# Towards a Dimension-Free Understanding of Adaptive Linear Control

arXiv: https://arxiv.org/abs/2103.10620 · HTML: https://mlsanity.com/paper/2103.10620/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 67 references that are in the map.
- First posted on arXiv 19 Mar 2021.

## Builds on (most cited first, 4 of 4)

- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [On Adaptive Linear-Quadratic Regulators](https://mlsanity.com/paper/1806.10749/) (2018)
- [On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces](https://mlsanity.com/paper/2011.04622/) (2020)
- [Non-Asymptotic Analysis of Robust Control from Coarse-Grained Identification](https://mlsanity.com/paper/1707.04791/) (2017)

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

- [Stabilizing Dynamical Systems via Policy Gradient Methods](https://mlsanity.com/paper/2110.06418/) (2021)
- [Sparsity in Partially Controllable Linear Systems](https://mlsanity.com/paper/2110.06150/) (2021)

## Abstract (arXiv)

We study the problem of adaptive control of the linear quadratic regulator for systems in very high, or even infinite dimension. We demonstrate that while sublinear regret requires finite dimensional inputs, the ambient state dimension of the system need not be bounded in order to perform online control. We provide the first regret bounds for LQR which hold for infinite dimensional systems, replacing dependence on ambient dimension with more natural notions of problem complexity. Our guarantees arise from a novel perturbation bound for certainty equivalence which scales with the prediction error in estimating the system parameters, without requiring consistent parameter recovery in more stringent measures like the operator norm. When specialized to finite dimensional settings, our bounds recover near optimal dimension and time horizon dependence.