Stochastic Nonlinear Control via Finite-dimensional Spectral Dynamic Embedding

This paper proposes an approach, Spectral Dynamics Embedding Control (SDEC), to optimal control for nonlinear stochastic systems. This method reveals an infinite-dimensional feature representation induced by the system's nonlinear stochastic dynamics, enabling a linear representation of the state-action value function. For practical implementation, this representation is approximated using finite-dimensional truncations, specifically via two prominent kernel approximation methods: random feature truncation and Nystrom approximation. To characterize the effectiveness of these approximations, we provide an in-depth theoretical analysis to characterize the approximation error arising from the finite-dimension truncation and statistical error due to finite-sample approximation in both policy evaluation and policy optimization. Empirically, our algorithm performs favorably against existing stochastic control algorithms on several benchmark problems.