Linear convergence of a policy gradient method for some finite horizon continuous time control problems
This provides a theoretical justification for reinforcement learning heuristics like entropy regularization, addressing a gap in continuous control for practitioners.
The paper tackles the lack of provably convergent policy gradient methods for continuous space-time control problems with nonlinear state dynamics by proposing proximal gradient algorithms for finite-time horizon stochastic control, proving linear convergence to a stationary point under suitable conditions.
Despite its popularity in the reinforcement learning community, a provably convergent policy gradient method for continuous space-time control problems with nonlinear state dynamics has been elusive. This paper proposes proximal gradient algorithms for feedback controls of finite-time horizon stochastic control problems. The state dynamics are nonlinear diffusions with control-affine drift, and the cost functions are nonconvex in the state and nonsmooth in the control. The system noise can degenerate, which allows for deterministic control problems as special cases. We prove under suitable conditions that the algorithm converges linearly to a stationary point of the control problem, and is stable with respect to policy updates by approximate gradient steps. The convergence result justifies the recent reinforcement learning heuristics that adding entropy regularization or a fictitious discount factor to the optimization objective accelerates the convergence of policy gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations.