为随机插值生成模型的数值方法提供了有限时间收敛保障。
Finite-Time Convergence Analysis of ODE-based Generative Models for Stochastic Interpolants
- 基于常微分方程数值积分，分析了前向欧拉与海恩法的收敛性。
- 给出了总变差距离下的有限时间误差界，证明方法有效。
- 提出优化调度方案提升计算效率，适合生成模型研究者。
随机插值提供了一种稳健的框架，可连续地将样本从任意数据分布转换到目标分布，在生成建模中具有重要潜力。尽管如此，针对实际数值算法的有限时间收敛性保证仍缺乏系统研究。本文针对由随机插值导出的常微分方程（ODE）的数值实现，开展有限时间收敛性分析。具体而言，我们为两种广泛应用的数值积分器——一阶前向欧拉法和二阶海恩法——建立了总变差距离下的新有限时间误差界。此外，对特定随机插值构造的迭代复杂度分析，进一步提出了优化调度策略以提升计算效率。理论结果通过数值实验验证，充分支持所推导的误差界与复杂度分析。
原文摘要 · Abstract (English)
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence guarantees for practical numerical schemes remain largely unexplored. In this work, we address the finite-time convergence analysis of numerical implementations for ordinary differential equations (ODEs) derived from stochastic interpolants. Specifically, we establish novel finite-time error bounds in total variation distance for two widely used numerical integrators: the first-order forward Euler method and the second-order Heun's method. Furthermore, our analysis on the iteration complexity of specific stochastic interpolant constructions provides optimized schedules to enhance computational efficiency. Our theoretical findings are corroborated by numerical experiments, which validate the derived error bounds and complexity analyses.
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