C$^2$FG: Control Classifier-Free Guidance via Score Discrepancy Analysis

Jiayang Gao; Tianyi Zheng; Jiayang Zou; Fengxiang Yang; Shice Liu; Luyao Fan; Zheyu Zhang; Hao Zhang

cs.LGMar 9, 2026

C $^2$ FG: Control Classifier-Free Guidance via Score Discrepancy Analysis

Authors: Jiayang Gao, Tianyi Zheng, Jiayang Zou, Fengxiang Yang, Shice Liu, Luyao Fan, Zheyu Zhang, Hao Zhang, +4 more

Organizations: Shanghai Jiao Tong University, Shanghai, China · vivo BlueImage Lab, vivo Mobile Communication Co., Ltd., China

Abstract

Classifier-Free Guidance (CFG) is a cornerstone of modern conditional diffusion models, yet its reliance on the fixed or heuristic dynamic guidance weight is predominantly empirical and overlooks the inherent dynamics of the diffusion process. In this paper, we provide a rigorous theoretical analysis of the Classifier-Free Guidance. Specifically, we establish strict upper bounds on the score discrepancy between conditional and unconditional distributions at different timesteps based on the diffusion process. This finding explains the limitations of fixed-weight strategies and establishes a principled foundation for time-dependent guidance. Motivated by this insight, we introduce \textbf{Control Classifier-Free Guidance (C $^2$ FG)}, a novel, training-free, and plug-in method that aligns the guidance strength with the diffusion dynamics via an exponential decay control function. Extensive experiments demonstrate that C $^2$ FG is effective and broadly applicable across diverse generative tasks, while also exhibiting orthogonality to existing strategies.

Figures & tables

Panel of Figure 1 — Figure 1: Following [ 47 ] , (a) and (b) present results for $t\geq t_{0}>0$ . (a) shows that the MSE of conditional score and unconditional score can be bounded by a function which tends to 0 when $t\to+\infty$ ; (b) shows that the normalized cosine similarity between the two vectors decreases over reverse time, indicating that their directions gradually diverge in the reasoning process.

Figure 2: Noise to Image Process of C 2 FG : Dynamic guidance weight $\omega(t)$ adaptively balances conditional and unconditional outputs at each timestep t during generation, guided by theoretical bounds on the score function. Moreover, we can choose to add the method of [ 25 ] , where we fix the $\omega(t)=1$ at the beginning of generation or when $t$ tends to 0.

Figure 3: A two-dimensional distribution featuring two classes represented by gray and orange regions. Approximately 99% of the probability mass is inside the shown contours. (a) Ground truth samples from the orange class. (b) EDM2 ( $\omega=1$ ) produces some outliers. (c) $\beta$ -CFG ( $\alpha=\beta=2,\omega=1$ ) produces more outliers. (d) C 2 FG ( $\omega_{0}=1,\lambda=0.6$ ) generates fewer outliers and better matches the target distribution.

ImageNet
Model ( 256 $\times$ 256 ), 50k samples, 250 inference timesteps	FID $\downarrow$	IS $\uparrow$	sFID $\downarrow$	Prec $\uparrow$	Rec $\uparrow$
DiT-XL/2 ( $\omega=1.5$ , ODE sampler)	2.29	276.8	4.6	0.83	0.57
DiT-XL/2 (Rectified Diffusion, $\omega=1.5$ , ODE)	2.13	/	/	0.83	0.58
DiT-XL/2 + Ours ( $\omega_{0}=1,\lambda=\ln 2$ , ODE)	2.07	291.5	4.6	0.83	0.59
SiT-XL/2 (REPA)( $\omega=1.35$ , SDE)	1.80	284.0	4.5	0.81	0.61
SiT-XL/2 (REPA) + Ours ( $\omega_{0}=1,\lambda=1$ , SDE)	1.51	315.0	4.6	0.80	0.62

Table 1: Quantitative Comparison. Comparison of different evaluation metrics on Class-Conditional ImageNet datasets with different diffusion architectures and inference steps.

Latent Space (MS-COCO)
Model	FID $\downarrow$
U-ViT( $\omega=2$ )	5.37
U-ViT+Ours ( $\omega_{0}=2,\lambda=0.2$ )	5.28
Model	CLIP $\uparrow$
SD15( $\omega=5$ )	31.8
SD15+Ours ( $\omega_{0}=5,\lambda=0.2$ )	31.9

Table 2: Evaluation of C 2 FG on MS-COCO (U-ViT) in latent space and ImageNet-64 (EDM2) in pixel space.

Figure 4: Qualitative Comparison. Qualitative comparison on Class-Conditional ImageNet datasets with different architectures and samplers. The sampler used and the number of inference steps are indicated in parentheses.

ImageNet( 256 $\times$ 256 )
Model 50 inference timesteps	FID $\downarrow$	sFID $\downarrow$	Prec $\uparrow$	Rec $\uparrow$
SiT-XL/2 (REPA)( $\omega=1.8$ , SDE)	3.36	4.5	0.86	0.54
SiT-XL/2 (REPA) + Ours ( $\omega_{0}=1.7,\lambda=0.15$ , SDE)	3.20	4.6	0.86	0.54
SiT-XL/2 (REPA)( $\omega=1.8$ , ODE)	3.46	4.5	0.86	0.54
SiT-XL/2 (REPA)+Ours ( $\omega_{0}=1.7,\lambda=0.15$ , ODE)	3.25	4.4	0.86	0.55
Model 20 inference timesteps

Table 3: Ablation Comparison. Comparison of different evaluation metrics on Class-Conditional ImageNet datasets with different architectures and fewer timesteps.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Comparison between reverse diffusion process by CFG and C 2 FG. Our C 2 FG guidance weight

\omega(t)

is a time-decay function.

Figure 6: Heatmaps of the logarithmic ratio ( $\log_{2}$ ) between conditional and unconditional predictions at selected timesteps. White indicates no difference (ratio=1), while red and blue highlight amplification and suppression, respectively. Stronger colors denote larger deviations between the two predictions.

Figure 7: Impact of the initial schedule weight

\omega_{0}

on IS–FID performance (with fixed

\lambda=1.0

, 250 inference steps).

Panel of Figure 8 — Figure 8: Comparison of IS–FID performance under different hyperparameter settings on DiT-XL/2 and ImageNet-256.

ImageNet( 256 $\times$ 256 )
Model 25 inference timesteps	FID $\downarrow$	IS $\uparrow$	Prec $\uparrow$	Rec $\uparrow$
DiT-XL/2 (baseline, $\omega=1.5$ )	11.88	192.13	0.7176	0.6651
DiT-XL/2+ $\beta$ -CFG ( $\omega=1.5,a=b=2.0$ )	8.42	254.42	0.8038	0.6026
DiT-XL/2 + RAAG ( $\omega_{\max}=18.0,\alpha=12.0$ )	21.57	444.57	0.8360	0.2636
DiT-XL/2 + Ours ( $\omega_{0}=1.0,\lambda=1.0$ )	7.70	294.03	0.7994	0.6357

Appendix

Table 4: Performance comparison between our method and existing adaptive CFG approaches on ImageNet-256.

Panel of Figure 9 — Figure 9: (a) demonstrates the impact of initial weight $\omega_{0}$ ; (b) illustrates how different $\lambda$ values affect the decay profile.

ImageNet(256 $\times$ 256), 50k samples, 250 SDE inference timesteps
Model	FID $\downarrow$
REPA (Fixed CFG = 1.35)	1.80
REPA ( $\lambda=\log 2$ )	1.68
REPA ( $\lambda=1(\log e)$ )	1.51
REPA ( $\lambda=\log 3$ )	1.58

Appendix

Table 5: Comparison between the different effect of

\lambda

, fixing

\omega_{0}=1.0

ImageNet(64 $\times$ 64)
Model	FID $\downarrow$	$\textnormal{FD}_{\textnormal{DINOv2}}\downarrow$
EDM2-S-autoguidance ( $\omega=1.7$ )	1.044	56.3
EDM2-S-autoguidance+Ours ( $\omega_{0}=0.9,\lambda=0.7$ )	1.028	52.7
ImageNet (512 $\times$ 512), 10k samples
EDM2-S-autoguidance ( $\omega=1.4$ )	5.27	121.2
EDM2-S-autoguidance+Ours ( $\omega_{0}=0.9,\lambda=0.5$ )	5.15	116.7

Appendix

Table 6: We evaluated conditional image generation on ImageNet with EDM2 and Autoguidance.

Figure 10: Comparison between results during the denoising process of C 2 FG and Baseline.

Compare	Fixed SD1.5,MS-COCO				Fixed SiT,ImageNet		Compare T2I models (CLIP $\uparrow$ )
Method	CFG	CFG++	$\beta$ -CFG	C 2 FG (4,1)	FDG	C 2 FG (1.7,0.15)	Models	Flux (1.5,1)	SD3 (5,1)
FID(10k) $\downarrow$	19.32	18.87	16.74	16.71	6.15	3.20	CFG	31.4	31.4
CLIP(10k) $\uparrow$	32.0	32.0	31.7	32.0	–	–	C 2 FG	31.5	31.5

Appendix

Table 7: Additional Comparisons. Left: Comparisons with dynamic guidance methods on SD1.5 (MS-COCO) and SiT (ImageNet). Right: Results on modern T2I models (Flux, SD3).

Figure 11: Visual Comparison. C 2 FG produces images that better align with the text prompt than standard CFG, yielding more faithful details, consistent with the quantitative gains in Table 7 .

Panel of Figure 12 — Figure 12: Additional results for C 2 FG.

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Classifier-Free Guidance Diffusion Sampling Diffusion-Based Image Generation Diffusion Timestep Scheduling

Open paper AskPDF arXiv

Classifier-free guidance (CFG) is the default mechanism for conditional generation in diffusion models, but the distribution sampled by its deterministic guided dynamics is not captured by the usual product-distribution heuristic

p_0^ωq_0^{1-ω}

. We analyze CFG through the probability flow ODE and derive exact analytic path-integral representations of the induced distributions for both constant and time-dependent guidance. The resulting formulas show that CFG modifies

p_{t_0}

by an exponential path-integral correction, and that a time-dependent schedule enters this correction through the weight

ω(t)-1

. This characterization explains how score discrepancies accumulate along sampling trajectories and motivates Distribution-Guided CFG (DG-CFG), a schedule that balances timestep contributions while accounting for signal strength and low-noise score-error amplification. A toy model with analytic scores closely verifies the predicted distributions. Across Stable Diffusion~~1.5, Stable Diffusion~~2.1, and Stable DiffusionXL, DG-CFG yields a stronger diversity--fidelity trade-off and robustly mitigates the saturation and quality degradation caused by strong constant or heuristic guidance. Complete NFE experiments on Stable Diffusion1.5 and Stable Diffusion~2.1 confirm that these gains persist across sampling budgets, while fixed-quality experiments on both backbones show that DG-CFG reaches target metrics with fewer sampling steps.

Abstract

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