Value-gradient iteration with quadratic approximate value functions

We propose a method for designing policies for convex stochastic control problems characterized by random linear dynamics and convex stage cost. We consider policies that employ quadratic approximate value functions as a substitute for the true value function. Evaluating the associated control policy involves solving a convex problem, typically a quadratic program, which can be carried out reliably in real-time. Such policies often perform well even when the approximate value function is not a particularly good approximation of the true value function. We propose value-gradient iteration, which fits the gradient of value function, with regularization that can include constraints reflecting known bounds on the true value function. Our value-gradient iteration method can yield a good approximate value function with few samples, and little hyperparameter tuning. We find that the method can find a good policy with computational effort comparable to that required to just evaluate a control policy via simulation.