---
title: "Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.07665v1"
permalink: "https://feed.astralor.com/items/itm_e27f11bfb0515872fb65d506"
published: "2026-07-08T17:21:51.000Z"
score: 73
channel: "ai"
tags: ["Classifier-Free Guidance","DDIM","Stable Diffusion","Diffusion Models"]
---

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

## AI commentary

> This paper is worth following, especially for researchers optimizing diffusion model samplers. It uses elegant mathematics to reveal the root cause of CFG failure at high guidance and provides a zero-cost fix. While not a panacea, the repair directly improves stability and FID in applications demanding strong conditioning, such as precise text-to-image generation.

## At a glance

Classifier-free guidance (CFG) is the standard technique to strengthen class conditioning in diffusion models, yet at high guidance it suffers from oversaturation and instability. Through an asymptotic-preserving numerical analysis, this paper shows that DDIM is the unique fitted operator for the unguided terminal layer, but guidance re-stiffens the discriminative subspace to exponent 1+w, breaking the fit. The authors propose a zero-extra-NFE repair: replacing CFG's direction term with r^(1+w)-r, removing divergence at sigma_min→0. On CIFAR-10 and Stable Diffusion 1.5, it stabilizes high guidance and improves FID, though not universally.

## Key points

- Oversaturation of CFG at high guidance stems from DDIM no longer fitting the guided operator, causing residual divergence.
- Asymptotic analysis reveals three ordered step-size thresholds in the guidance clock, and one-step oversaturation is a solver artifact.
- A one-coefficient zero-extra-NFE repair replaces CFG's direction with r^(1+w)-r, removing sigma_min divergence.
- On CIFAR-10 and Stable Diffusion 1.5, the repair achieves 9/9 point-FID wins over CFG on tested grids, but is not a universal quality improvement.

## Full text

Classifier-free guidance (CFG) is the standard way to strengthen class-conditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes, symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer, exact on the final small-sigma stretch of sampling, we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1+w. DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as sigma_min goes to zero. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint: a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair: replace CFG's w(r-1) by r^(1+w)-r on the guidance direction. On the calibration model's discriminative crossover, this removes CFG's sigma_min-divergent blow-up and is first-order accurate against the exact guided flow as sigma_min goes to zero. 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, gives 9/9 point-FID wins over CFG on the tested grid, and preserves classifier-proxy target accuracy in the hard-cell blocks. We report the limits alongside: it is not a universal image-quality win, and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.

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