Complexity of Derivative-Free Policy Optimization for Structured H_∞ Control

The applications of direct policy search in reinforcement learning and continuous control have received increasing attention. In this work, we present novel theoretical results on the complexity of derivative-free policy optimization on an important class of robust control tasks, namely the structured H_∞ synthesis with static output feedback. Optimal H_∞ synthesis under structural constraints leads to a constrained nonconvex nonsmooth problem and is typically addressed using subgradient-based policy search techniques that are built upon the concept of Goldstein subdifferential or other notions of enlarged subdifferential. In this paper, we study the complexity of finding (δ, ϵ)-stationary points for such nonsmooth robust control design tasks using policy optimization methods which can only access the zeroth-order oracle (i.e. the H_∞ norm of the closed-loop system). First, we study the exact oracle setting and identify the coerciveness of the cost function to prove high-probability feasibility/complexity bounds for derivative-free policy optimization on this problem. Next, we derive a sample complexity result for the multi-input multi-output (MIMO) H_∞-norm estimation. We combine this with our analysis to obtain the first sample complexity of model-free, trajectory-based, zeroth-order policy optimization on finding (δ, ϵ)-stationary points for structured H_∞ control. Numerical results are also provided to demonstrate our theory.