Global Convergence of Direct Policy Search for State-Feedback $\mathcal{H}_\infty$ Robust Control: A Revisit of Nonsmooth Synthesis with Goldstein Subdifferential

• URL: http://arxiv.org/abs/2210.11577v1
• Date: Thu, 20 Oct 2022 20:33:20 GMT
• Title: Global Convergence of Direct Policy Search for State-Feedback $\mathcal{H}_\infty$ Robust Control: A Revisit of Nonsmooth Synthesis with Goldstein Subdifferential
• Abstract summary: We show that direct policy search is guaranteed find the global robust $mathcal_infty$ state search problem. All stationary points are a minimum design solution for this nonsmooth optimization problem. Based on these properties, we show that Goldstein's subgradient method and its implementable variants can be guaranteed to stay in the non feasible set.
• Score: 3.9391112596932243
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Direct policy search has been widely applied in modern reinforcement learning and continuous control. However, the theoretical properties of direct policy search on nonsmooth robust control synthesis have not been fully understood. The optimal $\mathcal{H}_\infty$ control framework aims at designing a policy to minimize the closed-loop $\mathcal{H}_\infty$ norm, and is arguably the most fundamental robust control paradigm. In this work, we show that direct policy search is guaranteed to find the global solution of the robust $\mathcal{H}_\infty$ state-feedback control design problem. Notice that policy search for optimal $\mathcal{H}_\infty$ control leads to a constrained nonconvex nonsmooth optimization problem, where the nonconvex feasible set consists of all the policies stabilizing the closed-loop dynamics. We show that for this nonsmooth optimization problem, all Clarke stationary points are global minimum. Next, we identify the coerciveness of the closed-loop $\mathcal{H}_\infty$ objective function, and prove that all the sublevel sets of the resultant policy search problem are compact. Based on these properties, we show that Goldstein's subgradient method and its implementable variants can be guaranteed to stay in the nonconvex feasible set and eventually find the global optimal solution of the $\mathcal{H}_\infty$ state-feedback synthesis problem. Our work builds a new connection between nonconvex nonsmooth optimization theory and robust control, leading to an interesting global convergence result for direct policy search on optimal $\mathcal{H}_\infty$ synthesis.

Related papers

• Algorithmic Optimality Guarantees for Nonsmooth $H_\infty$ Output-Feedback Policy Search [24.350063807321945]
  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.
  arXiv  Detail & Related papers  (2026-09-05T17:36:26Z)
• Augmented Lagrangian Method for Last-Iterate Convergence for Constrained MDPs [7.237620866644061]
  We study policy optimization for infinite-horizon, discounted constrained Markov decision processes (CMDPs)<n>We use the classic inexact augmented Lagrangian ($textttAL$) method from constrained optimization, and propose a general framework with provable last-iterate convergence for CMDPs.<n>We generalize these results to handle log-linear policies, and demonstrate that an efficient, projected variant of $textttPQA$ can achieve last-iterate convergence.
  arXiv  Detail & Related papers  (2026-05-12T07:51:18Z)
• Revisiting Policy Gradients for Restricted Policy Classes: Escaping Myopic Local Optima with $k$-step Policy Gradients [8.64427265159929]
  This work revisits standard policy gradient methods used on restricted policy classes.<n>We propose a generalized $k$-step policy gradient method that couples the randomness within a $k$-step time window.<n>We show this new method is theoretically guaranteed to converge to a solution that is exponentially close in performance to the optimal deterministic policy.
  arXiv  Detail & Related papers  (2026-05-11T17:49:09Z)
• Convergence and Sample Complexity of First-Order Methods for Agnostic Reinforcement Learning [66.4260157478436]
  We study reinforcement learning in the policy learning setting.<n>The goal is to find a policy whose performance is competitive with the best policy in a given class of interest.
  arXiv  Detail & Related papers  (2025-07-06T14:40:05Z)
• Rethinking the Global Convergence of Softmax Policy Gradient with Linear Function Approximation [52.772454746132276]
  We show that the approximation error in modeling problem-dependent quantities is irrelevant to the algorithm's global convergence.<n>We prove that $textttLin-SPG$ with any arbitrary constant learning rate can ensure global convergence to the optimal policy.
  arXiv  Detail & Related papers  (2025-05-06T04:03:06Z)
• Nearly Optimal Sample Complexity of Offline KL-Regularized Contextual Bandits under Single-Policy Concentrability [49.96531901205305]
  We propose the emphfirst algorithm with $tildeO(epsilon-1)$ sample complexity under single-policy concentrability for offline contextual bandits. Our proof leverages the strong convexity of the KL regularization, and the conditional non-negativity of the gap between the true reward and its pessimistic estimator. We extend our algorithm to contextual dueling bandits and achieve a similar nearly optimal sample complexity.
  arXiv  Detail & Related papers  (2025-02-09T22:14:45Z)
• Low-Switching Policy Gradient with Exploration via Online Sensitivity Sampling [23.989009116398208]
  We design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an $varepsilon$-optimal policy with only $widetildeO(fractextpoly(d)varepsilon3)$ samples.
  arXiv  Detail & Related papers  (2023-06-15T23:51:46Z)
• Towards Painless Policy Optimization for Constrained MDPs [46.12526917024248]
  We study policy optimization in an infinite horizon, $gamma$-discounted constrained Markov decision process (CMDP) Our objective is to return a policy that achieves large expected reward with a small constraint violation. We propose a generic primal-dual framework that allows us to bound the reward sub-optimality and constraint violation for arbitrary algorithms.
  arXiv  Detail & Related papers  (2022-04-11T15:08:09Z)
• Globally Convergent Policy Search over Dynamic Filters for Output Estimation [64.90951294952094]
  We introduce the first direct policy search algorithm convex which converges to the globally optimal $textitdynamic$ filter. We show that informativity overcomes the aforementioned degeneracy.
  arXiv  Detail & Related papers  (2022-02-23T18:06:20Z)
• Homotopic Policy Mirror Descent: Policy Convergence, Implicit Regularization, and Improved Sample Complexity [40.2022466644885]
  homotopic policy mirror descent (HPMD) method for solving discounted, infinite horizon MDPs with finite state and action space. We report three properties that seem to be new in the literature of policy gradient methods.
  arXiv  Detail & Related papers  (2022-01-24T04:54:58Z)
• Understanding the Effect of Stochasticity in Policy Optimization [86.7574122154668]
  We show that the preferability of optimization methods depends critically on whether exact gradients are used. Second, to explain these findings we introduce the concept of committal rate for policy optimization. Third, we show that in the absence of external oracle information, there is an inherent trade-off between exploiting geometry to accelerate convergence versus achieving optimality almost surely.
  arXiv  Detail & Related papers  (2021-10-29T06:35:44Z)
• On the Convergence and Sample Efficiency of Variance-Reduced Policy Gradient Method [38.34416337932712]
  Policy gives rise to a rich class of reinforcement learning (RL) methods, for example the REINFORCE. Yet the best known sample complexity result for such methods to find an $epsilon$-optimal policy is $mathcalO(epsilon-3)$, which is suboptimal. We study the fundamental convergence properties and sample efficiency of first-order policy optimization method.
  arXiv  Detail & Related papers  (2021-02-17T07:06:19Z)
• CRPO: A New Approach for Safe Reinforcement Learning with Convergence Guarantee [61.176159046544946]
  In safe reinforcement learning (SRL) problems, an agent explores the environment to maximize an expected total reward and avoids violation of certain constraints. This is the first-time analysis of SRL algorithms with global optimal policies.
  arXiv  Detail & Related papers  (2020-11-11T16:05:14Z)
• Global Convergence of Policy Gradient for Linear-Quadratic Mean-Field Control/Game in Continuous Time [109.06623773924737]
  We study the policy gradient method for the linear-quadratic mean-field control and game. We show that it converges to the optimal solution at a linear rate, which is verified by a synthetic simulation.
  arXiv  Detail & Related papers  (2020-08-16T06:34:11Z)
• Provably Efficient Safe Exploration via Primal-Dual Policy Optimization [105.7510838453122]
  We study the Safe Reinforcement Learning (SRL) problem using the Constrained Markov Decision Process (CMDP) formulation. We present an provably efficient online policy optimization algorithm for CMDP with safe exploration in the function approximation setting.
  arXiv  Detail & Related papers  (2020-03-01T17:47:03Z)
• Naive Exploration is Optimal for Online LQR [49.681825576239355]
  We show that the optimal regret scales as $widetildeTheta(sqrtd_mathbfu2 d_mathbfx T)$, where $T$ is the number of time steps, $d_mathbfu$ is the dimension of the input space, and $d_mathbfx$ is the dimension of the system state. Our lower bounds rule out the possibility of a $mathrmpoly(logT)$-regret algorithm, which had been
  arXiv  Detail & Related papers  (2020-01-27T03:44:54Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.