Online Learning of Decentralized Linear Quadratic Regulator with Sublinear Regret

Core Concepts

An online learning algorithm that adaptively designs a decentralized linear quadratic regulator when the system model is unknown, achieving a regret that scales sublinearly with the time horizon.

Abstract

The paper proposes an online learning algorithm for designing a decentralized linear quadratic regulator (LQR) when the system model is unknown. The key contributions are:

1. The algorithm uses a disturbance-feedback representation of the state-feedback controllers, which is coupled with an online convex optimization (OCO) algorithm that has memory and delayed feedback. This allows the algorithm to respect the prescribed information pattern in the decentralized setting.

2. Under the assumption of a stable system or a known stabilizing controller, the algorithm achieves an expected regret that scales as √T with the time horizon T for the case of partially nested information pattern. This matches the regret bound for the centralized LQR case.

3. For more general information patterns where the optimal decentralized controller is unknown even if the system model is known, the regret of the proposed controller is shown with respect to a linear sub-optimal controller.

4. The theoretical findings are validated through numerical experiments.

The key steps of the algorithm are:

1. In the first phase, the algorithm uses a least squares method to estimate the unknown system matrices A and B from a single system trajectory.

2. In the second phase, the algorithm uses an OCO algorithm with memory and delayed feedback to adaptively design a decentralized control policy, leveraging the estimated system model from the first phase.

The regret analysis shows that the proposed algorithm achieves the optimal √T regret scaling, despite the additional challenge of the decentralized information constraints.

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Key Insights Distilled From

Learning Decentralized Linear Quadratic Regulator with $\sqrt{T}$ Regret

by Lintao Ye,Mi... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2210.08886.pdf

Deeper Inquiries

How would the algorithm and analysis change if the system is not assumed to be stable or a known stabilizing controller is not available

If the system is not assumed to be stable or a known stabilizing controller is not available, the algorithm and analysis would need to be adjusted to account for the instability of the system. In this case, the decentralized online control algorithm would need to incorporate additional measures to ensure stability in the control policy design. This could involve introducing constraints or regularization terms in the optimization problem to prevent the system from becoming unstable. The analysis would also need to consider the impact of system instability on the regret bounds and performance of the algorithm.

Can the algorithm be extended to handle more general information patterns beyond the partially nested case, without relying on a sub-optimal controller for the regret analysis

Yes, the algorithm can be extended to handle more general information patterns beyond the partially nested case without relying on a sub-optimal controller for the regret analysis. This extension would involve modifying the decentralized control policy design to accommodate different information patterns in the networked system. The algorithm could be adapted to incorporate more complex communication structures and information flow constraints, allowing for a more flexible and robust decentralized control solution. The regret analysis would need to be adjusted to account for the changes in the information patterns and ensure that the algorithm performs effectively in these scenarios.

What are the potential applications of this decentralized online learning framework beyond the linear quadratic regulator problem

The decentralized online learning framework proposed in the context of the linear quadratic regulator problem has various potential applications in real-world systems and industries. Some of the applications include:

Smart Grids: The framework can be used for decentralized control of power distribution networks, optimizing energy consumption and grid stability.
Autonomous Vehicles: Implementing decentralized control algorithms in autonomous vehicle networks can improve coordination and decision-making among vehicles.
Industrial Automation: Decentralized control can enhance the efficiency and reliability of manufacturing processes by distributing control tasks among different subsystems.
Multi-Robot Systems: Decentralized control is crucial for coordinating actions and tasks among multiple robots in applications such as search and rescue missions or warehouse management.
Healthcare Systems: Decentralized control can be applied in healthcare systems to optimize patient care processes, resource allocation, and treatment planning in hospitals or clinics.