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Direct policy search has achieved great empirical success in reinforcementlearning. Recently, there has been increasing interest in studying itstheoretical properties for continuous control, and fruitful results have beenestablished for linear quadratic regulator (LQR) and linear quadratic Gaussian(LQG) control that are smooth and nonconvex. In this paper, we consider thestandard $H_\infty$ robust control for output feedback systems and investigatethe global optimality of direct policy search. Unlike LQR or LQG, the$H_\infty$ cost function is nonsmooth in the policy space. Despite the lack ofsmoothness and convexity, our main result shows that for a class ofnon-degenerated stabilizing controllers, all Clarke stationary points of$H_\infty$ robust control are globally optimal and there is no spurious localminimum. Our proof technique is motivated by the idea of differentiable convexliftings (DCL), and we extend DCL to analyze the nonsmooth and nonconvex$H_\infty$ robust control via convex reformulation. Our result sheds some lighton the analysis of direct policy search for solving nonsmooth and nonconvexrobust control problems.

On the Global Optimality of Direct Policy Search for Nonsmooth $H_\infty$ Output-Feedback Control