In this talk, we discuss a data-driven regression framework for the computation of high-dimensional optimal feedback laws. We propose a causality-free approach for approximating the value function of deterministic control problems via Pontryagin's Maximum Principle. A cloud of open-loop solves and the augmented information from the adjoints are used to perform a LASSO regression for a polynomial model of the value function. This allows to compute a reduced complexity representation of the optimal feedback map.