Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems

We study the problem of adaptive control in partially observable linear dynamical systems. We propose a novel algorithm, adaptive control online learning algorithm (AdaptOn), which efficiently explores the environment, estimates the system dynamics episodically and exploits these estimates to design effective controllers to minimize the cumulative costs. Through interaction with the environment, AdaptOn deploys online convex optimization to optimize the controller while simultaneously learning the system dynamics to improve the accuracy of controller updates. We show that when the cost functions are strongly convex, after T times step of agent-environment interaction, AdaptOn achieves regret upper bound of polylog(T). To the best of our knowledge, AdaptOn is the first algorithm which achieves polylog(T) regret in adaptive control of unknown partially observable linear dynamical systems which includes linear quadratic Gaussian (LQG) control

Adaptive Control and Regret Minimization in Linear Quadratic Gaussian (LQG) Setting

We study the problem of adaptive control in partially observable linear quadratic Gaussian control systems, where the model dynamics are unknown a priori. We propose LqgOpt, a novel reinforcement learning algorithm based on the principle of optimism in the face of uncertainty, to effectively minimize the overall control cost. We employ the predictor state evolution representation of the system dynamics and deploy a recently proposed closed-loop system identification method, estimation, and confidence bound construction. LqgOpt efficiently explores the system dynamics, estimates the model parameters up to their confidence interval, and deploys the controller of the most optimistic model for further exploration and exploitation. We provide stability guarantees for LqgOpt and prove the regret upper bound of O(√T) for adaptive control of linear quadratic Gaussian (LQG) systems, where T is the time horizon of the problem

Adaptive Control and Regret Minimization in Linear Quadratic Gaussian (LQG) Setting

We study the problem of adaptive control in partially observable linear quadratic Gaussian control systems, where the model dynamics are unknown a priori. We propose LqgOpt, a novel reinforcement learning algorithm based on the principle of optimism in the face of uncertainty, to effectively minimize the overall control cost. We employ the predictor state evolution representation of the system dynamics and deploy a recently proposed closed-loop system identification method, estimation, and confidence bound construction. LqgOpt efficiently explores the system dynamics, estimates the model parameters up to their confidence interval, and deploys the controller of the most optimistic model for further exploration and exploitation. We provide stability guarantees for LqgOpt and prove the regret upper bound of O(√T) for adaptive control of linear quadratic Gaussian (LQG) systems, where T is the time horizon of the problem

Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems

We study the problem of adaptive control in partially observable linear dynamical systems. We propose a novel algorithm, adaptive control online learning algorithm (AdaptOn), which efficiently explores the environment, estimates the system dynamics episodically and exploits these estimates to design effective controllers to minimize the cumulative costs. Through interaction with the environment, AdaptOn deploys online convex optimization to optimize the controller while simultaneously learning the system dynamics to improve the accuracy of controller updates. We show that when the cost functions are strongly convex, after T times step of agent-environment interaction, AdaptOn achieves regret upper bound of polylog(T). To the best of our knowledge, AdaptOn is the first algorithm which achieves polylog(T) regret in adaptive control of unknown partially observable linear dynamical systems which includes linear quadratic Gaussian (LQG) control

Statistical Learning For System Identification, Estimation, And Control

Despite the recent widespread success of machine learning, we still do not fully understand its fundamental limitations. Going forward, it is crucial to better understand learning complexity, especially in critical decision making applications, where a wrong decision can lead to catastrophic consequences. In this thesis, we focus on the statistical complexity of learning unknown linear dynamical systems, with focus on the tasks of system identification, prediction, and control. We are interested in sample complexity, i.e. the minimum number of samples required to achieve satisfactory learning performance. Our goal is to provide finite-sample learning guarantees, explicitly highlighting how the learning objective depends on the number of samples. A fundamental question we are trying to answer is how system theoretic properties of the underlying process can affect sample complexity. Using recent advances in statistical learning, high-dimensional statistics, and control theoretic tools, we provide finite-sample guarantees in the following settings. i) System Identification. We provide the first finite-sample guarantees for identifying a stochastic partially-observed system; this problem is also known as the stochastic system identification problem. ii) Prediction. We provide the first end-to-end guarantees for learning the Kalman Filter, i.e. for learning to predict, in an offline learning architecture. We also provide the first logarithmic regret guarantees for the problem of learning the Kalman Filter in an online learning architecture, where the data are revealed sequentially. iii) Difficulty of System Identification and Control. Focusing on fully-observed systems, we investigate when learning linear systems is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension. We show that there actually exist classes of linear systems, which are hard to learn. Such classes include indirectly excited systems with large degree of indirect excitation. Similar conclusions hold for both the problem of system identification and the problem of learning to control