Linear Systems can be Hard to Learn

by   Anastasios Tsiamis, et al.

In this paper, we investigate when system identification is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Most prior research in the finite sample regime falls in this category, focusing on systems that are directly excited by process noise. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension, regardless of the identification algorithm. Using tools from minimax theory, we show that classes of linear systems can be hard to learn. Such classes include, for example, under-actuated or under-excited systems with weak coupling among the states. Having classified some systems as easy or hard to learn, a natural question arises as to what system properties fundamentally affect the hardness of system identifiability. Towards this direction, we characterize how the controllability index of linear systems affects the sample complexity of identification. More specifically, we show that the sample complexity of robustly controllable linear systems is upper bounded by an exponential function of the controllability index. This implies that identification is easy for classes of linear systems with small controllability index and potentially hard if the controllability index is large. Our analysis is based on recent statistical tools for finite sample analysis of system identification as well as a novel lower bound that relates controllability index with the least singular value of the controllability Gramian.


page 1

page 2

page 3

page 4


Sample Complexity Lower Bounds for Linear System Identification

This paper establishes problem-specific sample complexity lower bounds f...

Finite-Sample Analysis for SARSA and Q-Learning with Linear Function Approximation

Though the convergence of major reinforcement learning algorithms has be...

Learning to Control Linear Systems can be Hard

In this paper, we study the statistical difficulty of learning to contro...

Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm

In this paper, we investigate the sample complexity of policy evaluation...

Minimax Learning of Ergodic Markov Chains

We compute the finite-sample minimax (modulo logarithmic factors) sample...

Empirical or Invariant Risk Minimization? A Sample Complexity Perspective

Recently, invariant risk minimization (IRM) was proposed as a promising ...

Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers

A common lens to theoretically study neural net architectures is to anal...