On the Generalization and Approximation Capacities of Neural Controlled Differential Equations

Neural Controlled Differential Equations (NCDE) 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 roughness of the sampling 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 sampling-dependant generalization bound for this class of predictors. 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 NCDE. Our theoretical results are validated through a series of experiments.