PathGuide: Dynamic Classifier-Free Guidance via On-Policy Transport Alignment

Avishag Nevo; Tamir Hazan

cs.LGAug 29, 2026

PathGuide: Dynamic Classifier-Free Guidance via On-Policy Transport Alignment

Authors: Avishag Nevo, Tamir Hazan

Organizations: Technion - Israel Institute of Technology

Abstract

While modern generative models excel at modeling complex data, precise inference-time control in conditional generation remains a critical challenge. Classifier-free guidance (CFG) is a primary mechanism for such control, yet it is typically treated as a static tuning parameter. In flow-based models, however, the guidance scale fundamentally dictates the velocity field and the resulting probability path, making guidance selection a dynamic path-optimization problem. We introduce PathGuide, a framework that reformulates scalar CFG selection as an on-policy transport problem. Leveraging the weak form of the continuity equation, we derive a selection criterion with a direct path-correctness interpretation: we prove that if the guided field is weakly equivalent to the exact conditional field along the generated rollout, the sampler's path coincides with the target conditional law. For scalar CFG, this criterion yields a strictly quadratic local objective with an efficient, closed-form selector for each solver interval. PathGuide enables optimal guidance scales to be computed and used online during generation or fitted offline as a reusable piecewise-constant schedule. We validate our method on low-resolution image manifolds and controlled settings across various continuous-time flow constructions, demonstrating that this transport-based selector improves path alignment and sample fidelity over both fixed and state-of-the-art adaptive guidance baselines.

Figures & tables

Figure 1: Comparison of rollout trajectories against the exact conditional marginal

p_{t_{i}}(\cdot\mid y)

. (a) Online samples at the start (

t_{0}

) and near the end (

t_{T-1}

) of a

T=20

step rollout, together with the quadratic local objective and the selected value at the midpoint time

t_{T/2}

. (b) Quantitative discrepancy measured by

\mathrm{MMD}

\mathrm{KL}

, and

\mathrm{W}_{2}

. The true-velocity rollout serves as a discretization reference, since finite-step integration can deviate from the analytic marginal path even when the exact conditional velocity is used. While the plain conditional learned field (

\omega\equiv 1

) accumulates drift, our practical selector effectively calibrates the rollout to track the true path more closely.

		KL $\downarrow$		$\mathrm{W}_{2}\downarrow$		MMD $\downarrow$
Variant	Method	$T=200$	$T=500$	$T=200$	$T=500$	$T=200$	$T=500$
RF ( Liu et al., 2022 )	Plain CFG	$0.0044$	$0.0044$	$0.0060$	$0.0060$	$0.0006$	$0.0006$
	CFG-Zero ∗	$0.0046$	$0.0045$	$0.0064$	$0.0062$	$0.0006$	$0.0006$
	CFG-MP	$6.2232$	$1.8740$	$9.5319$	$2.9592$	$0.6234$	$0.2987$
	R-CFG++	$0.0191$	$0.0185$	$0.0248$	$0.0243$	$0.0020$	$0.0019$
	PathGuide (Ours)	$0.0039$	$0.0038$	$0.0056$	$0.0055$	$0.0005$	$0.0005$

Table 1: Generation quality against guidance baselines for all flow variants. Each metric is reported for

T=200

and

T=500

; lower is better. All methods are evaluated using

2^{14}

generated samples per seed. Our method is fitted once using

M=N=2^{14}

, independently of the evaluation samples. Each row reports the best configuration selected from that method’s hyperparameter sweep, using the selection rule described in Appendix E.5 . Full sweeps including standard deviations are in Appendix H .

RF ( Liu et al., 2022 )		OT ( Lipman et al., 2023 )
Method	FID $\downarrow$	Method	FID $\downarrow$
Plain CFG	$15.802$	Plain CFG	$7.578$
CFG-Zero ∗	$26.534$	CFG-Zero ∗	$11.327$
CFG-MP	$30.951$	CFG-MP	$24.475$
R-CFG++	$16.979$	R-CFG++	$15.038$
PathGuide (Ours)	$\mathbf{15.699}$	PathGuide (Ours)	$\mathbf{7.375}$

Table 2: MNIST image-generation quality for RF and OT flow variants. We report FID computed with torch-fidelity==0.4.0 . Lower is better. All methods are evaluated over three evaluation seeds using

2^{13}

generated samples per seed, with

M=2^{11}

N=2^{12}

, and

T=50

. Each row reports the configuration with the lowest mean FID within that method’s sweep. Full sweeps are reported in Table 16 . On matched inference latents, the paired PathGuide-minus-tuned-CFG FID difference is

-0.103

(95% CI

[-0.186,-0.021]

) for RF and

-0.175

(

[-0.250,-0.101]

) for OT.

Figure 2: Reusing a schedule fitted at grid

T

on coarser inference grids

T^{-}\leq T

. Each curve fixes the fitting grid and varies the deployed inference grid.

\mathrm{degradation}=100\cdot(\mathrm{coarse\ result}-\mathrm{full\ result})/\mathrm{full\ result},

each result is averaged over the fitting seeds. Shaded regions show one standard deviation over the inference seeds. Lower is better for all metrics.

Figure 3: Schedule diagnostic for the VP flow, class $0$ . The background shows the analytic local-objective landscape over $(t_{i},\omega)$ . Curves show fitted schedules for different solver grids together with the oracle schedule. The practical schedules preserve the same global shape across resolutions and better follow low-objective regions as the grid is refined.

Figure 4: Monte Carlo scaling on the diagonal

M=N

. Shaded regions show one standard deviation over fitting seeds. Downstream metrics and schedule distance stabilize at moderate budgets, while runtime continues to grow.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

x_{0}^{(n)}\sim p_{0},\qquad n=1,\ldots,N.

(54)

Appendix

Algorithm 1 Practical interval-wise guidance selection

Variant	Conditioning	Path sample $x_{t}$	Velocity label
RF [ Liu et al., 2022 ]	$x_{1},y$	$(1-t)x_{0}+tx_{1}$	$x_{1}-x_{0}$
OT [ Lipman et al., 2023 ]	$x_{1},y$	$\bigl(1-(1-\sigma_{\min})t\bigr)x_{0}+tx_{1}$	$x_{1}-(1-\sigma_{\min})x_{0}$
I-CFM [ Tong et al., 2024 ]	$z=(x_{0},x_{1}),y$	$(1-t)x_{0}+tx_{1}+\sigma_{\mathrm{icfm}}\varepsilon$	$x_{1}-x_{0}$
VP [ Song et al., 2021b ]	$x_{1},y$	$\bar{\alpha}_{t}x_{1}+\sqrt{1-\bar{\alpha}_{t}^{2}}\,x_{0}$	$\partial_{t}x_{t}$

Appendix

Table 3: Conditional paths used for experimental validation. The time variable is always generation time

t\in[0,1]

, from prior to endpoint.

Item	Value / convention
Endpoint law	Eq. ( 81 )
Source law	$\mathcal{N}(0,\mathbf{I})$
Dimension	$d=2$
Class schedules	fitted separately for each $y\in\mathcal{Y}$
Main solver	fixed-grid Euler
Solver grid	$0=t_{0}<t_{1}<\cdots<t_{T}=1$

Appendix

Table 4: Reproducibility parameters for the Gaussian-mixture experimental validation.

Method	Sweep grid
Plain CFG [ Zheng et al., 2023 ]	$\omega\in\{1.0,1.25,1.5,2.0,3.0\}$
CFG-Zero ∗ [ Fan et al., 2025 ]	$\omega\in\{1.0,1.25,1.5,2.0,3.0\}$ , $K_{\mathrm{zero}}\in\{1,2\}$
CFG-MP [ Cai et al., 2026 ]	$\omega\in\{1.0,1.25,1.5,2.0,3.0\}$ , $K_{\mathrm{proj}}\in\{1,2\}$ (GM); $K_{\mathrm{proj}}=3$ (MNIST)
Rectified-CFG++ [ Saini et al., 2025 ]	$\lambda_{\max}\in\{0.5,1.0,2.0,3.0\}$ , $\sigma\in\{0.0,0.05\}$

Appendix

Table 5: Baseline sweep grids.

Figure 5: Sensitivity analysis of

L

, the number of test functions used to estimate the selector. This graph aggregates over all flow variants, classes, fitting and inference seeds. Shaded regions show one standard deviation over fitting seeds. Downstream metrics and schedule variance stabilize at moderate

L

$(L_{\rm lin},L_{\rm quad},L_{\rm cubic})$	Flow	Paired $\Delta$ FID	95% CI	Mean schedule diff.
$(2048,1024,1024)$	RF	$-0.0011$	$[-0.0047,0.0038]$	$0.0033\pm 0.0038$
$(2048,1024,1024)$	OT	$-0.0220$	$[-0.0321,-0.0133]$	$0.0045\pm 0.0057$
$(1024,2048,1024)$	RF	$-0.2499$	$[-0.3860,-0.0626]$	$0.0103\pm 0.0142$
$(1024,2048,1024)$	OT	$-0.5663$	$[-1.1632,0.1931]$	$0.0123\pm 0.0156$
$(1024,1024,2048)$	RF	$-0.2544$	$[-0.3980,-0.0702]$	$0.0100\pm 0.0131$
$(1024,1024,2048)$	OT	$-0.5924$	$[-1.1851,0.1669]$	$0.0144\pm 0.0176$

Appendix

Table 6: Equal-budget probe allocations on MNIST. Paired

\Delta

FID is variant minus default; mean schedule difference is the mean absolute scale difference over the

50

solver intervals.

Ablation	Variant	$\mathrm{KL}\downarrow$	$\mathrm{W}_{2}\downarrow$	$\mathrm{MMD}\downarrow$	Time $\downarrow$
reuse	online	0.0059 $\pm$ 0.0041	0.0075 $\pm$ 0.0048	0.0009 $\pm$ 0.0005	13.58 $\pm$ 5.67
	offline, same $T$	0.0060 $\pm$ 0.0042	0.0076 $\pm$ 0.0049	0.0009 $\pm$ 0.0005	13.58 $\pm$ 5.66
weights	posterior	0.0059 $\pm$ 0.0041	0.0074 $\pm$ 0.0048	0.0008 $\pm$ 0.0005	13.51 $\pm$ 5.72
	uniform	0.0060 $\pm$ 0.0042	0.0075 $\pm$ 0.0049	0.0009 $\pm$ 0.0005	13.42 $\pm$ 5.61
stability	floor off	0.0065 $\pm$ 0.0048	0.0079 $\pm$ 0.0053	0.0009 $\pm$ 0.0006	13.52 $\pm$ 5.73
	floor on, $\eta=10^{-2}$	0.0059 $\pm$ 0.0041	0.0075 $\pm$ 0.0048	0.0009 $\pm$ 0.0005	13.57 $\pm$ 5.60

Appendix

Table 7: Aggregate ablation results over all flow variants. For each ablation variant and metric, we pool all raw measurements across the four flow backbones (RF, I-CFM, OT, and VP), fitting seeds, evaluation seeds, and classes. Each cell reports mean

\pm

sample standard deviation. Lower is better.

Figure 6: Full $(M,N)$ sweep for downstream metrics. Each cell shows the percentage gap from the best observed budget for that metric; lighter is better and the star marks the best cell. The heatmaps show that increasing $M$ gives the main early improvement, while very large $N$ has smaller marginal effect once $M$ is moderate.

Figure 7: Online path discrepancy to the exact conditional marginal

p_{t_{i}}(\cdot\mid y)

, measured along

\hat{p}_{t_{i}}(\cdot\mid y;\vec{\omega}_{i})

\mathrm{MMD}\text{-}\mathrm{RBF}^{2}

\mathrm{KL}

, and

\mathrm{W}_{2}

. Lower is better. Panel (a) shows VP, and panel (b) shows I-CFM. The true-velocity rollout provides a discretization reference. In both cases, the practical/oracle selector stays closer to this reference than the learned conditional rollout

\omega\equiv 1

, indicating improved path alignment.

Flow	MMD	$W_{2}$	KL	Mean error	Cov. error
RF	$0.57\,[0.49,0.65]$	$0.59\,[0.52,0.65]$	$0.55\,[0.48,0.62]$	$0.61\,[0.47,0.74]$	$0.47\,[0.40,0.54]$
I-CFM	$0.66\,[0.56,0.77]$	$0.65\,[0.55,0.74]$	$0.64\,[0.53,0.75]$	$0.62\,[0.47,0.76]$	$0.58\,[0.48,0.68]$
OT	$0.50\,[0.38,0.61]$	$0.55\,[0.46,0.65]$	$0.56\,[0.45,0.66]$	$0.59\,[0.47,0.71]$	$0.41\,[0.30,0.53]$
VP	$0.46\,[0.23,0.67]$	$0.50\,[0.28,0.70]$	$0.54\,[0.33,0.72]$	$0.16\,[-0.13,0.45]$	$0.44\,[0.26,0.60]$

Appendix

Table 8: Correlation between the held-out finite residual and the resulting path discrepancy, at fixed time and fixed rollout state. Mean Spearman correlation [95% CI].

Flow	Pointwise projection: $W_{2}^{2}$ / KL / MMD 2	PathGuide: $W_{2}^{2}$ / KL / MMD 2
RF	$.0592{\pm}.0034$ / $.0444{\pm}.0050$ / $.00532{\pm}.00071$	$.0135{\pm}.0056$ / $.0109{\pm}.0040$ / $.00180{\pm}.00065$
I-CFM	$.0580{\pm}.0034$ / $.0440{\pm}.0045$ / $.00539{\pm}.00069$	$.0130{\pm}.0055$ / $.0108{\pm}.0039$ / $.00178{\pm}.00065$
OT	$.0441{\pm}.0214$ / $.0329{\pm}.0178$ / $.00433{\pm}.00134$	$.0069{\pm}.0046$ / $.0053{\pm}.0028$ / $.00100{\pm}.00040$
VP	$.0056{\pm}.0031$ / $.0040{\pm}.0022$ / $.00066{\pm}.00019$	$.0058{\pm}.0031$ / $.0040{\pm}.0024$ / $.00067{\pm}.00020$

Appendix

Table 9: Pointwise projection versus the weak-form selector under identical endpoint information. Mean

\pm

std over 18 paired settings; lower is better.

Flow	$\Delta$ KL $T{=}200/500$	$\Delta W_{2}$ $T{=}200/500$	$\Delta$ MMD 2 $T{=}200/500$
RF	$+.03264/+.03446$	$+.04097/+.04310$	$+.001005/+.001104$
I-CFM	$+.01161/+.01247$	$+.01303/+.01397$	$+.000593/+.000650$
OT	$+.00950/+.01033$	$+.01060/+.01148$	$+.000471/+.000528$
VP	$+.000001/+.000006$	$+.000139/+.000145$	$+.000004/+.000004$

Appendix

Table 10: Effect of relaxing the admissible lower bound from

\omega_{\min}=1

0

. Positive value means

\omega_{\min}=1

is better.

Figure 8: Relative degradation under coarser inference. Each entry reports the percentage increase in the metric when a schedule fitted at $T$ is deployed at $T^{-}$ , relative to evaluating the same schedule at its native grid $T$ . Lower is better; $0\%$ corresponds to native-grid evaluation.

Method	Offline fit cost	Sampling cost	Memory	NFE-equivalent
Conditional-only	none	$TC_{\mathrm{cond}}$	$\mathcal{O}(N_{\mathrm{gen}}d)$	$T$
Constant CFG	sweep over $\Omega_{\mathrm{CFG}}$	$T(C_{\mathrm{cond}}+C_{\mathrm{uncond}})$	$\mathcal{O}(N_{\mathrm{gen}}d)$	$2T$
Proposed, offline	Eq. ( 86 )	$T(C_{\mathrm{cond}}+C_{\mathrm{uncond}})$	$\mathcal{O}((N+M)d)$	$2T$
Proposed, online	none	Eq. ( 86 ) during sampling	$\mathcal{O}((N+M)d)$	$2T$ plus estimator work

Appendix

Table 11: Cost accounting for fitted schedules. Offline fitting is amortized across future deployments; deployment uses the stored schedule and therefore has the same NFE-equivalent as standard CFG.

Workload	$\omega\equiv 1$	CFG / stored PathGuide	PathGuide fitting
GMM, $N=M=2^{14}$ , $L=2^{12}$ , $T=500$	10.0 s / 34 MB / $T$	19.7 s / 34 MB / $2T$	137 s / 333 MB / $2T$
MNIST, $N=2^{12}$ , $M=2^{11}$ , $L=2^{12}$ , $T=50$	29.9 s / 315 MB / $T$	59.6 s / 327 MB / $2T$	230 s / 578 MB / $2T$

Appendix

Table 12: Measured cost per workload. Each cell reports wall-clock time / peak memory / backbone NFE. Deployment with a stored schedule matches plain CFG on all three; only fitting is additional, and it is amortized across later deployments.

Experiment family	Runs	Total time	Mean time
Baseline sweeps	163	226.9h	1.4h
Correction-field diagnostics	173	27.8h	9.6min
Online path alignment	11	22.8h	2.1h
Fit-at- $T$ , infer-at-coarser- $T$	136	145.3h	1.1h
Monte Carlo scaling	87	236.5h	2.7h
Schedule heatmap overlays	86	60.3h	42.1min

Appendix

Table 13: Compute resources for the experiment families reported in the paper. All experiments were run on the same workstation. Runtime is reported as accumulated single-GPU time.

Variant	Method	Configuration	KL $\downarrow$	$W_{2}\downarrow$	MMD $\downarrow$
RF	Plain CFG	$\omega=1$	$0.0044_{0.0013}$	$0.0060_{0.0016}$	$0.0006_{0.0001}$
		$\omega=1.25$	$0.0441_{0.0005}$	$0.0657_{0.0025}$	$0.0054_{0.0003}$
		$\omega=1.5$	$0.1298_{0.0015}$	$0.1961_{0.0052}$	$0.0175_{0.0005}$
	CFG-Zero ∗	$\omega=1$ , $K_{\mathrm{zero}}=1$	$0.0046_{0.0013}$	$0.0064_{0.0018}$	$0.0006_{0.0001}$
		$\omega=1$ , $K_{\mathrm{zero}}=2$	$0.0050_{0.0014}$	$0.0072_{0.0019}$	$0.0006_{0.0001}$
		$\omega=1.25$ , $K_{\mathrm{zero}}=2$	$0.0080_{0.0005}$	$0.0127_{0.0013}$	$0.0013_{0.0002}$

Appendix

Table 14: Appendix sweep results for baseline configurations at

T=200

. For each flow variant and method, we report up to three configurations from the hyperparameter sweep. Lower is better for all metrics. All entries are evaluated over 3 inference seeds using

2^{14}

generated samples per seed, and report mean

\pm

standard deviation. The mean is averaged over classes per evaluation seed; the standard deviation is the sample std over the 3 per-seed averages.

Variant	Method	Configuration	KL $\downarrow$	$W_{2}\downarrow$	MMD $\downarrow$
RF	Plain CFG	$\omega=1$	$0.0044_{0.0012}$	$0.0060_{0.0016}$	$0.0006_{0.0001}$
		$\omega=1.25$	$0.0429_{0.0005}$	$0.0645_{0.0026}$	$0.0053_{0.0003}$
		$\omega=1.5$	$0.1277_{0.0016}$	$0.1939_{0.0052}$	$0.0172_{0.0005}$
	CFG-Zero ∗	$\omega=1$ , $K_{\mathrm{zero}}=1$	$0.0045_{0.0013}$	$0.0062_{0.0017}$	$0.0006_{0.0001}$
		$\omega=1$ , $K_{\mathrm{zero}}=2$	$0.0046_{0.0013}$	$0.0064_{0.0017}$	$0.0006_{0.0001}$
		$\omega=1.25$ , $K_{\mathrm{zero}}=2$	$0.0093_{0.0005}$	$0.0150_{0.0014}$	$0.0015_{0.0002}$

Appendix

Table 15: Appendix sweep results for baseline configurations at

T=500

. For each flow variant and method, we report up to three configurations from the hyperparameter sweep. Lower is better for all metrics. All entries are evaluated over 3 inference seeds using

2^{14}

generated samples per seed, and report mean

\pm

standard deviation. The mean is averaged over classes per evaluation seed; the standard deviation is the sample std over the 3 per-seed averages.

Variant	Method	Configuration	FID $\downarrow$
RF	Plain CFG	$\omega=1.0$	$15.802\pm 0.464$
		$\omega=1.25$	$16.342\pm 0.506$
		$\omega=1.5$	$18.567\pm 0.640$
		$\omega=2.0$	$26.028\pm 0.718$
	CFG-Zero ∗	$\omega=1.0$ , $K_{\mathrm{zero}}=1$	$29.627\pm 0.494$
		$\omega=1.25$ , $K_{\mathrm{zero}}=1$	$26.805\pm 0.610$

Appendix

Table 16: Full MNIST image-generation sweep results for RF and OT flow variants at

T=50

. FID is computed with torch-fidelity==0.4.0 ; lower is better. All entries are evaluated over three evaluation seeds using

2^{13}

generated samples per seed, with

M=2^{11}

and

N=2^{12}

. Entries report mean

\pm

sample standard deviation over three evaluation seeds.

Method	$\mathrm{KL}\downarrow$	$\mathrm{MMD}\text{-}\mathrm{RBF}^{2}\downarrow$	$\mathrm{W}_{2}\downarrow$
$\omega\equiv 1$	$0.01261\pm 0.00184$	$0.00271\pm 0.00090$	$0.01755\pm 0.00420$
Practical schedule	$0.01232\pm 0.00216$	$0.00260\pm 0.00073$	$0.01569\pm 0.00340$
Exact field $u$	$0.01147\pm 0.00127$	$0.00258\pm 0.00059$	$0.01516\pm 0.00309$

Appendix

Table 17: Endpoint path discrepancy at

t_{i}\approx 1

for

T=20

, our method is evaluated online where

M=N=2^{11}

against the plain conditional learned field under flow type is VP. We report mean

\pm

sample standard deviation. Lower is better for all metrics.

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