Analytic Distribution of Classifier-Free Guidance for Schedule Design

arXiv:2607.19725v1 Announce Type: new Abstract: 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^omega q_0^{1-omega}. 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 omega(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. On Stable Diffusion~1.5, DG-CFG improves generation and yields a stronger diversity--fidelity trade-off across guidance strengths, with especially clear gains when strong guidance causes saturation and quality degradation in constant and heuristic schedules. Across NFE budgets, DG-CFG reaches fixed image-quality targets with fewer sampling steps, reducing the sampling cost needed to achieve target metrics.

Source: arXiv cs.LG | 2026-07-23