Closing the Null Space: Guidance-Aware Quantization for Classifier-Free Diffusion

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.

Classifier-free guidance (CFG) is the default mechanism for conditional generation in diffusion models, but the distribution sampled by its deterministic guided dynamics is not captured by the usual product-distribution heuristic $p_0^ωq_0^{1-ω}$. We analyze CFG through the probability flow ODE and derive exact analytic path-integral representations of the induced distributions for both constant and time-dependent guidance. The resulting formulas show that CFG modifies $p_{t_0}$ by an exponential path-integral correction, and that a time-dependent schedule enters this correction through the weight $ω(t)-1$. This characterization explains how score discrepancies accumulate along sampling trajectories and motivates Distribution-Guided CFG (DG-CFG), a schedule that balances timestep contributions while accounting for signal strength and low-noise score-error amplification. A toy model with analytic scores closely verifies the predicted distributions. Across Stable Diffusion1.5, Stable Diffusion2.1, and Stable DiffusionXL, DG-CFG yields a stronger diversity--fidelity trade-off and robustly mitigates the saturation and quality degradation caused by strong constant or heuristic guidance. Complete NFE experiments on Stable Diffusion1.5 and Stable Diffusion~2.1 confirm that these gains persist across sampling budgets, while fixed-quality experiments on both backbones show that DG-CFG reaches target metrics with fewer sampling steps.

Classifier-free guidance (CFG) is usually kept on throughout masked diffusion language model decoding, although its benefit varies across prompts and over time. We study when CFG is actually needed by comparing, from any partial output, the probability of eventual constraint satisfaction under continued CFG and under base-only continuation. Their difference defines the remaining value of guidance. Guidance dependence is highly prompt-specific. Many prompts already succeed without CFG, while for others it provides no measurable benefit or can be harmful. For prompts that do benefit, the gain is often concentrated early. We define the commitment horizon $\astar$ as the earliest point from which switching all remaining decoding to the base model reduces final success by no more than a chosen tolerance. Under the base model, the corresponding success probability, or committor, is a martingale. To first order, CFG's per-step effect is governed by the covariance between the guidance logit direction and the successor committor. This gives a local account of when guidance can help, but it does not by itself locate the horizon. Among prompts with an observed preterminal horizon, $\astar$ is usually early and varies more within constraint families than between them. Freezing each prompt at its own cross-fitted horizon is noninferior to full CFG on all 13 subtasks at the prespecified margin, even while many tokens remain masked. This separates commitment from realization. The boundary also identifies a later region in which higher parallelism adds only a small cost in constraint success, although fluency still degrades with parallel width. For failed trajectories, reopening committed positions improves recovery in both failure modes.