Global exponential convergence of gradient methods over the nonconvex landscape of the linear quadratic regulator

Abstract

In large-scale and model-free settings, first-order algorithms are often used in an attempt to find the optimal control action without identifying the underlying dynamics. The convergence properties of these algorithms remain poorly understood because of nonconvexity. In this paper, we revisit the continuous-time linear quadratic regulator problem and take a step towards demystifying the efficiency of gradient-based strategies. Despite the lack of convexity, we establish a linear rate of convergence to the globally optimal solution for the gradient descent algorithm. The key component of our analysis is that we relate the gradient-flow dynamics associated with the nonconvex formulation to that of a convex reparameterization. This allows us to provide convergence guarantees for the nonconvex approach from its convex counterpart.