Experimental Designs for Heteroskedastic Variance

Justin David Naggar Weltz, Tanner Fiez, Alexander Volfovsky, Eric Laber, Blake Mason, houssam nassif, Lalit K Jain

Keywords: Heteroskedastic Variance, Linear Bandits, Experimental design

TL;DR: We make novel contributions to heteroskedastic variance estimation and experimental design in flexible settings.

Abstract:

Most linear experimental design problems assume homogeneous variance, while the presence of heteroskedastic noise is present in many realistic settings.
Let a learner have access to a finite set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$ that can be probed to receive noisy linear responses of the form $y=x^{\top}\theta^{\ast}+\eta$.
Here $\theta^{\ast}\in \mathbb{R}^d$ is an unknown parameter vector, and $\eta$ is independent mean-zero $\sigma_x^2$-sub-Gaussian noise defined by a flexible heteroskedastic variance model, $\sigma_x^2 = x^{\top}\Sigma^{\ast}x$. Assuming that $\Sigma^{\ast}\in \mathbb{R}^{d\times d}$ is an unknown matrix, we propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters, $\sigma_x^2$.
We demonstrate this method on two adaptive experimental design problems under heteroskedastic noise, fixed confidence transductive best-arm identification and level-set identification and prove the first instance-dependent lower bounds in these settings.
Lastly, we construct near-optimal algorithms and demonstrate the large improvements in sample complexity gained from accounting for heteroskedastic variance in these designs empirically.

Submission Number: 14607