# The Fundamental Limitations of Learning Linear-Quadratic Regulators

arXiv: https://arxiv.org/abs/2303.15637 · HTML: https://mlsanity.com/paper/2303.15637/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2023.

## Built on by (most cited first, 2 of 2)

- [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023)
- [Active Learning for Control-Oriented Identification of Nonlinear Systems](https://mlsanity.com/paper/2404.09030/) (2024)

## Abstract (arXiv)

We present a local minimax lower bound on the excess cost of designing a linear-quadratic controller from offline data. The bound is valid for any offline exploration policy that consists of a stabilizing controller and an energy bounded exploratory input. The derivation leverages a relaxation of the minimax estimation problem to Bayesian estimation, and an application of Van Trees' inequality. We show that the bound aligns with system-theoretic intuition. In particular, we demonstrate that the lower bound increases when the optimal control objective value increases. We also show that the lower bound increases when the system is poorly excitable, as characterized by the spectrum of the controllability gramian of the system mapping the noise to the state and the H_∞ norm of the system mapping the input to the state. We further show that for some classes of systems, the lower bound may be exponential in the state dimension, demonstrating exponential sample complexity for learning the linear-quadratic regulator offline.