Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it remains unclear in particular how the irregularity of the time series affects their predictions. By merging the rich theory of controlled differential equations (CDE) and Lipschitz-based measures of the complexity of deep neural nets, we take a first step towards the theoretical understanding of NCDE. Our first result is a generalization bound for this class of predictors that depends on the regularity of the time series data. In a second time, we leverage the continuity of the flow of CDEs to provide a detailed analysis of both the sampling-induced bias and the approximation bias. Regarding this last result, we show how classical approximation results on neural nets may transfer to NCDEs. Our theoretical results are validated through a series of experiments.

翻译：神经控制微分方程（NCDEs）是处理非均匀采样时间序列监督学习任务的最前沿工具（Kidger, 2020）。然而，目前尚未有对其性能的理论分析，尤其不清楚时间序列的非均匀性如何影响其预测结果。通过融合控制微分方程（CDE）的丰富理论与基于Lipschitz的深度神经网络复杂度度量，我们为理解NCDE奠定了初步理论基础。我们的第一个结果是针对这类预测器的泛化界，该界限依赖于时间序列数据的正则性。其次，我们利用CDE流的连续性，对采样偏差和逼近偏差进行了详细分析。关于后一个结果，我们展示了经典神经网络逼近结果如何迁移至NCDE。通过一系列实验验证了理论结果的有效性。