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This paper presents an approach, Spectral Dynamics Embedding Control (SDEC),to optimal control for nonlinear stochastic systems. This method leverages aninfinite-dimensional feature to linearly represent the state-action valuefunction and exploits finite-dimensional truncation approximation for practicalimplementation. To characterize the effectiveness of these finite dimensionalapproximations, we provide an in-depth theoretical analysis to characterize theapproximation error induced by the finite-dimension truncation and statisticalerror induced by finite-sample approximation in both policy evaluation andpolicy optimization. Our analysis includes two prominent kernel approximationmethods: truncations onto random features and Nystrom features. We alsoempirically test the algorithm and compare the performance with Koopman-based,iLQR, and energy-based methods on a few benchmark problems.

Stochastic Nonlinear Control via Finite-dimensional Spectral Dynamic Embedding