We study the problem of adaptive control of the stochastic linear quadratic regulator (LQR) with constraints that must be satisfied at every time step. Prior work on the multidimensional problem has shown $\tilde{O}(T^{2/3})$ regret and satisfaction of robust constraints, leaving open the question of whether $\tilde{O}(\sqrt{T})$ regret can be attained in the constrained LQR setting. We contribute to this problem by showing $\tilde{O}(\sqrt{T})$ regret and satisfaction of chance constraints. This type of constraints allow us to handle unbounded noise and also enable analytical techniques not directly applicable to robust constraints. Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe. Our theoretical analysis establishes regret and constraint guarantees via a key lemma that bounds the system covariance in terms of the chosen policy. This covariance-based analysis is in contrast with the cost-to-go based analysis that is typically used in adaptive LQR.
We consider adaptive control of the Linear Quadratic Regulator (LQR), where an unknown linear system is controlled subject to quadratic costs. Leveraging recent developments in the estimation of linear systems and in robust controller synthesis, we present the first provably polynomial time algorithm that provides high probability guarantees of sub-linear regret on this problem. We further study the interplay between regret minimization and parameter estimation by proving a lower bound on the expected regret in terms of the exploration schedule used by any algorithm. Finally, we conduct a nume…
The problem of regret minimization for online adaptive control of linear-quadratic systems is studied. In this problem, the true system transition parameters (matrices $A$ and $B$) are unknown, and the objective is to design and analyze algorithms that generate control policies with sublinear regret. Recent studies show that when the system parameters are fully unknown, there exists a choice of these parameters such that any algorithm that only uses data from the past system trajectory at best achieves a square root of time horizon regret bound, providing a hard fundamental limit on the achiev…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；SEAS Harvard University（哈佛大学工程与应用科学学院）
We study the adaptive control of an unknown linear system with a quadratic cost function subject to safety constraints on both the states and actions. The challenges of this problem arise from the tension among safety, exploration, performance, and computation. To address these challenges, we propose a polynomial-time algorithm that guarantees feasibility and constraint satisfaction with high probability under proper conditions. Our algorithm is implemented on a single trajectory and does not require system restarts. Further, we analyze the regret of our learning algorithm compared to the opti…
Learning Decentralized Linear Quadratic Regulators with $\sqrt{T}$ Regret
School of Artificial Intelligence and Automation at the Huazhong University of Science and Technology（华中科技大学人工智能与自动化学院）；College of Petroleum Engineering, Yangtze University（长江大学石油工程学院）；Elmore Family School of Electrical and Computer Engineering at Purdue University（普渡大学埃尔莫尔家族电气与计算机工程学院）
This paper addresses safe distributed online optimization over an unknown set of linear safety constraints. A network of agents aims at jointly minimizing a global, time-varying function, which is only partially observable to each individual agent. Therefore, agents must engage in local communications to generate a safe sequence of actions competitive with the best minimizer sequence in hindsight, and the gap between the two sequences is quantified via dynamic regret. We propose distributed safe online gradient descent (D-Safe-OGD) with an exploration phase, where all agents estimate the const…
We study the constrained linear quadratic regulator with unknown dynamics, addressing the tension between safety and exploration in data-driven control techniques. We present a framework which allows for system identification through persistent excitation, while maintaining safety by guaranteeing the satisfaction of state and input constraints. This framework involves a novel method for synthesizing robust constraint-satisfying feedback controllers, leveraging newly developed tools from system level synthesis. We connect statistical results with cost sub-optimality bounds to give non-asymptoti…
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We study optimal regret bounds for control in linear dynamical systems under adversarially changing strongly convex cost functions, given the knowledge of transition dynamics. This includes several well studied and fundamental frameworks such as the Kalman filter and the linear quadratic regulator. State of the art methods achieve regret which scales as $O(\sqrt{T})$, where $T$ is the time horizon. We show that the optimal regret in this setting can be significantly smaller, scaling as $O(\text{poly}(\log T))$. This regret bound is achieved by two different efficient iterative methods, online …
We study the problem of safe online convex optimization, where the action at each time step must satisfy a set of linear safety constraints. The goal is to select a sequence of actions to minimize the regret without violating the safety constraints at any time step (with high probability). The parameters that specify the linear safety constraints are unknown to the algorithm. The algorithm has access to only the noisy observations of constraints for the chosen actions. We propose an algorithm, called the {Safe Online Projected Gradient Descent} (SO-PGD) algorithm, to address this problem. We s…
In this paper, we study the dynamic regret of online linear quadratic regulator (LQR) control with time-varying cost functions and disturbances. We consider the case where a finite look-ahead window of cost functions and disturbances is available at each stage. The online control algorithm studied in this paper falls into the category of model predictive control (MPC) with a particular choice of terminal costs to ensure the exponential stability of MPC. It is proved that the regret of such an online algorithm decays exponentially fast with the length of predictions. The impact of inaccurate pr…
This paper presents local asymptotic minimax regret lower bounds for adaptive Linear Quadratic Regulators (LQR). We consider affinely parametrized $B$-matrices and known $A$-matrices and aim to understand when logarithmic regret is impossible even in the presence of structural side information. After defining the intrinsic notion of an uninformative optimal policy in terms of a singularity condition for Fisher information we obtain local minimax regret lower bounds for such uninformative instances of LQR by appealing to van Trees' inequality (Bayesian Cramér-Rao) and a representation of regret…
The Australian National University（澳大利亚国立大学）；The University of Texas at Dallas（德克萨斯大学达拉斯分校）
In this paper, we propose and analyze a new method for online linear quadratic regulator (LQR) control with a priori unknown time-varying cost matrices. The cost matrices are revealed sequentially with the potential for future values to be previewed over a short window. Our novel method involves using the available cost matrices to predict the optimal trajectory, and a tracking controller to drive the system towards it. We adopted the notion of dynamic regret to measure the performance of this proposed online LQR control method, with our main result being that the (dynamic) regret of our metho…
Safe and Efficient Online Convex Optimization with Linear Budget Constraints and Partial Feedback
Research Institute of China Electronics Technology Group（中国电子科技集团公司研究所）；School of Information Science and Technology（信息科学与技术学院）；ShanghaiTech University（上海科技大学）
The Linear Quadratic Regulator (LQR) framework considers the problem of regulating a linear dynamical system perturbed by environmental noise. We compute the policy regret between three distinct control policies: i) the optimal online policy, whose linear structure is given by the Ricatti equations; ii) the optimal offline linear policy, which is the best linear state feedback policy given the noise sequence; and iii) the optimal offline policy, which selects the globally optimal control actions given the noise sequence. We fully characterize the optimal offline policy and show that it has a r…
We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within $\varepsilon$ of the optimal LQR controller, each step of policy evaluation requires at most $(n…
We study the impact of predictions in online Linear Quadratic Regulator control with both stochastic and adversarial disturbances in the dynamics. In both settings, we characterize the optimal policy and derive tight bounds on the minimum cost and dynamic regret. Perhaps surprisingly, our analysis shows that the conventional greedy MPC approach is a near-optimal policy in both stochastic and adversarial settings. Specifically, for length-$T$ problems, MPC requires only $O(\log T)$ predictions to reach $O(1)$ dynamic regret, which matches (up to lower-order terms) our lower bound on the require…
University of Pennsylvania（宾夕法尼亚大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Lund University（隆德大学）；Google Brain（谷歌大脑）
Machine and reinforcement learning (RL) are increasingly being applied to plan and control the behavior of autonomous systems interacting with the physical world. Examples include self-driving vehicles, distributed sensor networks, and agile robots. However, when machine learning is to be applied in these new settings, the algorithms had better come with the same type of reliability, robustness, and safety bounds that are hallmarks of control theory, or failures could be catastrophic. Thus, as learning algorithms are increasingly and more aggressively deployed in safety critical settings, it i…
Online optimization has recently opened avenues to study optimal control for time-varying cost functions that are unknown in advance. Inspired by this line of research, we study the distributed online linear quadratic regulator (LQR) problem for linear time-invariant (LTI) systems with unknown dynamics. Consider a multi-agent network where each agent is modeled as a LTI system. The network has a global time-varying quadratic cost, which may evolve adversarially and is only partially observed by each agent sequentially. The goal of the network is to collectively (i) estimate the unknown dynamic…
Stanford University（斯坦福大学）；The University of Chicago（芝加哥大学）
We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon $T$ with fixed and known cost matrices $Q,R$, but unknown and non-stationary dynamics $\{A_t, B_t\}$. The sequence of dynamics matrices can be arbitrary, but with a total variation, $V_T$, assumed to be $o(T)$ and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all $t$, we present an algorithm that achieves the optimal dynamic regret of $\tilde{\mathcal{O}}\left(V_T^{2/5}T^{3/5}\right)$. With piece-wise c…
California Institute of Technology（加州理工学院）；Purdue University（普渡大学）
Thompson Sampling (TS) is an efficient method for decision-making under uncertainty, where an action is sampled from a carefully prescribed distribution which is updated based on the observed data. In this work, we study the problem of adaptive control of stabilizable linear-quadratic regulators (LQRs) using TS, where the system dynamics are unknown. Previous works have established that $\tilde O(\sqrt{T})$ frequentist regret is optimal for the adaptive control of LQRs. However, the existing methods either work only in restrictive settings, require a priori known stabilizing controllers, or ut…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
We study the problem of adaptive control of the linear quadratic regulator for systems in very high, or even infinite dimension. We demonstrate that while sublinear regret requires finite dimensional inputs, the ambient state dimension of the system need not be bounded in order to perform online control. We provide the first regret bounds for LQR which hold for infinite dimensional systems, replacing dependence on ambient dimension with more natural notions of problem complexity. Our guarantees arise from a novel perturbation bound for certainty equivalence which scales with the prediction err…
KTH Royal Institute of Technology（皇家理工学院（KTH））；Stockholm, Sweden（瑞典斯德哥尔摩）
We consider the problem of online learning in Linear Quadratic Control systems whose state transition and state-action transition matrices $A$ and $B$ may be initially unknown. We devise an online learning algorithm and provide guarantees on its expected regret. This regret at time $T$ is upper bounded (i) by $\widetilde{O}((d_u+d_x)\sqrt{d_xT})$ when $A$ and $B$ are unknown, (ii) by $\widetilde{O}(d_x^2\log(T))$ if only $A$ is unknown, and (iii) by $\widetilde{O}(d_x(d_u+d_x)\log(T))$ if only $B$ is unknown and under some mild non-degeneracy condition ($d_x$ and $d_u$ denote the dimensions of…
The filtering problem of causally estimating a desired signal from a related observation signal is investigated through the lens of regret optimization. Classical filter designs, such as $\mathcal H_2$ (Kalman) and $\mathcal H_\infty$, minimize the average and worst-case estimation errors, respectively. As a result $\mathcal H_2$ filters are sensitive to inaccuracies in the underlying statistical model, and $\mathcal H_\infty$ filters are overly conservative since they safeguard against the worst-case scenario. We propose instead to minimize the \emph{regret} in order to design filters that pe…
