  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.
