# Rethinking Classifier-Free Guidance in On-Policy Diffusion Distillation

> Negative Branch Asymmetry

- **Source:** [arXiv](https://arxiv.org/abs/2607.24731)
- **Published:** 2026-07-29
- **Permalink:** https://picx.dev/p/dQLCvu
- **Whiteboard:** https://picx.dev/p/dQLCvu/image

## Summary

## Summary (Overview)

- On-policy distillation (OPD) of diffusion models under classifier-free guidance (CFG) is shown to be **under-identified at the branch level**: matching only the CFG-composed velocity allows positive- and negative-branch errors to cancel each other, creating infinitely many degenerate solutions.
- This ambiguity becomes harmful when the teacher’s negative branch contains **privileged information** unavailable to the student, leading to a failure mode called **Negative Branch Asymmetry (NBA)**: the positive-branch error decreases while the negative-branch error increases during training.
- The authors propose **Positive–Direction Matching (PDM)**, a branch-aware objective that separately constrains the positive prediction and the CFG conditional direction, eliminating cross-branch error compensation.
- PDM is applied to **dense-to-sparse video control** (pose, depth, scribble), where naive guided matching exhibits severe sensitivity to inference guidance scale, while PDM and a simpler baseline (Independent Branch Matching, IBM) remain stable and improve control fidelity.
- Key insight: NBA is diagnosed by **excess degradation relative to the teacher or a branch-aware student** under guidance-scale shifts, not by an absolute CFG-free performance drop.

## Introduction and Theoretical Foundation

Classifier-free guidance (CFG) (Ho & Salimans, 2022) is the standard inference mechanism for modern diffusion models. The denoising velocity is a composition of two branch predictions:

$$
\tilde{v} = \gamma v^+ + (1 - \gamma) v^-
$$

where $v^+$ is the positive branch (under target condition) and $v^-$ is the negative branch (under null or negative condition), with guidance scale $\gamma > 1$.

On-policy distillation (OPD) trains a student model by querying a teacher along the student’s own denoising trajectory. The natural extension of OPD to CFG-based models is to match the composed velocities:

$$
\mathcal{L}_{\text{naive}} = \mathbb{E}_{x_t^S \sim p_\theta, t} \left[ w(t) \| \gamma e^+ + (1-\gamma) e^- \|_2^2 \right]
$$

where $e^+ = v_T^+ - v_S^+$ and $e^- = v_T^- - v_S^-$ are the positive- and negative-branch errors.

This objective is **under-identified** at the branch level: the condition $\gamma e^+ + (1-\gamma) e^- = 0$ admits infinitely many non-zero pairs $(e^+, e^-)$, allowing the two errors to compensate without changing the composed prediction. The paper investigates when this ambiguity becomes harmful.

## Methodology

### Negative Branch Asymmetry (NBA)

Under **privileged negative conditioning** (e.g., teacher’s negative branch retains reference images or dense control signals unavailable to the student), the teacher’s negative target cannot be directly reproduced by the student. The composed objective then induces **antagonistic branch-error dynamics**: reducing $\|e^+\|_2$ while increasing $\|e^-\|_2$. This failure mode is named **NBA**.

### PDM: Positive–Direction Matching

To prevent cross-branch error compensation, PDM supervises the predictions **before CFG composition**. Define the CFG conditional direction:

$$
d_M = v_M^+ - v_M^-
$$

The PDM loss is:

$$
\ell_{\text{PDM}} = \|e^+\|_2^2 + \lambda \| d_T - d_S \|_2^2 = \|e^+\|_2^2 + \lambda \| e^+ - e^- \|_2^2
$$

where $\lambda > 0$ controls direction-matching strength. Zero PDM loss implies $e^+ = e^- = 0$, eliminating compensation freedom.

### Baseline: Independent Branch Matching (IBM)

IBM directly matches both branches with per-branch weights:

$$
\ell_{\text{IBM}} = \gamma^2 \|e^+\|_2^2 + (\gamma-1)^2 \|e^-\|_2^2
$$

Like PDM, it removes cross-branch interaction but uses a different parameterization.

### Efficient Supervision for Video

For dense-to-sparse video control, the authors supervise only the first $K$ student-visited states, reducing cost while maintaining performance:

$$
\mathcal{L}_{\text{OPD}}^{(K)} = \mathbb{E} \left[ \frac{1}{K} \sum_{k=0}^{K-1} w(t_k) \ell(x_{t_k}^S, t_k) \right]
$$

## Empirical Validation / Results

### Branch-Error Dynamics (Figure 2)

Two image-domain distillation settings are contrasted:

- **Shared negative conditioning** (text-rendering): both branch errors decrease jointly under all objectives (benign regime).
- **Privileged negative conditioning** (reference-conditioned): naive matching reduces $\|e^+\|_2$ while increasing $\|e^-\|_2$ — NBA. PDM suppresses this negative-branch degradation.

### Intrinsic vs. Distillation-Induced CFG Sensitivity

**Table 1** (text-rendering) shows that naive and PDM students closely track the teacher’s intrinsic CFG response — no excess sensitivity introduced.

| Method | γ=1 | γ=2 | γ=3 | γ=4 | γ=4.5 |
|--------|-----|-----|-----|-----|-------|
| Teacher | 75.24 | 92.91 | 94.16 | 94.43 | 94.09 |
| OPD (Naive) | 73.87 | 92.72 | 94.52 | 94.43 | 94.11 |
| OPD (PDM) | 74.48 | 93.73 | 93.99 | 94.02 | 94.38 |

In **reference-conditioned distillation** (Figure 3), naive matching departs from the teacher at off-scale γ, producing visual distortion and style drift, while PDM remains stable.

### Main Results on Dense-to-Sparse Video Control

**Table 2** reports results at γ=5 across three modalities. PDM provides the most consistent improvements over naive OPD, SFT, and off-policy distillation.

| Pose-Control | MPJPE↓ All/Key | PCK@0.2↑ All/Key | PCK@0.1↑ All/Key | FID↓ | FVD↓ |
|--------------|----------------|-------------------|-------------------|------|------|
| Teacher | 3.03/2.82 | 96.02/96.46 | 91.15/92.23 | 14.13 | 56.01 |
| OPD (Naive) | 4.43/3.14 | 91.84/95.25 | 80.28/89.35 | 13.49 | 65.12 |
| OPD (PDM) | **4.13/2.83** | **92.98/96.12** | **82.48/91.03** | 13.52 | **62.77** |

Similar patterns hold for depth and scribble modalities.

### Guidance-Scale Generalization

**Table 3** shows that naive OPD degrades sharply when inference γ differs from training γ=5 (e.g., MPJPE All 8.98 at γ=1), while PDM and IBM remain stable.

| Method | γ=5 | γ=3 | γ=1 |
|--------|-----|-----|-----|
| Naive (MPJPE↓) | 4.62 | 4.94 | 8.98 |
| PDM (MPJPE↓) | **4.27** | **4.17** | **4.48** |
| Naive (FVD↓) | 67.44 | 134.09 | 507.70 |
| PDM (FVD↓) | 58.65 | 55.85 | 60.97 |

### Ablation Studies

**Table 4** (direction-matching weight λ): λ=1 provides best control fidelity; larger λ improves distributional quality but weakens control.

**Table 5** (supervision horizon K): K=8 gives strongest control fidelity while being substantially cheaper than full-trajectory (K=50) supervision.

## Theoretical and Practical Implications

- **Theoretical**: The paper identifies a previously unrecognized failure mode in CFG-based OPD — NBA — and shows that branch-level ambiguity is not always harmful but becomes problematic under privileged negative conditioning. This provides a principled understanding of when and why naive guided matching fails.
- **Practical**: PDM is a simple, drop-in replacement for the naive OPD loss that requires no architectural change. It improves robustness to inference guidance-scale shifts and control fidelity in dense-to-sparse video control. The efficient supervision strategy (supervising only early steps) makes OPD practical for video models.
- **Broader impact**: The findings apply to any OPD setting where the teacher’s CFG schema retains privileged information in the negative branch (e.g., reference images, dense control signals, auxiliary modalities). Branch-aware objectives should be preferred in such cases.

## Conclusion

The paper demonstrates that CFG-composed OPD is under-identified at the branch level, leading to NBA under privileged negative conditioning. PDM resolves this by separately supervising the positive prediction and CFG conditional direction. Experiments on text-rendering, reference-conditioned distillation, and dense-to-sparse video control confirm that PDM improves guidance-scale robustness and knowledge transfer.

**Limitations**: The relative behavior of PDM and IBM is not theoretically characterized; PDM’s advantage is empirical. Future work may extend NBA analysis to other guidance mechanisms and teacher–student conditioning asymmetries.

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