arXiv:2510.17608stat.MLcs.LG2025-10被引 8

在弱对数凹性下,为概率流ODE提供非渐近误差界。

Non-asymptotic error bounds for probability flow ODEs under weak log-concavity

  • 基于弱对数凹性和得分函数的Lipschitz连续性,推导收敛界。
  • 首次涵盖初始化误差、得分近似误差与离散化影响的完整误差分析。
  • 适用于高斯混合等非对数凹分布,适合关注理论保障的研究者。

基于得分的生成建模通过概率流常微分方程(ODE)在诸多实际场景中表现优异。然而,现有收敛性保证大多依赖于目标分布的严格正则性假设,如强对数凹性或有界支撑。本文在较弱假设——弱对数凹性与得分函数的Lipschitz连续性下,建立了概率流ODE在2-Wasserstein距离下的非渐近收敛界。该框架可处理非对数凹分布(如高斯混合模型),并显式考虑初始化误差、得分近似误差及基于指数积分器的离散化效应。解决了扩散生成建模中的关键理论难题,将收敛理论拓展至更现实的数据分布与实用的ODE求解器。我们为采样算法的效率与正确性提供了具体理论保证,补足了扩散模型的实证成功与严谨理论之间的鸿沟。此外,明确的收敛速率对选择超参数(如离散化步长)具有实际指导意义。

原文摘要 · Abstract (English)

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution -- such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

生成模型概率流收敛性分析

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