Analytic Distribution of Classifier-Free Guidance for Schedule Design

• URL: http://arxiv.org/abs/2607.19725v1
• Date: Wed, 22 Jul 2026 03:53:07 GMT
• Title: Analytic Distribution of Classifier-Free Guidance for Schedule Design
• Abstract summary: We analyze CFG through the flow ODE and derive exact analytic path-integral representations of the induced distributions for both constant and time-dependent guidance.<n>This characterization explains how score discrepancies accumulate along sampling trajectories and motivates Distribution-GuidedG (DG-CFG)<n>On Stable Diffusion1.5, DG-CFG improves generation and yields a stronger diversity--fidelity trade-off across guidance strengths.
• Score: 3.2049543530625098
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $p_0^ωq_0^{1-ω}$. 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. 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.

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