An Elementary Approach to Scheduling in Generative Diffusion Models

Abstract

An elementary approach to characterizing the impact of noise scheduling and time discretization in generative diffusion models is developed. We first utilize the Cram\'er-Rao bound to identify the Gaussian setting as a fundamental performance limit, necessitating its study as a reference. Building on this insight, we consider a simplified model in which the source distribution is a multivariate Gaussian with a given covariance matrix, together with the deterministic reverse sampling process. The explicit closed-form evolution trajectory of the distributions across reverse sampling steps is derived, and consequently, the Kullback-Leibler (KL) divergence between the source distribution and the reverse sampling output is obtained. The effect of the number of time discretization steps on the convergence of this KL divergence is studied via the Euler-Maclaurin expansion. An optimization problem is formulated, and its solution noise schedule is obtained via calculus of variations, shown to follow a tangent law whose coefficient is determined by the eigenvalues of the source covariance matrix. For an alternative scenario, more realistic in practice, where pretrained models have been obtained for some given noise schedules, the KL divergence also provides a measure to compare different time discretization strategies in reverse sampling. Experiments across different datasets and pretrained models demonstrate that the time discretization strategy selected by our approach consistently outperforms baseline and search-based strategies, particularly when the budget on the number of function evaluations is very tight.