Noise Schedule Design for Diffusion Models: An Optimal Control Perspective

We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process, which evolves according to an ODE, and whose control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which $\tilde{\mathcal{O}}(d/n)$ sampling error is achievable, where $d$ is the data dimension and $n$ is the number of discretization steps. While existing theoretical works also prove that $\tilde{\mathcal{O}}(d/n)$ sampling error bounds are achievable, these results hold for specific noise schedules that differ from those used in practice. Under additional assumptions on the data distribution, we derive a family of noise schedules that recovers the functional forms of schedules widely used in practice. We show that these schedules achieve the $\tilde{\mathcal{O}}(d/n)$ error bound when their parameters are chosen according to our analysis. Our theory offers a principled approach to noise schedule design, yielding explicit functional forms and guidance for selecting their parameters.