Online A-Optimal Design and Active Linear Regression

TL;DR

This paper introduces an online active sampling algorithm for heteroscedastic linear regression, optimizing experiment design to minimize estimation error with theoretical guarantees and empirical validation.

Contribution

It proposes a novel active sampling algorithm for heteroscedastic linear regression with theoretical regret bounds and empirical validation, advancing online experiment design methods.

Findings

Achieved an $ ext{O}(T^{-2})$ regret bound in basis covariate settings.

Validated theoretical results through numerical experiments.

Improved upon existing algorithms for online A-optimal design.

Abstract

We consider in this paper the problem of optimal experiment design where a decision maker can choose which points to sample to obtain an estimate $\hat{\beta}$ of the hidden parameter $\beta^{\star}$ of an underlying linear model. The key challenge of this work lies in the heteroscedasticity assumption that we make, meaning that each covariate has a different and unknown variance. The goal of the decision maker is then to figure out on the fly the optimal way to allocate the total budget of $T$ samples between covariates, as sampling several times a specific one will reduce the variance of the estimated model around it (but at the cost of a possible higher variance elsewhere). By trying to minimize the $\ell^2$-loss $\mathbb{E} [\lVert\hat{\beta}-\beta^{\star}\rVert^2]$ the decision maker is actually minimizing the trace of the covariance matrix of the problem, which corresponds then to online A-optimal design. Combining techniques from bandit and convex optimization we propose a new active sampling algorithm and we compare it with existing ones. We provide theoretical guarantees of this algorithm in different settings, including a $\mathcal{O}(T^{-2})$ regret bound in the case where the covariates form a basis of the feature space, generalizing and improving existing results. Numerical experiments validate our theoretical findings.