On the Sample Complexity of the Linear Quadratic Regulator

  • Sarah Dean
  • Horia Mania
  • Nikolai Matni
  • Benjamin RechtEmail author
  • Stephen Tu


This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage 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 enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.


Optimal control Robust control System identification Statistical learning theory Reinforcement learning System level synthesis 

Mathematics Subject Classification

49N05 93D09 93D21 93E12 93E35 



We thank Ross Boczar, Qingqing Huang, Laurent Lessard, Michael Littman, Manfred Morari, Andrew Packard, Anders Rantzer, Daniel Russo, and Ludwig Schmidt for many helpful comments and suggestions. We also thank the anonymous referees for making several suggestions that have significantly improved the paper and its presentation.

Supplementary material


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Copyright information

© SFoCM 2019

Authors and Affiliations

  • Sarah Dean
    • 1
  • Horia Mania
    • 1
  • Nikolai Matni
    • 2
  • Benjamin Recht
    • 1
    Email author
  • Stephen Tu
    • 1
  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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