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Model-free reinforcement learning attempts to find an optimal control actionfor an unknown dynamical system by directly searching over the parameter spaceof controllers. The convergence behavior and statistical properties of theseapproaches are often poorly understood because of the nonconvex nature of theunderlying optimization problems and the lack of exact gradient computation. Inthis paper, we take a step towards demystifying the performance and efficiencyof such methods by focusing on the standard infinite-horizon linear quadraticregulator problem for continuous-time systems with unknown state-spaceparameters. We establish exponential stability for the ordinary differentialequation (ODE) that governs the gradient-flow dynamics over the set ofstabilizing feedback gains and show that a similar result holds for thegradient descent method that arises from the forward Euler discretization ofthe corresponding ODE. We also provide theoretical bounds on the convergencerate and sample complexity of the random search method with two-point gradientestimates. We prove that the required simulation time for achieving$\epsilon$-accuracy in the model-free setup and the total number of functionevaluations both scale as $\log \, (1/\epsilon)$.

Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem