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 $H_\infty$ synthesis with static output feedback. Optimal $H_\infty$ synthesis under structural constraints leads to a constrained nonconvex nonsmooth problem and is typicallyaddressed 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 $(\delta,\epsilon)$-stationary points for such nonsmooth robust control design tasks using policy optimization methods which can only access the zeroth-order oracle (i.e. the $H_\infty$ 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) $H_\infty$-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 $(\delta,\epsilon)$-stationary points for structured $H_\infty$ control. Numerical results are also provided to demonstrate our theory.

Direct policy search serves as one of the workhorses in modern reinforcement learning (RL), and its applications in continuous control tasks have recently attracted increasing attention. In this work, we investigate the convergence theory of policy gradient (PG) methods for learning the linear risk-sensitive and robust controller. In particular, we develop PG methods that can be implemented in a derivative-free fashion by sampling system trajectories, and establish both global convergence and sample complexity results in the solutions of two fundamental settings in risk-sensitive and robust control: the finite-horizon linear exponential quadratic Gaussian, and the finite-horizon linear-quadratic disturbance attenuation problems. As a by-product, our results also provide the first sample complexity for the global convergence of PG methods on solving zero-sum linear-quadratic dynamic games, a nonconvex-nonconcave minimax optimization problem that serves as a baseline setting in multi-agent reinforcement learning (MARL) with continuous spaces. One feature of our algorithms is that during the learning phase, a certain level of robustness/risk-sensitivity of the controller is preserved, which we termed as the implicit regularization property, and is an essential requirement in safety-critical control systems.

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_\infty synthesis with static output feedback. Optimal H_\infty synthesis under structural constraints leads to a constrained nonconvex nonsmooth problem and is typicallyaddressed 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 (\delta,\epsilon) -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_\infty 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_\infty -norm estimation.

Direct policy search serves as one of the workhorses in modern reinforcement learning (RL), and its applications in continuous control tasks have recently attracted increasing attention. In this work, we investigate the convergence theory of policy gradient (PG) methods for learning the linear risk-sensitive and robust controller. In particular, we develop PG methods that can be implemented in a derivative-free fashion by sampling system trajectories, and establish both global convergence and sample complexity results in the solutions of two fundamental settings in risk-sensitive and robust control: the finite-horizon linear exponential quadratic Gaussian, and the finite-horizon linear-quadratic disturbance attenuation problems. As a by-product, our results also provide the first sample complexity for the global convergence of PG methods on solving zero-sum linear-quadratic dynamic games, a nonconvex-nonconcave minimax optimization problem that serves as a baseline setting in multi-agent reinforcement learning (MARL) with continuous spaces. One feature of our algorithms is that during the learning phase, a certain level of robustness/risk-sensitivity of the controller is preserved, which we termed as the implicit regularization property, and is an essential requirement in safety-critical control systems.