Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control (2102.04393v1)

Abstract: This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: 1) connectivity of the set of stabilizing controllers $\mathcal{C}_n$; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that 1) the set of stabilizing controllers $\mathcal{C}_n$ has at most two path-connected components and they are diffeomorphic under a mapping defined by a similarity transformation; 2) there might exist many \emph{strictly suboptimal stationary points} of the LQG cost function over $\mathcal{C}_n$ and these stationary points are always \emph{non-minimal}; 3) all \emph{minimal} stationary points are globally optimal and they are identical up to a similarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.

• The paper establishes that the set of stabilizing controllers has at most two path-connected components, facilitating effective gradient search strategies.
• The paper reveals that all minimal stationary points are globally optimal while many suboptimal saddles exist, clarifying the structure of the optimization landscape.
• The paper demonstrates that understanding this landscape enhances policy gradient methods in LQG control, improving convergence in model-free learning.

Analyzing the Optimization Landscape of Linear Quadratic Gaussian Control

The paper provides a deep dive into the optimization landscape of Linear Quadratic Gaussian (LQG) control by examining the structure and connectivity of the set of stabilizing controllers ($\mathcal{C}_n$) and the characteristics of stationary points within the problem domain. LQG control, a fundamental optimal control problem, has been traditionally approached with algebraic methods and Riccati equations for solving the control synthesis. However, this paper revisits LQG from a modern optimization perspective, laying down insights that are pivotal for understanding the implications on the performance of direct policy gradient methods.

Key Contributions and Findings

1. Connectivity of Stabilizing Controllers:

• The research establishes that under standard assumptions, the set $\mathcal{C}_n$ is composed of at most two path-connected components. This implies that, while $\mathcal{C}_n$ can be disconnected, it maintains at most two path-connected subsets that are diffeomorphic under a similarity transformation, showing internal symmetry and suggesting potential pathways for gradient-based local search algorithms to operate uniformly across components.
• A significant contribution is the identification of conditions under which $\mathcal{C}_n$ is path-connected, particularly when reduced-order controllers ($\mathcal{C}_{n-1}$) exist. This characteristic becomes crucial for seamless local search navigation within model-free reinforcement learning methods, which rely heavily on gradient descent processes.

2. Structure of Stationary Points:

• The paper ventures beyond connectivity, shedding light on the structure of stationary points in the optimization landscape. It finds that similar to stability see in LQR, LQG under certain conditions possesses a rich structure with many strictly suboptimal stationary points, which can act as non-minimal saddles.
• Furthermore, it advocates that all minimal stationary points in $\mathcal{C}_n$ are globally optimal, and any controller achieving a minimal representation necessarily brings the system to its best-performance configuration. This implies that optimal solutions can be identified by recognizing minimal (controllable and observable) nature in stationary points, paving the way for attaining global optimal solutions without extensive computation.

Implications for Reinforcement Learning

The paper's findings are instrumental in reinforcing the applicability of policy gradient methods for LQG control systems. The connectivity insights suggest that model-free learning approaches, often iterative and gradient-based, can effectively navigate the feasible terrain without the risk of being trapped in isolated components, as long as path-connectedness can be established or reduced-order controllers verified as feasible. Moreover, understanding stationary point structures allows these algorithms to benefit from an internal check on minimal realizability, enhancing prediction accuracy and efficiency.

Future Research Directions

The paper opens pathways for future research, particularly:

• Exploring gradient descent algorithms informed by these insights, ensuring convergence to optimal solutions by leveraging structural properties of the LQG landscape.
• Investigating broadening the parameter space for dynamical controllers, including utilizing distinct controller parameterizations that could further streamline optimization methods.
• Engaging deeper with symmetry properties could result in tractable solutions for even more complex control scenarios.

Overall, the paper serves as a stepping stone towards integrating classical control theories with contemporary optimization algorithms, marking an evolutionary leap in both theoretical understanding and practical applications in LQG control systems.