Accelerated Optimization Landscape of Linear-Quadratic Regulator

TL;DR

This paper introduces accelerated optimization methods for solving the Linear-Quadratic Regulator (LQR) problem, providing convergence guarantees and improved complexity bounds for both state-feedback and output-feedback cases.

Contribution

It develops a first-order accelerated framework for LQR, including a hybrid dynamic system approach for SLQR and a Hessian-free method for OLQR, with theoretical convergence analysis.

Findings

Exponential convergence to optimal feedback gain in SLQR

Discretized Nesterov-type method preserves continuous-time convergence rate

OLQR method achieves $ ilde{O}( ext{poly}(1/ ext{epsilon}))$ complexity, better than vanilla gradient descent

Abstract

Linear-quadratic regulator (LQR) is a landmark problem in the field of optimal control, which is the concern of this paper. Generally, LQR is classified into state-feedback LQR (SLQR) and output-feedback LQR (OLQR) based on whether the full state is obtained. It has been suggested in existing literature that both SLQR and OLQR could be viewed as \textit{constrained nonconvex matrix optimization} problems in which the only variable to be optimized is the feedback gain matrix. In this paper, we introduce a first-order accelerated optimization framework of handling the LQR problem, and give its convergence analysis for the cases of SLQR and OLQR, respectively. Specifically, a Lipschiz Hessian property of LQR performance criterion is presented, which turns out to be a crucial property for the application of modern optimization techniques. For the SLQR problem, a continuous-time hybrid dynamic system is introduced, whose solution trajectory is shown to converge exponentially to the optimal feedback gain with Nesterov-optimal order $1-\frac{1}{\sqrt{\kappa}}$ ($\kappa$ the condition number). Then, the symplectic Euler scheme is utilized to discretize the hybrid dynamic system, and a Nesterov-type method with a restarting rule is proposed that preserves the continuous-time convergence rate, i.e., the discretized algorithm admits the Nesterov-optimal convergence order. For the OLQR problem, a Hessian-free accelerated framework is proposed, which is a two-procedure method consisting of semiconvex function optimization and negative curvature exploitation. In a time $\mathcal{O}(\epsilon^{-7/4}\log(1/\epsilon))$, the method can find an $\epsilon$-stationary point of the performance criterion; this entails that the method improves upon the $\mathcal{O}(\epsilon^{-2})$ complexity of vanilla gradient descent. Moreover, our method provides the second-order guarantee of stationary point.