Experimental Designs for Heteroskedastic Variance-猫眼课题宝

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发表时间：

2023-10

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ArXiv

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其他

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作者：

Justin Weltz;Tanner Fiez;Alex Volfovsky;Eric B. Laber;Blake Mason;Houssam Nassif;Lalit Jain

尽管在许多实际设置中存在异方差噪声，但大多数线性实验设计问题都假设齐次方差。让学习者能够访问一组有限的测量向量 $\mathcal{X}\subset \mathbb{R}^d$，可以探测这些向量以接收 $y=x^{\top}\theta^{\ast}+\eta$ 形式的噪声线性响应。这里$\theta^{\ast}\in \mathbb{R}^d$是一个未知参数向量，$\eta$是由灵活的异方差模型定义的独立均值零$\sigma_x^2$-亚高斯噪声，$\sigma_x^2 = x^{\top}\Sigma^{\ast}x$。假设 $\Sigma^{\ast}\in \mathbb{R}^{d\times d}$ 是一个未知矩阵，我们提出、分析和实证评估方差参数 $\sigma_x^2$ 的统一边界估计误差的新颖设计。我们通过异方差噪声下的两个自适应实验设计问题、固定置信传导最佳臂识别和水平集识别证明了该方法的优点，并证明了这些设置中的第一个依赖于实例的下界。最后，我们构建了接近最优的算法，并根据经验证明了通过考虑这些设计中的异方差方差而获得的样本复杂性的巨大改进。

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 $y=x^{\top}\theta^{\ast}+\eta$. 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 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.

Experimental Designs for Heteroskedastic Variance

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