Algorithmic Optimality Guarantees for Nonsmooth H_∞ Output-Feedback Policy Search

We study continuous-time full-order dynamic output-feedback H_∞ policy search, a nonconvex and nonsmooth problem. Direct policy search is a central paradigm in reinforcement learning and continuous control, but rigorous guarantees remain scarce in robust output-feedback settings. The H_∞ problem is a canonical benchmark because it captures disturbance attenuation and robustness while exposing the hard nonsmooth geometry of policy-space optimization. We prove that on the exact identity-gauge slice of the extended convex lift, varepsilon-stationarity yields O(varepsilon)-suboptimality on compact exact slices, which in turn yields convergence-rate guarantees for nonsmooth policy-search methods. This result addresses the finite-time optimality-gap question raised by Guo and Hu [2022] in the more general dynamic output-feedback H_∞ policy-search setting. We further use the established value equivalence supplied by extended convex lifting to formulate a nonstrict-feasibility bisection method with one final strict-feasibility recovery step, yielding an explicit varepsilon-optimal stabilizing controller. These results provide a quantitative and algorithmic strengthening of prior qualitative optimality theory for nonsmooth H_∞ policy search.