# Sample Complexity Lower Bounds for Linear System Identification

## TL;DR

This paper derives problem-specific lower bounds on the sample complexity for linear system identification, highlighting the inherent difficulty based on system properties and providing tight bounds for uncontrolled systems.

## Contribution

It introduces explicit, system-dependent lower bounds on sample complexity in the PAC framework for both uncontrolled and controlled linear systems, advancing understanding of identification hardness.

## Key findings

- Lower bounds depend on the system's controllability gramian.
- Bounds are tight for many uncontrolled systems.
- Insights into control policy design with minimal samples.

## Abstract

This paper establishes problem-specific sample complexity lower bounds for linear system identification problems. The sample complexity is defined in the PAC framework: it corresponds to the time it takes to identify the system parameters with prescribed accuracy and confidence levels. By problem-specific, we mean that the lower bound explicitly depends on the system to be identified (which contrasts with minimax lower bounds), and hence really captures the identification hardness specific to the system. We consider both uncontrolled and controlled systems. For uncontrolled systems, the lower bounds are valid for any linear system, stable or not, and only depend of the system finite-time controllability gramian. A simplified lower bound depending on the spectrum of the system only is also derived. In view of recent finitetime analysis of classical estimation methods (e.g. ordinary least squares), our sample complexity lower bounds are tight for many systems. For controlled systems, our lower bounds are not as explicit as in the case of uncontrolled systems, but could well provide interesting insights into the design of control policy with minimal sample complexity.