Policy Optimization for $\mathcal{H}_2$ Linear Control with $\mathcal{H}_\infty$ Robustness Guarantee: Implicit Regularization and Global Convergence

Policy optimization (PO) is a key ingredient for reinforcement learning (RL). For control design, certain constraints are usually enforced on the policies to optimize, accounting for either the 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 $\mathcal{H}_\infty$-norm constraint that guarantees the system robustness, are difficult to enforce as the PO methods proceed. Recently, policy gradient methods have been shown to converge to the global optimum of linear quadratic regulator (LQR), a classical optimal control problem, without regularizing/projecting the control iterates onto the stabilizing set (Fazel et al., 2018), its (implicit) feasible set. This striking result is built upon the coercive property of the cost, ensuring that the iterates remain feasible as the cost decreases. In this paper, we study the convergence theory of PO for $\mathcal{H}_2$ linear control with $\mathcal{H}_\infty$-norm robustness guarantee. One significant new feature of this problem is the lack of coercivity, i.e., the cost may have finite value around the feasible set boundary, breaking the existing analyses for LQR. Interestingly, we show that two PO methods enjoy the implicit regularization property, i.e., the iterates preserve the $\mathcal{H}_\infty$ robustness constraint 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.