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Sparsity in Partially Controllable Linear Systems
Optimization and Control · Submitted 12 Oct 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main results establish structural conditions and finite-sample guarantees for learning to control such systems.
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The abstract, sentence by sentence
A fundamental concept in control theory is that of controllability, where any system state can be reached through an appropriate choice of control inputs. Indeed, a large body of classical and modern approaches are designed for controllable linear dynamical systems. However, in practice, we often encounter systems in which a large set of state variables evolve exogenously and independently of the control inputs; such systems are only partially controllable. The focus of this work is on a large class of partially controllable linear dynamical systems, specified by an underlying sparsity pattern. Our main results establish structural conditions and finite-sample guarantees for learning to control such systems. In particular, our structural results characterize those state variables which are irrelevant for optimal control, an analysis which departs from classical control techniques. Our algorithmic results adapt techniques from high-dimensional statistics -- specifically soft-thresholding and semiparametric least-squares -- to exploit the underlying sparsity pattern in order to obtain finite-sample guarantees that significantly improve over those based on certainty-equivalence. We also corroborate these theoretical improvements over certainty-equivalent control through a simulation study.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A fundamental concept in control theory is that of controllability, where any system state can be reached through an appropriate choice of control inputs. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | Indeed, a large body of classical and modern approaches are designed for controllable linear dynamical systems. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | However, in practice, we often encounter systems in which a large set of state variables evolve exogenously and independently of the control inputs; such systems are only partially controllable. | background 0.88 | 0.08 | 0.03 | asserted 0.92 |
| s3 | The focus of this work is on a large class of partially controllable linear dynamical systems, specified by an underlying sparsity pattern. | contribution 0.90 | 0.44 | 0.07 | asserted 0.97 |
| s4 | Our main results establish structural conditions and finite-sample guarantees for learning to control such systems. | contribution 0.61 | 0.95 | 0.81 | asserted 1.00 |
| s5 | In particular, our structural results characterize those state variables which are irrelevant for optimal control, an analysis which departs from classical control techniques. | result 0.76 | 0.94 | 0.19 | asserted 1.00 |
| s6 | Our algorithmic results adapt techniques from high-dimensional statistics -- specifically soft-thresholding and semiparametric least-squares -- to exploit the underlying sparsity pattern in order to obtain finite-sample guarantees that significantly improve over those based on certainty-equivalence. | method 0.84 | 0.93 | 0.20 | asserted 1.00 |
| s7 | We also corroborate these theoretical improvements over certainty-equivalent control through a simulation study. | result 0.47 | 0.95 | 0.09 | asserted 0.98 |
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The paper
Original abstract
A fundamental concept in control theory is that of controllability, where any system state can be reached through an appropriate choice of control inputs. Indeed, a large body of classical and modern approaches are designed for controllable linear dynamical systems. However, in practice, we often encounter systems in which a large set of state variables evolve exogenously and independently of the control inputs; such systems are only partially controllable. The focus of this work is on a large class of partially controllable linear dynamical systems, specified by an underlying sparsity pattern. Our main results establish structural conditions and finite-sample guarantees for learning to control such systems. In particular, our structural results characterize those state variables which are irrelevant for optimal control, an analysis which departs from classical control techniques. Our algorithmic results adapt techniques from high-dimensional statistics -- specifically soft-thresholding and semiparametric least-squares -- to exploit the underlying sparsity pattern in order to obtain finite-sample guarantees that significantly improve over those based on certainty-equivalence. We also corroborate these theoretical improvements over certainty-equivalent control through a simulation study.
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How the literature uses it
2 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Compares with it 1 paper
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Unlike our work, [ref] assumes that the inherent system noise induces sufficient exploration, and hence does not address the exploration problem.
Sample-Efficient Reinforcement Learning in the Presence of Exogenous Information · Related Work · 0.97
Cites it as background 2 papers
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The sample complexity of system identification was studied extensively in the setting of fully observed linear systems [ref] as well as partially-observed systems [ref] .
Learning to Control Linear Systems can be Hard · Related Work · 0.98
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Also related to this line of research is [ref] , which considers a linear control setting with exogenous observations.
Sample-Efficient Reinforcement Learning in the Presence of Exogenous Information · Related Work · 0.85
What it cites · 3 sentences without a confident label
Uses the method or tool of 2 papers
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Instead of using classical least-squares, our approach is based on a reduction to semiparametric least-squares [ref] , which, as we will see, results in a sample complexity of $1/ϵ^2 + d/ϵ$ for entrywise estimation.
Locally Robust Semiparametric Estimation · Positive definite covariance matrix · 0.96
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Instead of using classical least-squares, our approach is based on a reduction to semiparametric least-squares [ref] , which, as we will see, results in a sample complexity of $1/ϵ^2 + d/ϵ$ for entrywise estimation.
Orthogonal Statistical Learning · Positive definite covariance matrix · 0.96
Cites as background 6 papers
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Recently, there has been a surge of interest in the learnability of LQ [ref] .
Linear Systems can be Hard to Learn · Related Work · 1.00
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Both approaches have been utilized in the context of decision making, the former via dimension-free linear control [ref] and the Kernelized Nonlinear Regulator [ref] , and the latter both in RL [ref] and some works on continuous control [ref] .
Active Learning for Nonlinear System Identification with Guarantees · Introduction · 0.89
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Both approaches have been utilized in the context of decision making, the former via dimension-free linear control [ref] and the Kernelized Nonlinear Regulator [ref] , and the latter both in RL [ref] and some works on continuous control [ref] .
Information Theoretic Regret Bounds for Online Nonlinear Control · Introduction · 0.89
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