Minimax Optimality of the Probability Flow ODE for Diffusion Models
Changxiao Cai, Gen Li
TL;DR
The paper establishes the first end-to-end minimax-type guarantees for deterministic probability flow ODE samplers in diffusion models under mild target-distribution assumptions. By introducing a smooth regularized score estimator that jointly controls the $L^2$ score error and the mean Jacobian error, and by performing a refined analysis of the discretized probability flow ODE, the authors show that the resulting sampler attains the near-minimax total-variation rate $n^{- rac{eta}{d+2eta}}$ up to logarithmic factors (with suitable choices of $K$ and regularization) for subgaussian targets with Hölder smooth densities of order $eta \leq 2$. The framework does not require density lower bounds or Lipschitz/smooth scores, and it accounts for all sources of error in end-to-end sampling, including discretization and score estimation. This work thus validates the practical effectiveness of probability flow ODE samplers and positions them as statistically near-optimal methods for a broad class of high-dimensional distributions, with potential extensions to higher-order solvers and broader distribution classes.
Abstract
Score-based diffusion models have become a foundational paradigm for modern generative modeling, demonstrating exceptional capability in generating samples from complex high-dimensional distributions. Despite the dominant adoption of probability flow ODE-based samplers in practice due to their superior sampling efficiency and precision, rigorous statistical guarantees for these methods have remained elusive in the literature. This work develops the first end-to-end theoretical framework for deterministic ODE-based samplers that establishes near-minimax optimal guarantees under mild assumptions on target data distributions. Specifically, focusing on subgaussian distributions with $β$-Hölder smooth densities for $β\leq 2$, we propose a smooth regularized score estimator that simultaneously controls both the $L^2$ score error and the associated mean Jacobian error. Leveraging this estimator within a refined convergence analysis of the ODE-based sampling process, we demonstrate that the resulting sampler achieves the minimax rate in total variation distance, modulo logarithmic factors. Notably, our theory comprehensively accounts for all sources of error in the sampling process and does not require strong structural conditions such as density lower bounds or Lipschitz/smooth scores on target distributions, thereby covering a broad range of practical data distributions.