The Approximation Ratio for the Risk of Myopic Bayesian Active Learning for Linear Regression

Stephen Mussmann

The Approximation Ratio for the Risk of Myopic Bayesian Active Learning for Linear Regression

Machine Learning 2026-07-07 v1

Authors: Stephen Mussmann

Abstract

Active learning studies the fundamental question: what data should we choose to observe? The greedy algorithm in optimal experiment design is a common heuristic and also equivalent to myopic Bayesian active learning for linear regression, the common framework where long-term planning is replaced with the one-step optimal choice. In this work, we prove a first-of-its-kind approximation ratio for the greedy algorithm's risk that is tight up to an absolute constant. The approximation ratio is linear in the maximum initial leverage score (MILS), a newly identified quantity fundamental to the greedy algorithm's performance. Finally, we illustrate the results with simple numerical simulations.

Keywords

active learning Bayesian linear regression greedy algorithm approximation ratio leverage score

Cite

@article{arxiv.2607.06642,
  title  = {The Approximation Ratio for the Risk of Myopic Bayesian Active Learning for Linear Regression},
  author = {Stephen Mussmann},
  journal= {arXiv preprint arXiv:2607.06642},
  year   = {2026}
}