Experimental Designs for Heteroskedastic Variance
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916398228307968 |
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| author | Weltz, Justin Fiez, Tanner Volfovsky, Alexander Laber, Eric Mason, Blake Nassif, Houssam Jain, Lalit |
| author_facet | Weltz, Justin Fiez, Tanner Volfovsky, Alexander Laber, Eric Mason, Blake Nassif, Houssam Jain, Lalit |
| contents | Most linear experimental design problems assume homogeneous variance although 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 $y=x^{\top}θ^{\ast}+η$. Here $θ^{\ast}\in \mathbb{R}^d$ is an unknown parameter vector, and $η$ is independent mean-zero $σ_x^2$-sub-Gaussian noise defined by a flexible heteroskedastic variance model, $σ_x^2 = x^{\top}Σ^{\ast}x$. Assuming that $Σ^{\ast}\in \mathbb{R}^{d\times d}$ is an unknown matrix, we propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters, $σ_x^2$. We demonstrate the benefits of this method with 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04390 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Experimental Designs for Heteroskedastic Variance Weltz, Justin Fiez, Tanner Volfovsky, Alexander Laber, Eric Mason, Blake Nassif, Houssam Jain, Lalit Statistics Theory Most linear experimental design problems assume homogeneous variance although 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 $y=x^{\top}θ^{\ast}+η$. Here $θ^{\ast}\in \mathbb{R}^d$ is an unknown parameter vector, and $η$ is independent mean-zero $σ_x^2$-sub-Gaussian noise defined by a flexible heteroskedastic variance model, $σ_x^2 = x^{\top}Σ^{\ast}x$. Assuming that $Σ^{\ast}\in \mathbb{R}^{d\times d}$ is an unknown matrix, we propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters, $σ_x^2$. We demonstrate the benefits of this method with 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. |
| title | Experimental Designs for Heteroskedastic Variance |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2310.04390 |