Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time $T_1$ by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires $T_1\to\infty$. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as $T_1$ diverges; from a practical viewpoint, a large $T_1$ increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time $T_1$. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the $L^2$ loss on the score approximation, which is the quantity minimized in practice.

翻译：基于分数的生成模型是从复杂数据分布中采样的强大工具。其核心思想在于：(i) 通过向数据添加噪声执行时间长度为 $T_1$ 的前向过程，(ii) 估计其分数函数，以及 (iii) 利用该估计运行反向过程。由于反向过程以前向过程的平稳分布作为初始化，现有分析范式要求 $T_1\to\infty$。然而这存在问题：从理论角度看，对于给定精度的分数近似，收敛保证会随 $T_1$ 发散而失效；从实践角度看，较大的 $T_1$ 会增加计算开销并导致误差传播。本文通过考虑流行的预测-校正方案变体来解决这一问题：在前向过程结束后，我们首先通过非精确朗之万动力学估计最终分布，随后逆转该过程。我们的关键技术贡献在于提供了仅需前向过程运行至固定有限时间 $T_1$ 的收敛保证。所提出的边界对输入维度和目标分布的次高斯范数展现出温和的对数依赖，对数据的假设要求极低，且仅需控制实践中最小化的分数近似 $L^2$ 损失。