Policy Optimization for H2 Linear Control with H Robustness Guarantee: Implicit Regularization and Global Convergence

Kaiqing Zhang, Bin Hu, Tamer Başar

Research output: Contribution to journalConference articlepeer-review


Policy optimization (PO) is a key ingredient for modern reinforcement learning (RL). For control design, certain constraints are usually enforced on the policies to optimize, accounting for stability, robustness, or safety concerns on the system. Hence, PO is by nature a constrained (nonconvex) optimization in most cases, whose global convergence is challenging to analyze in general. More importantly, some constraints that are safety-critical, e.g., the closed-loop stability, or the H-norm constraint that guarantees the system robustness, can be difficult to enforce on the controller being learned as the PO methods proceed. In this paper, we study the convergence theory of PO for H2 linear control with H robustness guarantee. This general framework includes risk-sensitive linear control as a special case. One significant new feature of this problem, in contrast to the standard H2 linear control, namely, linear quadratic regulator (LQR) problems, is the lack of coercivity of the cost function. This makes it challenging to guarantee the feasibility, namely, the H robustness, of the iterates. Interestingly, we propose two PO algorithms that enjoy the implicit regularization property, i.e., the iterates preserve the H robustness, as if they are regularized by the algorithms. Furthermore, convergence to the globally optimal policies with globally sublinear and locally (super-)linear rates are provided under certain conditions, despite the nonconvexity of the problem. To the best of our knowledge, our work offers the first results on the implicit regularization property and global convergence of PO methods for robust/risk-sensitive control.

Original languageEnglish (US)
Pages (from-to)179-190
Number of pages12
JournalProceedings of Machine Learning Research
StatePublished - 2020
Externally publishedYes
Event2nd Annual Conference on Learning for Dynamics and Control, L4DC 2020 - Berkeley, United States
Duration: Jun 10 2020Jun 11 2020


  • H robust control
  • Reinforcement learning
  • global convergence
  • implicit regularization
  • policy optimization

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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