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This paper analyzes classifier-free guidance (CFG) using a numerical analysis perspective, identifying that CFG destabilizes diffusion and flow-matching samplers at high guidance scales. The authors propose a "terminal-fitted repair" for CFG, which modifies a single coefficient in the guidance direction. Empirically, this repair acts as a high-guidance stabilizer, reducing residual amplification and saturation, and achieving significant FID wins on CIFAR-10 and Stable Diffusion benchmarks without increasing computational cost.

The paper addresses the following bottlenecks in classifier-free guidance (CFG):

Oversaturation and Destabilization: At large guidance scales, CFG causes oversaturation and destabilization in diffusion and flow-matching samplers.
Solver Artifacts: The observed oversaturation and norm inflation at large guidance scales can be a solver-induced component, appearing even when the exact guided flow is the reference.
Loss of Fitted Operator Property: Guidance re-stiffens the discriminative subspace to an anomalous exponent, causing the deterministic DDIM step to no longer be fitted, leading to divergence of the guided residual as the noise scale approaches zero.

A layer classification of the guided terminal exponent, identifying how guidance re-stiffens the discriminative subspace.
A guided clock barrier with three ordered step-size thresholds, explaining oversaturation as an endpoint and providing a parameter-free prediction for the terminal edge of limited-interval guidance.
A "spectrum-free fitted repair" for CFG, which is a first-order consistent, sign-preserving, and terminal-exact modification that removes the omin-divergent blow-up and is first-order accurate.
An empirical validation demonstrating that the proposed fitted CFG acts as a high-guidance stabilizer, improving residual amplification and clipping, and achieving FID wins on learned CIFAR-10 checkpoints and Stable Diffusion 1.5.

The proposed method, termed "fitted CFG," modifies the guidance update in diffusion and flow-matching samplers. The core idea is to replace the standard CFG update with a "guided fitted step" that rectifies the anomalous exponent introduced by guidance.

Key Equations: The standard CFG update for the velocity field ηω is given by: ηω = ηc + w γ where ηc is the conditional denoiser, ηu is the unconditional denoiser, w is the guidance scale, and γ = (ηu - ηc)/σ is the discriminative direction.

The proposed fitted CFG update for the state xn+1 is: Xn+1 = Dc + r (xn - Dc) + (r1+w - r)(Du – Dc) where Dc is the conditional denoiser, Du is the unconditional denoiser, r = ση+1/ση is the step size ratio, and (r1+w - r) is the modified coefficient for the guidance direction. This modification ensures terminal exactness and removes the divergence observed in standard CFG.

Inference: The fitted CFG is implemented as a drop-in replacement for the standard CFG coefficient, requiring no additional network function evaluations (NFEs).

Architecture: Not stated in the paper.
Key Equations:
- Guided flow: dx/dσ = ηω = ηc + w γ, where γ = (ηu - ηc)/σ.
- Fitted CFG update: Xn+1 = Dc + r (xn - Dc) + (r1+w - r)(Du – Dc).
Loss/Objective: The paper focuses on the properties of the sampler's discretization and its agreement with the exact guided flow, rather than a specific loss function for training.
Optimization: Not stated in the paper.
Augmentation: Not stated in the paper.
Data Handling: The analysis is performed on a commuting class-subset Gaussian model for theoretical insights and on learned CIFAR-10 EDM checkpoints and Stable Diffusion 1.5 for empirical validation.
Implementation Details: The fitted CFG is implemented as a single-line modification to the CFG update, using the same two denoiser evaluations per step.

Datasets:
- Commuting class-subset Gaussian model (for theoretical analysis).
- Learned CIFAR-10 EDM checkpoints.
- Stable Diffusion 1.5.
Metrics:
- Residual amplification (amp med, amp p95).
- Clipping fraction (clip A, clip p95).
- FID (Fréchet Inception Distance).
- KID (Kernel Inception Distance).
- Target accuracy (classifier target-label accuracy).
- Saturation (pixel saturation).
- CLIP image-text alignment.
Baselines:
- Classifier-Free Guidance (CFG).
- Tuned limited-interval guidance.
- Terminal limited-interval guidance.
Numeric Results:
- On CIFAR-10 at high guidance (w=6.5, 8), fitted CFG significantly reduces residual amplification and clipping compared to CFG. For example, at w=8, N=8, class-0 amp p95 falls from 13.35 (CFG) to 1.71 (fitted CFG).
- Fitted CFG achieves the best FID in all tested blocks on CIFAR-10 (e.g., 25.22 vs 32.44 for CFG on 5k (A)).
- A nine-cell grid experiment shows fitted CFG wins FID in 9/9 cells and improves clipping p95 in 9/9 cells.
- On Stable Diffusion 1.5, fitted CFG sharply reduces saturation (p95 0.009 vs 0.134 for CFG at g=12) while preserving CLIP alignment better than terminal guidance shutdown.
Runtime/Training Cost: Fitted CFG requires zero additional NFE compared to standard CFG.

The paper does not claim an unqualified image-quality win, as KID sometimes favors CFG.
The learned diagnostics are consistent with the model ordering but do not definitively verify the sharp threshold law.
The 50k run is described as one hard cell, not a comprehensive large-sample result.
Fitted CFG is not uniformly a better integrator of the vanilla CFG field on learned checkpoints.
The class-subset hypothesis is a boundary condition; where the class is wider than the marginal, the continuous guided flow is unstable and no solver can repair it.
The theoretical results are based on a commuting class-subset Gaussian model, and extending these to learned geometry is future work.

Method evidence: "The repair is one coefficient. Rectifying the anomalous exponent 1 + w by a gauge change and pushing the o-clock step through it yields a one-coefficient modification of CFG that touches nothing else in the sampler-only the coefficient on the guidance direction Du - De changes: CFG: Xn+1 = Dc + r (xn - Dc) + w(r − 1) (Du – Dc), fitted: Xn+1 = Dc + r (xn – Dc) + (r1+w – r)(Du – Dc), with r = ση+1/ση." (Section: 2 THE GUIDED FLOW AND ITS TERMINAL LAYERS)
Result evidence: "On learned CIFAR-10 checkpoints-and, as a cross-domain smoke test, on Stable Diffusion 1.5 DDIM-it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation and gives 9/9 point-FID wins over CFG on the tested grid, while in the hard-cell blocks its classifier-proxy target accuracy stays close to CFG and terminal guidance shutdown loses much more." (Section: ABSTRACT)
Limitation evidence: "What is neither. We do not claim (i) an unqualified image-quality win—KID stays lowest for CFG at 5k and 50k, a split tied to CFG's oversaturation without proven causation; (ii) that CIFAR verifies the sharp threshold law-the learned diagnostics are only consistent with the model ordering; or (iii) a 50k KID win or many-cell large-sample result—the 50k run is one hard cell." (Section: 10)

Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance

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