Algorithmic Optimality Guarantees for Nonsmooth $H_\infty$ Output-Feedback Policy Search

• URL: http://arxiv.org/abs/2609.06197v1
• Date: Sat, 05 Sep 2026 17:36:26 GMT
• Title: Algorithmic Optimality Guarantees for Nonsmooth $H_\infty$ Output-Feedback Policy Search
• Abstract summary: We develop a robust $varepsilon$-stationarity benchmark for nonsmooth $H_infty$ policy search.<n>We prove that on the exact identity- slice of the extended convex lift, $O(varepsilon)$-suboptimality on compact exact slices yields convergence-rate guarantees for nonsmooth policy-search methods.
• Score: 24.350063807321945
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study continuous-time full-order dynamic output-feedback $H_\infty$ 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 $H_\infty$ 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 $H_\infty$ 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 $H_\infty$ policy search.

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