Regret Bound of Adaptive Control in Linear Quadratic Gaussian (LQG) Systems

One of the core challenges in the field of control theory and reinforcement learning is adaptive control. It is the problem of controlling dynamical systems when the dynamics of the systems are unknown to the decision-making agents. In adaptive control, agents interact with given systems in order to explore and control them while the long-term objective is to minimize the overall average associated costs. The agent has to balance between exploration and exploitation, learn the dynamics, strategize for further exploration, and exploit the estimation to minimize the overall costs. The sequential nature of agent-system interaction results in challenges in the system identifying, estimation, and control under uncertainty, and these challenges are magnified when the systems are partially observable, i.e. contain hidden underlying dynamics. In the linear systems, when the underlying dynamics are fully observable, the asymptotic optimality of estimation methods has been the topic of study in the last decades [Lai et al., 1982, Lai and Wei, 1987]. Recently, novel techniques and learning algorithms have been developed to study the finite-time behavior of adaptive control algorithms and shed light on the design of optimal methods [Peña et al., 2009, Fiechter, 1997, Abbasi-Yadkori and Szepesvári, 2011]. In particular, Abbasi-Yadkori and Szepesvári [2011] proposes to use the principle of optimism in the face of uncertainty (OFU) to balance exploration and exploitation in LQR, where the state of the system is observable.