# Finite Time Regret Bounds for Minimum Variance Control of Autoregressive Systems with Exogenous Inputs

arXiv: https://arxiv.org/abs/2305.16974 · HTML: https://mlsanity.com/paper/2305.16974/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 11 of 11)

- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently](https://mlsanity.com/paper/2002.08095/) (2020)
- [Safely Learning to Control the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/1809.10121/) (2018)
- [On Adaptive Linear-Quadratic Regulators](https://mlsanity.com/paper/1806.10749/) (2018)
- [Reinforcement Learning with Fast Stabilization in Linear Dynamical Systems](https://mlsanity.com/paper/2007.12291/) (2020)
- [Input Perturbations for Adaptive Control and Learning](https://mlsanity.com/paper/1811.04258/) (2018)
- [Thompson Sampling for Linear-Quadratic Control Problems](https://mlsanity.com/paper/1703.08972/) (2017)
- [Minimal Expected Regret in Linear Quadratic Control](https://mlsanity.com/paper/2109.14429/) (2021)
- [Augmented RBMLE-UCB Approach for Adaptive Control of Linear Quadratic Systems](https://mlsanity.com/paper/2201.10542/) (2022)

## Abstract (arXiv)

Minimum variance controllers have been employed in a wide-range of industrial applications. A key challenge experienced by many adaptive controllers is their poor empirical performance in the initial stages of learning. In this paper, we address the problem of initializing them so that they provide acceptable transients, and also provide an accompanying finite-time regret analysis, for adaptive minimum variance control of an auto-regressive system with exogenous inputs (ARX). Following [3], we consider a modified version of the Certainty Equivalence (CE) adaptive controller, which we call PIECE, that utilizes probing inputs for exploration. We show that it has a C log T bound on the regret after T time-steps for bounded noise, and Clog2 T in the case of sub-Gaussian noise. The simulation results demonstrate the advantage of PIECE over the algorithm proposed in [3] as well as the standard Certainty Equivalence controller especially in the initial learning phase. To the best of our knowledge, this is the first work that provides finite-time regret bounds for an adaptive minimum variance controller.