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

Jul 16, 2026

Classifier-free guidance (CFG) strengthens class conditioning in diffusion and flow-matching samplers but, at high guidance, oversaturates and destabilizes, motivating step inflation or limited-interval schedules. The work analyzes CFG via asymptotic-preserving numerical analysis, showing that guidance re-stiffens discriminative subspaces to an anomalous terminal exponent 1+w, breaking DDIM’s fitted-operator protection. A one-coefficient, terminal-exact repair modifies only the guidance-direction term, removing σmin-divergent blow-up and improving stability and residual behavior on tested models.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

Abstract

Introduction

Guidance re-stiffens discriminative directions

Two orthogonal sources of guidance pathologies

One-coefficient terminal repair

arXiv :2607 .07665v 1 [ cs .LG] 8 Jul 2026 GUIDANCE BREAKS THE FITTED OPERATOR: A TERMINAL-FITTED REPAIR FOR CLASSIFIER-FREE GUIDANCE Shiheng Zhang University of Washington [shzhang3@uw.edu](shzhang3@uw.edu) ABSTRACT Classifier-free guidance (CFG) is the standard way to strengthen classconditioning 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 [12] that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer—exact on the final, small-σ stretch of sampling—we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1 + w (guided coordinates contract like σ1+w rather than σ) . DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as σmin → 0. 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. This coefficient is the unique spectrum-free terminal-exact one, and it preserves the sign of every analyzed coordinate. On the calibration model’s discriminative crossover it removes CFG’s σmin-divergent blow-up and is first-order accurate against the exact guided flow as σmin → 0—asymptotic-preserving in σmin , though not uniform in w. 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. We report the limits alongside: it is not a universal image-quality win (KID can favor CFG; an interval can win FID in some cells), and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field. 1 INTRODUCTION Diffusion and flow-matching samplers integrate a probability-flow ODE whose velocity field stiffensas the noise scale σ → σmin: on a coordinate normal to the data manifold the exact flow contracts at exponent one, by σn+1/σn per step. A recent fitted-operator analysis of unguided samplers [12] shows that this terminal layer admits a unique fitted (layer-exact) one-step operator: a frozen-field Euler step—Euler with the velocity held at its start-of-step value—reproduces that exact contraction if and only if its integration variable is affine in σ . This singles out the σ-clock step, algebraically identical to deterministic DDIM [11], and makes it asymptotic-preserving (AP) on the exactly solvable calibration models studied there [12]—uniformly accurate as σmin → 0. This paper asks one question of that lens: what survives of DDIM’s fitted-operator protection once we turn on guidance? Guidance re-stiffens the discriminative subspace. Classifier-free guidance (CFG) [7] runs the sampler on an extrapolated denoiser Dw = (1 + w)Dc − wDu , that is, on a second velocity field ηw = ηc + w(ηc − ηu) (the η’s are noise-prediction fields; Section 2) with guidance scale g = µw (σ) λ = −log σ (terminal →) Figure 1: The guided terminal exponent µw (σ) by layer type on the commuting model (schematic; noise decreases to the right) . Guidance leaves the shared-normal exponent at one and freezes tangential directions, but re-stiffens the discriminative directions to 1 + w—the subspace on which DDIM+CFG exits the class of fitted operators. Here a, b are the class and marginal variances along the direction (Section 2) . 1 + w. This is

FAQs

Why does classifier-free guidance become unstable at large guidance scales?

At large guidance, CFG oversaturates and destabilizes the sampling dynamics, manifesting as terminal oversaturation and norm inflation. The paper links part of this behavior to solver discretization breaking the fitted-operator property.

What does the paper mean by DDIM’s “fitted operator” protection?

For unguided sampling, the deterministic DDIM step is identified as the unique fitted one-step operator for the terminal layer, preserving exact contraction under the right affine σ-clock condition. This protection holds asymptotically-preserving on solvable calibration models.

What is the proposed terminal-fitted repair and what problem does it address?

The repair replaces the CFG guidance-direction coefficient w(r−1) with a modified one-coefficient term that corrects the anomalous terminal exponent 1+w. It removes the σmin-divergent residual blow-up on coarse meshes while preserving sign structure and improving stability without extra function evaluations per step.

Jul 16, 2026

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