DeepAI

# Learning to Control Linear Systems can be Hard

In this paper, we study the statistical difficulty of learning to control linear systems. We focus on two standard benchmarks, the sample complexity of stabilization, and the regret of the online learning of the Linear Quadratic Regulator (LQR). Prior results state that the statistical difficulty for both benchmarks scales polynomially with the system state dimension up to system-theoretic quantities. However, this does not reveal the whole picture. By utilizing minimax lower bounds for both benchmarks, we prove that there exist non-trivial classes of systems for which learning complexity scales dramatically, i.e. exponentially, with the system dimension. This situation arises in the case of underactuated systems, i.e. systems with fewer inputs than states. Such systems are structurally difficult to control and their system theoretic quantities can scale exponentially with the system dimension dominating learning complexity. Under some additional structural assumptions (bounding systems away from uncontrollability), we provide qualitatively matching upper bounds. We prove that learning complexity can be at most exponential with the controllability index of the system, that is the degree of underactuation.

• 13 publications
• 8 publications
• 15 publications
• 36 publications
• 71 publications
01/27/2020

### Naive Exploration is Optimal for Online LQR

We consider the problem of online adaptive control of the linear quadrat...
04/02/2021

### Linear Systems can be Hard to Learn

In this paper, we investigate when system identification is statisticall...
09/29/2021

### Minimal Expected Regret in Linear Quadratic Control

We consider the problem of online learning in Linear Quadratic Control s...
12/03/2020

### Online learning with dynamics: A minimax perspective

We study the problem of online learning with dynamics, where a learner i...
11/13/2021

### Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds

Learning how to effectively control unknown dynamical systems is crucial...
03/19/2021

### Towards a Dimension-Free Understanding of Adaptive Linear Control

We study the problem of adaptive control of the linear quadratic regulat...