gg^\{\\text\{cov\-G\}\}collapses under gradient conflict\.gcov\-Gg^\{\\text\{cov\-G\}\}degrades sharply to71\.70%±0\.0471\.70\\%\\pm 0\.04PC on\[1,0\]\[1,0\]\.
- •GLASS\-FKS \(sample\-based\) avoids off\-manifold drift but is computationally costly and is highly sensitive to the particle countKK\. As shown in Table[6](https://arxiv.org/html/2605.20758#A5.T6), GLASS\-FKS maintains consistent Constraint Satisfaction \(CS\) scores across all compositional scenarios \(i\.e\.,\[0,0\]\[0,0\],\[1,0\]\[1,0\], and\[1,1\]\[1,1\]\), and does not have a severe performance drop under the conflicting\[1,0\]\[1,0\]setting\. However, when the number of particles is restricted \(e\.g\.,K=4K\{=\}4\), the variance increases\.
- •Guidance Matching suffers from confounding errors inherent in learning a guidance network from scratch, yielding a lower average PC of 84\.33%\. Moreover, GM requires over10710^\{7\}training samples per compositional reward \(approx\.20×20\\timesmore thangcarg^\{\\text\{car\}\}\)\.
- •PCGrad didn’t manage to correct off\-manifold drift\.
- •Ourgcarg^\{\\text\{car\}\}efficiently corrects off\-manifold drift while remaining compute\-light\.

##### Data efficiency via early stopping\.

To evaluate the impact of the conflict\-aware module ondata efficiency, we introduce an early\-stopping mechanism parameterized byϵ\\epsilon\. Training is halted when the proportion of generated samples with conflict score exceedingτ\\taudrops belowϵ\\epsilon, i\.e\.,P​\(score\>τ\)<ϵP\(\\text\{score\}\>\\tau\)<\\epsilon\. This criterion indicates thatx1x\_\{1\}has sufficiently resolved gradient conflicts\. Table[6](https://arxiv.org/html/2605.20758#A5.T6)reportsgcarg^\{\\text\{car\}\}underϵ∈\{0\.10,0\.05,0\.00\}\\epsilon\\in\\\{0\.10,0\.05,0\.00\\\}; the training dynamics are visualized in Figure[14](https://arxiv.org/html/2605.20758#A5.F14)\. As shown, the conflict score decreases stably across all settings, demonstrating thatgcarg^\{\\text\{car\}\}reliably learns to minimize gradient conflicts and rectify off\-manifold drift over time\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x13.png)Figure 14:Convergence of conflict scores ofgcarg^\{\\text\{car\}\}\.The figure tracks the fraction of online samples with a conflict score larger than the early\-stopping thresholdϵ\\epsilon\. This metric serves as an indicator for training stability\. Results are shown for targets \(a\)c=\[0,0\]c=\[0,0\], \(b\)c=\[1,0\]c=\[1,0\], and \(c\)c=\[1,1\]c=\[1,1\]across three early\-stopping threshold \(ϵ=0\.00,0\.05\\epsilon=0\.00,0\.05and0\.100\.10\), with a conflict thresholdτ=0\.50\\tau=0\.50\. The downward trend indicates that the conflict scores of online samples progressively decrease, demonstrating thatgcarg^\{\\text\{car\}\}effectively learns to minimize gradient conflicts and rectify off\-manifold drift\. Shaded areas represent the standard error across 5 random seeds\.

### E\.5Generative decision\-making as planners

#### E\.5\.1Inference\-time constraints

##### Static obstacle rewards\.

We formulate static obstacle avoidance as a differentiable, energy\-based reward functionrstatic​\(𝐱\)r\_\{\\text\{static\}\}\(\\mathbf\{x\}\), following\(luo2024potential\), which provides a smooth, bounded penalty landscape, thereby stabilizing the gradient\-based guidance∇𝐱rstatic​\(𝐱\)\\nabla\_\{\\mathbf\{x\}\}r\_\{\\text\{static\}\}\(\\mathbf\{x\}\)at inference time\.

Formally, the obstacles are defined as a set ofKKcenters\{𝐜k\}k=1K\\\{\\mathbf\{c\}\_\{k\}\\\}\_\{k=1\}^\{K\}\. The compositional static obstacle reward at state𝐱\\mathbf\{x\}is defined as:

rstatic​\(𝐱\)=−∑k=1Kexp⁡\(−‖𝐱−𝐜k‖2σ2\)r\_\{\\text\{static\}\}\(\\mathbf\{x\}\)=\-\\sum\_\{k=1\}^\{K\}\\exp\\left\(\-\\frac\{\\\|\\mathbf\{x\}\-\\mathbf\{c\}\_\{k\}\\\|^\{2\}\}\{\\sigma^\{2\}\}\\right\)\(47\)whereσ\\sigmadetermines the spatial decay rate of the repulsive potential, i\.e\., the influence diminishes as the distance from the center increases\. We setσ=2\.0\\sigma=2\.0, with the remaining settings unchanged\.

##### Static goal rewards\.

We consider instruction\-following scenarios \(e\.g\., “fetch an apple”\), where the objective is to reach specific spatial locations\. We formulate the guidance for reaching these goals using the same differentiable, energy\-based formulation as Equation \([47](https://arxiv.org/html/2605.20758#A5.E47)\)\.

Formally, we define the goals as a set ofKKcenters\{𝐠k\}k=1K\\\{\\mathbf\{g\}\_\{k\}\\\}\_\{k=1\}^\{K\}\. The compositional static goal reward at state𝐱\\mathbf\{x\}is defined as:

rgoal​\(𝐱\)=∑k=1Kexp⁡\(−‖𝐱−𝐠k‖2σ2\)r\_\{\\text\{goal\}\}\(\\mathbf\{x\}\)=\\sum\_\{k=1\}^\{K\}\\exp\\left\(\-\\frac\{\\\|\\mathbf\{x\}\-\\mathbf\{g\}\_\{k\}\\\|^\{2\}\}\{\\sigma^\{2\}\}\\right\)\(48\)whereσ\\sigmais the spatial decay rate\.

##### Dynamic obstacle rewards\.

We consider the dynamic agent avoidance task, where obstacles follow randomly generated linear trajectories\.

Formally, we define a set ofKKdynamic obstacles\. The generated robot trajectory is𝝉=\{𝐱1,…,𝐱H\}\\bm\{\\tau\}=\\\{\\mathbf\{x\}\_\{1\},\\dots,\\mathbf\{x\}\_\{H\}\\\}over a planning horizonHH\. Each obstaclekkmoves over a horizon ofNNsteps \(N≤HN\\leq H; i\.e\., its position𝐜k​\(t\)\\mathbf\{c\}\_\{k\}\(t\)updates for the firstNNsteps\) and remains stationary thereafter \(i\.e\.,𝐜k​\(t\)=𝐜k​\(N\)\\mathbf\{c\}\_\{k\}\(t\)=\\mathbf\{c\}\_\{k\}\(N\)fort\>Nt\>N\)\. The compositional reward for the entire trajectory𝝉\\bm\{\\tau\}is defined as:

rdynamic​\(𝝉\)=−∑t=1H∑k=1Kexp⁡\(−‖𝐱t−𝐜k​\(t\)‖2σ2\)r\_\{\\text\{dynamic\}\}\(\\bm\{\\tau\}\)=\-\\sum\_\{t=1\}^\{H\}\\sum\_\{k=1\}^\{K\}\\exp\\left\(\-\\frac\{\\\|\\mathbf\{x\}\_\{t\}\-\\mathbf\{c\}\_\{k\}\(t\)\\\|^\{2\}\}\{\\sigma^\{2\}\}\\right\)\(49\)where𝐱t\\mathbf\{x\}\_\{t\}denotes the agent state at time steptt, andσ\\sigmais the spatial decay rate\. We set the trajectory horizonH=48H=48and the obstacle trajectory horizonN=3N=3\.

##### Trajectory smoothness rewards\.

We set the trajectory smoothness cost\(urain2023se3\)\. Formally, given a trajectory𝝉=\{𝐱0,…,𝐱T\}\\bm\{\\tau\}=\\\{\\mathbf\{x\}\_\{0\},\\dots,\\mathbf\{x\}\_\{T\}\\\}, the smoothness reward is defined as:

rsmooth​\(𝝉\)=−∑t=0T−1‖𝐱t\+1−𝐱t‖2r\_\{\\text\{smooth\}\}\(\\bm\{\\tau\}\)=\-\\sum\_\{t=0\}^\{T\-1\}\\\|\\mathbf\{x\}\_\{t\+1\}\-\\mathbf\{x\}\_\{t\}\\\|^\{2\}\(50\)as the minimization of the relative distance between the neighbour points in the trajectory\. This reward can be thought as a spring making all the point in the trajectory be attracted between each other\.

#### E\.5\.2Hyperparameter

We provide the detailed hyperparameters used for the Maze2D experiments in Table[7](https://arxiv.org/html/2605.20758#A5.T7)\.

Table 7:Hyperparameters forgcarg^\{\\text\{car\}\}used in Maze2D\.
#### E\.5\.3Experimental results

We report all experimental results in Table[2](https://arxiv.org/html/2605.20758#S6.T2), the key observations are:

1. 1\.Inference\-time guidance applied to pre\-trained generative policy models is prone to off\-manifold drift, leading to poor prior preservation \(e\.g\., failing to reach the end point\) and constraint violations \(e\.g\., colliding with obstacles or maze walls\), as shown forgcov\-Gg^\{\\text\{cov\-G\}\}in Figure[16](https://arxiv.org/html/2605.20758#A5.F16)\.
2. 2\.PCGrad cannot recover from off\-manifold drift\.
3. 3\.GLASS\-FKS performs well on robot planning tasks\.
4. 4\.gcarg^\{\\text\{car\}\}consistently corrects off\-manifold drift across all settings, improving success rate and reducing constraint violations\.
5. 5\.MPPI is a strong planning baseline that refines generated paths from the base CFM model to satisfy runtime constraints\. Sometimes, it still suffers from prior preservation issues under compositional constraints\. Whengcarg^\{\\text\{car\}\}is applied on top of MPPI, MPPI \+gcarg^\{\\text\{car\}\}achieves the best overall performance, correcting off\-manifold drift while satisfying constraints\.

We further evaluate robustness to clutter by varying the number of static obstacles from 2 to 6 \(Figure[15](https://arxiv.org/html/2605.20758#A5.F15)\)\.gcov\-Gg^\{\\text\{cov\-G\}\}degrades sharply as the environment becomes more cluttered, with its success rate collapsing to12%12\\%at 6 obstacles, whereasgcarg^\{\\text\{car\}\}maintains34%34\\%in the same setting\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/fig/scalability_curves.png)Figure 15:Robustness to clustered environments on Maze2D\. We evaluate the safety and success rates by varying the number of static obstacles from 2 to 6\. Ourgcarg^\{\\text\{car\}\}\(red\) exhibits robustness even with 6 obstacles, whereasgcov\-Gg^\{\\text\{cov\-G\}\}suffers degradation\.![Refer to caption](https://arxiv.org/html/2605.20758v1/x14.png)Figure 16:Visualisation of guided trajectory generation under compositional constraints in Maze2D\.\(1\) static obstacles, \(2\) goal reachability, \(3\) dynamic obstacles, and \(4\) hybrid composition\. Observe thatgcov\-Gg^\{\\text\{cov\-G\}\}produces erratic, off\-manifold trajectories, whilegcarg^\{\\text\{car\}\}yields smooth, feasible trajectories\.![Refer to caption](https://arxiv.org/html/2605.20758v1/x15.png)Figure 17:Visual comparison of guided generation under increasing environmental complexity\.We scale the number of static obstacles to evaluate the solver’s ability to handle dense constraints\. As shown, traditional planning baselines like MPPI struggle with high\-dimensional constraint landscapes, often failing to find feasible paths\. In contrast,gcarg^\{\\text\{car\}\}effectively navigates through dense clutter, generating smooth, collision\-free trajectories that match the quality of those in simpler environments, highlighting its superior constraint\-satisfaction capabilities\.

### E\.6Generative decision\-making as policies

![Refer to caption](https://arxiv.org/html/2605.20758v1/x16.png)Figure 18:Architecture of the Base CFM Policy\.The conditioning context includes the goal state \(e\.g\., target placement coordinates\), the point cloud observation, and the robot state\. The observation \(4096 colored points\) is compressed via an encoder using a PointNet backbone trained from scratch\. The model outputs action chunks of horizonTTgenerated from noise\.#### E\.6\.1Base CFM policy

We implement a base Conditional Flow Matching \(CFM\) policy by adapting the PointFlowMatch architecture\(chisari2024learning\)for goal\-conditioned manipulation\. Specifically, we incorporate explicit goal conditioning, and improve success rates\. The detailed architecture is illustrated in Figure[18](https://arxiv.org/html/2605.20758#A5.F18)\.

Conditioning\.The policy is conditioned on a multimodal context vectorcc, constructed as follows:

1. 1\.Observation: Raw 3D point clouds \(N=4096N=4096\) with RGB features are fused from multi\-view cameras \(left, right, and gripper\)\. These are processed by a PointNet backbone to extract a dense feature vector\.
2. 2\.Proprio State: A vector containing the robot’s joint angles and gripper status\.
3. 3\.Goal: The 3D coordinates representing the target placement location \(e\.g\., the stacking position\)\.

These components are concatenated to form the conditioningcc\.

Output\.The model predicts action chunks of horizonTT\. The generative component is a Conditional 1D U\-Net that predicts the time\-dependent velocity fieldvθ​\(xt,t∣c\)v\_\{\\theta\}\(x\_\{t\},t\\mid c\)\. Here, the flow statext∈ℝT×7x\_\{t\}\\in\\mathbb\{R\}^\{T\\times 7\}represents the flattened action chunk sequence \(translation, rotation, and gripper action\)\. Trajectories are generated by integrating the learned ODE from a standard Gaussian distribution att=0t=0to the target action distribution att=1t=1\.

Training Objective\.Given an expert action chunk𝐀gt∈ℝT×7\\mathbf\{A\}\_\{\\text\{gt\}\}\\in\\mathbb\{R\}^\{T\\times 7\}\(denoted asx1x\_\{1\}\) and a random initial samplex0∼𝒩​\(0,I\)x\_\{0\}\\sim\\mathcal\{N\}\(0,I\), we samplet∼𝒰​\(0,1\)t\\sim\\mathcal\{U\}\(0,1\)and interpolatext=\(1−t\)​x0\+t​x1x\_\{t\}=\(1\-t\)x\_\{0\}\+tx\_\{1\}\. The model is trained to regress the target velocityvgt=x1−x0v^\{\\text\{gt\}\}=x\_\{1\}\-x\_\{0\}via mean\-squared error\.

Dataset and evaluation\.For each task \(PickCube and StackCube\), we collect 100 expert demonstrations to train the base CFM model\. The trained policy achieves 100% success rate on both training and test sets \(100 episodes with unseen random seeds\), confirming strong generalization\. The base CFM policy is lightweight yet sufficient to complete the manipulation tasks without constraints\. Our focus is on whether inference\-time guidance can satisfy runtime constraints while preserving the base flow prior and staying on the data manifold\.

Table 8:Comparison on ManiSkill2 StackCube and PickCub tasks\. Compositional reward settings: \(1\) static obstacle: two random static obstacles; \(2\) hybrid composition: two random static obstacles and trajectory smoothness\. Metrics include Inference Time \(ms/sample\), Violation \(mean constraint violations \#\), Success \(success rate %\), and Steps \(\#\)\. Results are averaged over 100 samples with conflict thresholdτ=0\.20\\tau=0\.20\. Note that we do not use inpainting, which allows us to better observe the capability of inference\-time alignment methods in preserving the base model prior\. For GLASS\-FKS, we useK=8K=8particles with a convergence coefficientρ=0\.95\\rho=0\.95, involving 24 internal steps per inference\. Thegcarg^\{\\text\{car\}\}method requires an online training period of20\.4±0\.420\.4\\pm 0\.4min for 8 training steps prior to inference\.Note:Bold textindicates the best performance\. Rows withgray backgroundsindicate methods that utilize ourgcarg^\{\\text\{car\}\}for conflict correction\.Purple superscriptsshow the performance change ofgcarg^\{\\text\{car\}\}overgcov\-Gg^\{\\text\{cov\-G\}\}, andteal superscriptsshow the change of PCGrad overgcov\-Gg^\{\\text\{cov\-G\}\}, where↑\\uparrowdenotes improvement and↓\\downarrowdenotes degradation\. For all metrics except Steps, we report the mean \(top row\) and standard deviation across 5 random seeds \(bottom row\)\.

#### E\.6\.2Experimental results

Table[8](https://arxiv.org/html/2605.20758#A5.T8)presents the comparative results under constrained settings\. In the challenging StackCube task,gcarg^\{\\text\{car\}\}reduces the violation rate from1\.21\.2to0\.10\.1\(static obstacles\) and from1\.81\.8to0\.40\.4\(hybrid composition\), while boosting the success rate from12%12\\%to72%72\\%and from9%9\\%to61%61\\%respectively\. In the PickCube task,gcarg^\{\\text\{car\}\}achieves perfect safety \(0\.00\.0violations\) in static environments and boosts the success rate from46%46\\%to94%94\\%in the static goal setting\. Notably, PCGrad degrades performance relative togcov\-Gg^\{\\text\{cov\-G\}\}across both tasks \(e\.g\., StackCube success drops to0%0\\%\), confirming that gradient surgery cannot handle high\-precision manipulation tasks under compositional constraints\.gcarg^\{\\text\{car\}\}achieves these gains efficiently, consistently converging in just 8 steps\.

Key observations are:

1. 1\.Adding inference\-time guidance to pre\-trained generative policy models is prone to OOD, and often fails to finish tasks, e\.g\., the failure shown in Figure[19](https://arxiv.org/html/2605.20758#A5.F19)ofgcov\-Gg^\{\\text\{cov\-G\}\}\.
2. 2\.PCGrad cannot recover from off\-manifold drift\.
3. 3\.GLASS\-FKS generally performs well, but struggles in high\-precision tasks such as StackCube \(i\.e\., stably and precisely placing one cube onto another\), due to its high transition variance\. On tasks such as conditional generation \(e\.g\., decision\-making tasks\), as long as the condition often appears in the dataset, GLASS\-FKS performs well because it is easier to obtain an accurate estimation ofgtg\_\{t\}\.
4. 4\.gcarg^\{\\text\{car\}\}corrects off\-manifold drift \(success rate↑\\uparrow\) and shows decreased violation rate\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x17.png)Figure 19:Visualization on ManiSkill2 PickCube task with conflict thresholdτ=0\.20\\tau=0\.20\. OOD: the trajectory leaves the data manifold, producing physically incoherent motions \(e\.g\., erratic spinning or tangled paths\); Fail:the trajectory stays on the manifold but fails the task \(e\.g\., does not reach the goal\)\.

### E\.7Text\-guided image manipulation

#### E\.7\.1Experimental details

To ensure fair comparisons, all text\-guided image manipulation experiments, including the training of the online guidance network and the inference latency measurements, were conducted on a dedicated local workstation\. The hardware specifications include an AMD EPYC 7543 Processor and a single NVIDIA RTX A5000 GPU \(24GB VRAM\)\. All algorithms and neural network architectures were implemented using the PyTorch framework with CUDA acceleration\.

Table 9:Hyperparameter ofgcarg^\{\\text\{car\}\}in image editing\.
#### E\.7\.2Inference\-time constraints

In our text\-to\-image generation experiment, we adopted the pipeline presented inliu2023flowgrad, utilizing the generative prior fromliu2023flow\. The terminal reward function is:

r​\(x1\)=CLIP​\(x1,T\),r\(x\_\{1\}\)=\\mathrm\{CLIP\}\(x\_\{1\},T\),\(51\)
Baseline configurations were aligned with those reported inliu2023flowgrad, and the complete results presented in Table[10](https://arxiv.org/html/2605.20758#A5.T10)reflect the same experimental conditions\. For quantitative comparison, we used the CelebA\-HQ dataset, randomly sampling 1,000 images, which were manipulated based on standard single text guidance \(i\.e\.,sad,angry,happy,smiling,curly hair\) and composed text guidance \(i\.e\.,sad \+ angry,sad \+ happy,sad \+ curly hair\)\.

#### E\.7\.3Evaluation metric

We evaluate our method using six quantitative metrics across two categories:

Text\-Image Alignment:We first use \(1\)CLIP\(Higher is better\), which measures basic text\-image alignment by calculating image and text embeddings separately and measuring the distance between them\. Because fine\-grained misalignments are often left undetected by standard multi\-modal models like CLIP\(singh2023divide\), we also report \(2\)BLIP\-ITM\(li2022blip\)\(Higher is better\)\. BLIP\-ITM utilizes cross\-attention between a ViT image encoder and a BERT\-base text processor to act as a strict binary classifier, predicting whether an image and prompt are an exact match\. Finally, we use \(3\)VQAScore\(vqascore\)\(Higher is better\) to evaluate complex compositional reasoning by reframing image evaluation as a visual question answering task using LLaVA\-1\.5\.

Image Quality and Preservation:To evaluate visual fidelity, we use \(1\)CLIP\-IQA\(wang2022exploring\)\(Higher is better\) to assess intrinsic visual quality and penalize blurry or artifact\-heavy generations\. To evaluate how well the original inputs are maintained, we report \(2\)LPIPS\(Lower is better\) for the preservation of overall image content, and \(3\)ID\(Higher is better\) for the preservation of subject identity\.

#### E\.7\.4Experimental results

Table 10:Comparison of methods on image quality metrics \(LPIPS, CLIP\-IQA, and ID\), text\-image alignment metrics \(CLIP, BLIP\-ITM, and VQAScore\), and computational efficiency for text\-guided face manipulation on CelebA\-HQ\. To demonstrate the imbalance issue in multi\-objective optimization \(i\.e\., optimizing for two prompts simultaneously\), we report the text\-image alignment metrics separately for the first prompt \(P1P\_\{1\}\), the second prompt \(P2P\_\{2\}\), and their average \(Avg\)\. A significant discrepancy betweenP1P\_\{1\}andP2P\_\{2\}indicates a severe optimization imbalance\. We report results separately for*composed text guidance*\(i\.e\.,sad \+ angry,sad \+ happy,sad \+ curly hair\)\.Note:Bold textindicates the best performance\. Rows withgray backgroundsindicate methods that use ourgcarg^\{\\text\{car\}\}for conflict correction\.Purple superscriptsshow the performance change ofgcarg^\{\\text\{car\}\}overgcov\-Gg^\{\\text\{cov\-G\}\}, andteal superscriptsshow the change of PCGrad overgcov\-Gg^\{\\text\{cov\-G\}\}, where↑\\uparrowdenotes improvement and↓\\downarrowdenotes degradation\. For all metrics, we report the mean \(top row\) and standard deviation \(bottom row\) across 5 random seeds\.

Table[10](https://arxiv.org/html/2605.20758#A5.T10)presents a detailed quantitative comparison\. We observe that compositional constraints significantly increase the difficulty of maintaining manifold adherence across all baselines, as conflicting objectives lead to higher LPIPS and lower ID scores\. FlowGrad achieves LPIPS of0\.2030\.203and ID of0\.6770\.677under composed prompts, and struggles to balance multiple objectives simultaneously \(i\.e\., the uneven text\-image alignment betweenP1P\_\{1\}andP2P\_\{2\}\)\.gcarg^\{\\text\{car\}\}outperformsgcov\-Gg^\{\\text\{cov\-G\}\}by a large margin in identity preservation \(0\.6810\.681vs\.0\.5430\.543\), proving its ability to rectify off\-manifold drift where approximate guidance fails\. Furthermore, the large gap betweenP1P\_\{1\}andP2P\_\{2\}scores for PCGrad \(BLIP\-ITM:0\.6500\.650vs\.0\.3370\.337; VQAScore:0\.7850\.785vs\.0\.3710\.371\) means that gradient surgery fails to balance multiple constraints, whereasgcarg^\{\\text\{car\}\}achieves consistent alignment across both prompts\. Visualization results are provided in Figure[20](https://arxiv.org/html/2605.20758#A5.F20)\.

Overall, the key observations are:

- •𝐠𝐜𝐨𝐯−𝐆\\mathbf\{g^\{cov\-G\}\}is prone to off\-manifold drift and has hallucinated generation; also it is too sensitive to the guidance scale\.
- •FlowGradfails to balance multiple constraints\.
- •PCGradattempts to resolve conflicts via gradient surgery but fails to balance multiple constraints, leaving some targets unfulfilled \(e\.g\., failing to generate an “angry” expression\)\. Furthermore, it cannot recover from off\-manifold drift\.
- •GLASS\-FKSfails to preserve the reference image\. This is largely due to the high variance of sampling \(i\.e\., ODE\-based transition sampling\) given a limited number of particles\. Specifically, estimatinggtg\_\{t\}requires samples from regions whereere^\{r\}is significantly higher than average, i\.e\., images already closely resembling the reference, which is unlikely to be achieved with a limited particle budget\.
- •Our𝐠car\\mathbf\{g^\{\\text\{car\}\}\}achieves superior compositional reward alignment across multiple prompts, corrects off\-manifold drift, and eliminates the hallucinated visual artifacts observed ingcov\-Gg^\{\\text\{cov\-G\}\}\.

##### More about GLASS\-FKS

In text\-guided image manipulation, GLASS\-FKS fails to preserve the reference image\. This is largely due to the high variance of sampling \(i\.e\., ODE\-based transition sampling\) given a limited number of particles\. Specifically, estimatinggtg\_\{t\}requires samples from regions whereere^\{r\}is significantly higher than average, i\.e\., images already closely resembling the reference, which is unlikely to be achieved with a limited particle budget\. This failure mode is similar to Monte Carlo guidance infeng2025guidance, where more advanced sampling techniques help GLASS\-FKS preserve more prior than Monte Carlo guidance but do not fully resolve the issue\. We also note that GLASS\-FKS’s original evaluation uses a stronger base model \(FLUX\) and a richer reward composition \(CLIP, Pick, HPSv2, ImageReward\), whereas our setting uses a Rectified Flow trained on CelebA\-HQ with CLIP score as the sole reward\.Richer reward composition likely provides more informative evaluation for particle steering, which helps explain the strong performance reported in the original paper\.

![Refer to caption](https://arxiv.org/html/2605.20758v1/x18.png)Figure 20:Additional visualization of text\-guided image manipulation\.This figure complements Figure[6](https://arxiv.org/html/2605.20758#S6.F6)by showing further results ofgcarg^\{\\text\{car\}\}on the CelebA\-HQ dataset under various composed text prompts\.