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 in Wasserstein distance which require to run the forward process only for a 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：分数生成模型（SGMs）是从复杂数据分布中采样的强大工具。其核心思想是：（i）通过向数据添加噪声运行正向过程至时间$T_1$，（ii）估计其分数函数，以及（iii）利用该估计运行逆向过程。由于逆向过程以正向过程的平稳分布初始化，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，对于给定的分数逼近精度，当$T_1$发散时收敛保证会失效；从实践角度看，大的$T_1$会增加计算成本并导致误差传播。本文通过考虑流行的预测-校正方案的一种变体来解决该问题：运行正向过程后，我们首先通过非精确朗之万动力学估计最终分布，然后反转该过程。我们的关键技术贡献在于提供了Wasserstein距离下的收敛保证，该保证仅要求正向过程运行有限时间$T_1$。我们的界对输入维度和目标分布的次高斯范数表现出温和的对数依赖，对数据假设要求极低，并且仅需控制分数逼近的$L^2$损失——这恰是实践中最小化的量。