Linear Quadratic Regulators (LQR) achieve enormous successful real-world applications. Very recently, people have been focusing on efficient learning algorithms for LQRs when their dynamics are unknown. Existing results effectively learn to control the unknown system using number of episodes depending polynomially on the system parameters, including the ambient dimension of the states. These traditional approaches, however, become inefficient in common scenarios, e.g., when the states are high-resolution images. In this paper, we propose an algorithm that utilizes the intrinsic system low-rank structure for efficient learning. For problems of rank-$m$, our algorithm achieves a $K$-episode regret bound of order $\widetilde{O}(m^{3/2} K^{1/2})$. Consequently, the sample complexity of our algorithm only depends on the rank, $m$, rather than the ambient dimension, $d$, which can be orders-of-magnitude larger.
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…
University of Notre Dame（圣母大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We study the problem of control policy design for decentralized state-feedback linear quadratic control with a partially nested information structure, when the system model is unknown. We propose a model-based learning solution, which consists of two steps. First, we estimate the unknown system model from a single system trajectory of finite length, using least squares estimation. Next, based on the estimated system model, we design a control policy that satisfies the desired information structure. We show that the suboptimality gap between our control policy and the optimal decentralized cont…
Gradient Episodic Memory is indeed a novel method for continual learning, which solves new problems quickly without forgetting previously acquired knowledge. However, in the process of studying the paper, we found there were some problems in the proof of the dual problem of Quadratic Program, so here we give our fixed version for this problem.
Approximate Thompson Sampling for Learning Linear Quadratic Regulators with $O(\sqrt{T})$ Regret
Department of Applied Artificial Intelligence, Seoul National University of Science and Technology（应用人工智能系，首尔科学技术大学）；Department of Electrical and Computer Engineering and ASRI, Seoul National University（电气电子工程系和ASRI，首尔国立大学）
University of California, Berkeley（加州大学伯克利分校）；California Institute of Technology（加州理工学院）
This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that …
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）
Risk-aware control, though with promise to tackle unexpected events, requires a known exact dynamical model. In this work, we propose a model-free framework to learn a risk-aware controller with a focus on the linear system. We formulate it as a discrete-time infinite-horizon LQR problem with a state predictive variance constraint. To solve it, we parameterize the policy with a feedback gain pair and leverage primal-dual methods to optimize it by solely using data. We first study the optimization landscape of the Lagrangian function and establish the strong duality in spite of its non-convex n…
We introduce a new problem setting for continuous control called the LQR with Rich Observations, or RichLQR. In our setting, the environment is summarized by a low-dimensional continuous latent state with linear dynamics and quadratic costs, but the agent operates on high-dimensional, nonlinear observations such as images from a camera. To enable sample-efficient learning, we assume that the learner has access to a class of decoder functions (e.g., neural networks) that is flexible enough to capture the mapping from observations to latent states. We introduce a new algorithm, RichID, which lea…
On the Optimization Landscape of Dynamic Output Feedback: A Case Study for Linear Quadratic Regulator
National University of Singapore（新加坡国立大学）；University of Science and Technology Beijing（北京科技大学）；Tsinghua University（清华大学）；University of California San Diego（加利福尼亚大学圣迭戈分校）
University of Southern California（南加州大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this paper, we take a step towards demystifying the performance and efficiency of such methods by focusing on the standard infinite-horizon linear quadratic regulator problem for continuous-time systems with u…
Model-Free Learning for the Linear Quadratic Regulator over Rate-Limited Channels
School of Artificial Intelligence and Automation, Huazhong University of Science and Technology（华中科技大学人工智能与自动化学院）；Department of Electrical and Computer Engineering, North Carolina State University（北卡罗来纳州立大学电气与计算机工程系）；The Elmore Family School of Electrical and Computer Engineering, Purdue University（普渡大学埃尔莫尔家族电气与计算机工程学院）
EDF R&D（法国电力公司研发部）；FIME（FIME实验室）；University of California, Berkeley（加州大学伯克利分校）；University of Oxford（牛津大学）
We study finite-time horizon continuous-time linear-quadratic reinforcement learning problems in an episodic setting, where both the state and control coefficients are unknown to the controller. We first propose a least-squares algorithm based on continuous-time observations and controls, and establish a logarithmic regret bound of order $O((\ln M)(\ln\ln M))$, with $M$ being the number of learning episodes. The analysis consists of two parts: perturbation analysis, which exploits the regularity and robustness of the associated Riccati differential equation; and parameter estimation error, whi…
Google Tel Aviv（谷歌特拉维夫）；Google Brain（谷歌大脑）；Tel-Aviv University（特拉维夫大学）
We present the first computationally-efficient algorithm with $\widetilde O(\sqrt{T})$ regret for learning in Linear Quadratic Control systems with unknown dynamics. By that, we resolve an open question of Abbasi-Yadkori and Szepesvári (2011) and Dean, Mania, Matni, Recht, and Tu (2018).
National University of Singapore（新加坡国立大学）；University of Science and Technology Beijing（北京科技大学）；The Ohio State University（俄亥俄州立大学）
The actor-critic (AC) reinforcement learning algorithms have been the powerhouse behind many challenging applications. Nevertheless, its convergence is fragile in general. To study its instability, existing works mostly consider the uncommon double-loop variant or basic models with finite state and action space. We investigate the more practical single-sample two-timescale AC for solving the canonical linear quadratic regulator (LQR) problem, where the actor and the critic update only once with a single sample in each iteration on an unbounded continuous state and action space. Existing analys…
We propose a single time-scale actor-critic algorithm to solve the linear quadratic regulator (LQR) problem. A least squares temporal difference (LSTD) method is applied to the critic and a natural policy gradient method is used for the actor. We give a proof of convergence with sample complexity $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1})^2)$. The method in the proof is applicable to general single time-scale bilevel optimization problem. We also numerically validate our theoretical results on the convergence.
School of Computer Science, Tel Aviv University（特拉维夫大学计算机科学学院）；Google Research（谷歌研究院）
We consider the task of learning to control a linear dynamical system under fixed quadratic costs, known as the Linear Quadratic Regulator (LQR) problem. While model-free approaches are often favorable in practice, thus far only model-based methods, which rely on costly system identification, have been shown to achieve regret that scales with the optimal dependence on the time horizon T. We present the first model-free algorithm that achieves similar regret guarantees. Our method relies on an efficient policy gradient scheme, and a novel and tighter analysis of the cost of exploration in polic…
University of Southern California（南加州大学）；T.U. Kaiserslautern（凯泽斯劳滕工业大学）
We study the variance of the REINFORCE policy gradient estimator in environments with continuous state and action spaces, linear dynamics, quadratic cost, and Gaussian noise. These simple environments allow us to derive bounds on the estimator variance in terms of the environment and noise parameters. We compare the predictions of our bounds to the empirical variance in simulation experiments.
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…
Criteo AI Lab（Criteo人工智能实验室）；Facebook AI Research（Facebook人工智能研究院）
We study the exploration-exploitation dilemma in the linear quadratic regulator (LQR) setting. Inspired by the extended value iteration algorithm used in optimistic algorithms for finite MDPs, we propose to relax the optimistic optimization of \ofulq and cast it into a constrained \textit{extended} LQR problem, where an additional control variable implicitly selects the system dynamics within a confidence interval. We then move to the corresponding Lagrangian formulation for which we prove strong duality. As a result, we show that an $ε$-optimistic controller can be computed efficiently by sol…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider the problem of learning in Linear Quadratic Control systems whose transition parameters are initially unknown. Recent results in this setting have demonstrated efficient learning algorithms with regret growing with the square root of the number of decision steps. We present new efficient algorithms that achieve, perhaps surprisingly, regret that scales only (poly)logarithmically with the number of steps in two scenarios: when only the state transition matrix $A$ is unknown, and when only the state-action transition matrix $B$ is unknown and the optimal policy satisfies a certain no…
Reinforcement learning (RL) has been successfully used to solve many continuous control tasks. Despite its impressive results however, fundamental questions regarding the sample complexity of RL on continuous problems remain open. We study the performance of RL in this setting by considering the behavior of the Least-Squares Temporal Difference (LSTD) estimator on the classic Linear Quadratic Regulator (LQR) problem from optimal control. We give the first finite-time analysis of the number of samples needed to estimate the value function for a fixed static state-feedback policy to within $\var…
University of Washington（华盛顿大学）；Duke University（杜克大学）
Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model 2) they are an "end-to-end" approach, directly optimizing the performance metric of interest 3) they inherently allow for richly parameterized policies. A notable drawback is that even in the most basic continuous control problem (that of linear quadratic regulators), these methods must solve a non-convex optimization problem, where little is understood about their efficiency f…
Northwestern University（西北大学）；University of Minnesota Twin Cities（明尼苏达大学双城分校）；Georgia Institute of Technology（佐治亚理工学院）
We study the global convergence of generative adversarial imitation learning for linear quadratic regulators, which is posed as minimax optimization. To address the challenges arising from non-convex-concave geometry, we analyze the alternating gradient algorithm and establish its Q-linear rate of convergence to a unique saddle point, which simultaneously recovers the globally optimal policy and reward function. We hope our results may serve as a small step towards understanding and taming the instability in imitation learning as well as in more general non-convex-concave alternating minimax o…
Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；Northwestern University（西北大学）；The University of Chicago Booth School of Business（芝加哥大学布斯商学院）
Multi-agent reinforcement learning has been successfully applied to a number of challenging problems. Despite these empirical successes, theoretical understanding of different algorithms is lacking, primarily due to the curse of dimensionality caused by the exponential growth of the state-action space with the number of agents. We study a fundamental problem of multi-agent linear quadratic regulator (LQR) in a setting where the agents are partially exchangeable. In this setting, we develop a hierarchical actor-critic algorithm, whose computational complexity is independent of the total number …
School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；Alan Turing Institute（艾伦·图灵研究所）；Department of Statistics, London School of Economics and Political Science（伦敦政治经济学院统计系）
We develop a probabilistic framework for analysing model-based reinforcement learning in the episodic setting. We then apply it to study finite-time horizon stochastic control problems with linear dynamics but unknown coefficients and convex, but possibly irregular, objective function. Using probabilistic representations, we study regularity of the associated cost functions and establish precise estimates for the performance gap between applying optimal feedback control derived from estimated and true model parameters. We identify conditions under which this performance gap is quadratic, impro…
We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The…
Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）；University of Minnesota（明尼苏达大学）；Northwestern University（西北大学）
Despite the empirical success of the actor-critic algorithm, its theoretical understanding lags behind. In a broader context, actor-critic can be viewed as an online alternating update algorithm for bilevel optimization, whose convergence is known to be fragile. To understand the instability of actor-critic, we focus on its application to linear quadratic regulators, a simple yet fundamental setting of reinforcement learning. We establish a nonasymptotic convergence analysis of actor-critic in this setting. In particular, we prove that actor-critic finds a globally optimal pair of actor (polic…
Pacific Northwest National Laboratory (PNNL)（太平洋西北国家实验室）
This paper discusses learning a structured feedback control to obtain sufficient robustness to exogenous inputs for linear dynamic systems with unknown state matrix. The structural constraint on the controller is necessary for many cyber-physical systems, and our approach presents a design for any generic structure, paving the way for distributed learning control. The ideas from reinforcement learning (RL) in conjunction with control-theoretic sufficient stability and performance guarantees are used to develop the methodology. First, a model-based framework is formulated using dynamic programm…
The principal task to control dynamical systems is to ensure their stability. When the system is unknown, robust approaches are promising since they aim to stabilize a large set of plausible systems simultaneously. We study linear controllers under quadratic costs model also known as linear quadratic regulators (LQR). We present two different semi-definite programs (SDP) which results in a controller that stabilizes all systems within an ellipsoid uncertainty set. We further show that the feasibility conditions of the proposed SDPs are \emph{equivalent}. Using the derived robust controller syn…
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…
The effectiveness of model-based versus model-free methods is a long-standing question in reinforcement learning (RL). Motivated by recent empirical success of RL on continuous control tasks, we study the sample complexity of popular model-based and model-free algorithms on the Linear Quadratic Regulator (LQR). We show that for policy evaluation, a simple model-based plugin method requires asymptotically less samples than the classical least-squares temporal difference (LSTD) estimator to reach the same quality of solution; the sample complexity gap between the two methods can be at least a fa…
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…
University of California Los Angeles（加利福尼亚大学洛杉矶分校）
Practitioners often rely on compute-intensive domain randomization to ensure reinforcement learning policies trained in simulation can robustly transfer to the real world. Due to unmodeled nonlinearities in the real system, however, even such simulated policies can still fail to perform stably enough to acquire experience in real environments. In this paper we propose a novel method that guarantees a stable region of attraction for the output of a policy trained in simulation, even for highly nonlinear systems. Our core technique is to use "bias-shifted" neural networks for constructing the co…
Department of Statistics, Harvard University（哈佛大学统计系）
The theory of reinforcement learning currently suffers from a mismatch between its empirical performance and the theoretical characterization of its performance, with consequences for, e.g., the understanding of sample efficiency, safety, and robustness. The linear quadratic regulator with unknown dynamics is a fundamental reinforcement learning setting with significant structure in its dynamics and cost function, yet even in this setting there is a gap between the best known regret lower-bound of $Ω_p(\sqrt{T})$ and the best known upper-bound of $O_p(\sqrt{T}\,\text{polylog}(T))$. The contrib…
RWTH Aachen University（亚琛工业大学）；Ingenieurgesellschaft Auto und Verkehr (IAV)（IAV汽车与交通工程公司）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Probabilistic models such as Gaussian processes (GPs) are powerful tools to learn unknown dynamical systems from data for subsequent use in control design. While learning-based control has the potential to yield superior performance in demanding applications, robustness to uncertainty remains an important challenge. Since Bayesian methods quantify uncertainty of the learning results, it is natural to incorporate these uncertainties into a robust design. In contrast to most state-of-the-art approaches that consider worst-case estimates, we leverage the learning method's posterior distribution i…
Near-Optimal Distributed Linear-Quadratic Regulator for Networked Systems
Mathematics and Computer Science Division, Argonne National Laboratory（阿贡国家实验室数学与计算机科学部）；California Institute of Technology（加州理工学院）；Department of Electrical and Computer Engineering, Carnegie Mellon University（卡内基梅隆大学电气与计算机工程系）；Department of Statistics, University of Chicago（芝加哥大学统计系）
Constrained Markov Decision Processes (CMDPs) formalize sequential decision-making problems whose objective is to minimize a cost function while satisfying constraints on various cost functions. In this paper, we consider the setting of episodic fixed-horizon CMDPs. We propose an online algorithm which leverages the linear programming formulation of finite-horizon CMDP for repeated optimistic planning to provide a probably approximately correct (PAC) guarantee on the number of episodes needed to ensure an $ε$-optimal policy, i.e., with resulting objective value within $ε$ of the optimal value …
