GUIDANCE BREAKS THE FITTED OPERATOR: A TERMINAL-FITTED REPAIR FOR CLASSIFIER-FREE GUIDANCE

Classifier-free guidance (CFG) is the industry standard for conditioning diffusion models, but it is known to suffer from oversaturation and instability at high guidance scales. This paper investigates whether these failures are inherent to the guided flow or if they are artifacts of the numerical solver used to integrate the probability-flow ODE. By applying a numerical-analysis lens, the author asks if the deterministic DDIM step—which is the optimal 'fitted operator' for unguided diffusion—remains effective when guidance is introduced.

The author models the guided terminal layer using a commuting Gaussian framework. They demonstrate that guidance re-stiffens the discriminative subspace (directions that separate the class from the marginal) to an anomalous exponent of 1+w. Because the standard DDIM update is not designed for this re-stiffened geometry, it fails to remain a 'fitted operator,' leading to residual divergence as the noise scale approaches zero. The paper derives a 'guided clock barrier' that identifies specific step-size thresholds where the sampler begins to reflect across the class manifold or amplify residuals.

The analysis reveals that the oversaturation observed in high-guidance CFG is largely a solver artifact rather than a property of the continuous guided law. To fix this, the author proposes a 'guided fitted step' that modifies only a single coefficient in the CFG update: replacing w(r-1) with r^(1+w)-r. This repair requires no additional network function evaluations (NFE) and is first-order accurate against the exact guided flow. Empirical tests on CIFAR-10 and Stable Diffusion 1.5 demonstrate that this modification acts as a high-guidance stabilizer, cutting residual amplification and consistently outperforming vanilla CFG in FID metrics on tested grids.

This work provides a theoretical foundation for why high-guidance sampling often breaks down and offers a simple, drop-in fix that improves stability without increasing computational cost. It shifts the focus from tuning guidance intervals heuristically to understanding the underlying numerical requirements of the guided probability-flow ODE.