Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiency

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.