Learning Random Feature Dynamics for Uncertainty Quantification

An inherent challenge of learning-based control tasks is posed by uncertainty due to finite training datasets. Even though there are principled tools to obtain confidence bounds for pointwise evaluation of learned dynamics models, it remains a challenging task to quantify the induced uncertainty in downstream quantities of interest due to the intrinsic recursive structure of dynamic systems. In this paper, we view the unknown one-step dynamics as a smooth function in a reproducing kernel Hilbert space and leverage random features for an approximate but highly structured parameterization of pointwise confidence bounds. As a result, we obtain downstream confidence bounds through an optimal control formulation under an uncertainty-aware random feature dynamics model. Our model is effectively a shallow neural network, which enables us to view the corresponding dynamic system as a deep neural network. Exploiting this perspective, we show that a Pontryagin’s minimum principle solution is equivalent to using the Frank-Wolfe algorithm on the induced neural network. Various numerical experiments on dynamics learning showcase the capacity of our methodology.