Abstract
Recent studies suggest utilizing generative models instead of traditional auto-regressive algorithms for time series forecasting (TSF) tasks. These non-auto-regressive approaches involving different generative methods, including GAN, Diffusion, and Flow Matching for time series, have empirically demonstrated high-quality generation capability and accuracy. However, we still lack an appropriate understanding of how it processes approximation and generalization. This paper presents the first theoretical framework from the perspective of flow-based generative models to relieve the knowledge of limitations. In particular, we provide our insights with strict guarantees from three perspectives: $\textbf{Approximation}$, $\textbf{Generalization}$ and $\textbf{Efficiency}$. In detail, our analysis achieves the contributions as follows: $\bullet$ By assuming a general data model, the fitting of the flow-based generative models is confirmed to converge to arbitrary error under the universal approximation of Diffusion Transformer (DiT). $\bullet$ Introducing a polynomial-based regularization for flow matching, the generalization error thus be bounded since the generalization of polynomial approximation. $\bullet$ The sampling for generation is considered as an optimization process, we demonstrate its fast convergence with updating standard first-order gradient descent of some objective.
Abstract (translated)
最近的研究表明，在时间序列预测（TSF）任务中，使用生成模型替代传统的自回归算法可以带来更好的效果。这些非自回归方法包括GAN、扩散和流匹配等不同的生成技术，并且在实践中展示了高质量的生成能力和准确性。然而，我们仍缺乏对它们如何进行近似处理以及如何泛化理解的知识。本文首次从基于流的生成模型的角度提出了一个理论框架来澄清其局限性知识。特别地，从以下三个角度提供了严格保证并给出了我们的见解：**近似**、**泛化** 和 **效率**。 具体来说，我们分析的主要贡献如下： - 假设了一个一般数据模型，基于流的生成模型在扩散变换器（DiT）通用逼近下被确认其拟合可以收敛到任意误差。 - 通过引入基于多项式的正则化方法用于流匹配，由于多项式近似的泛化特性，因此可以将泛化误差限制在一个范围内。 - 将生成过程中的采样视为一个优化过程，并展示了通过更新标准的一阶梯度下降目标函数能够实现快速收敛。 这些贡献共同提供了一个全面的理解框架，帮助我们更好地理解基于流的生成模型在时间序列预测任务中的表现及其潜在的应用前景。
URL
https://arxiv.org/abs/2503.14076
