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arXiv · 2412.20355

Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks

Carlos Misael Madrid Padilla·Oscar Hernan Madrid Padilla·Yik Lun Kei·Zhi Zhang·Yanzhen Chen
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Abstract

This paper addresses the problems of conditional variance estimation and confidence interval construction in nonparametric regression using dense networks with the Rectified Linear Unit (ReLU) activation function. We present a residual-based framework for conditional variance estimation, deriving nonasymptotic bounds for variance estimation under both heteroscedastic and homoscedastic settings. We relax the sub-Gaussian noise assumption, allowing the proposed bounds to accommodate sub-Exponential noise and beyond. Building on this, for a ReLU neural network estimator, we derive non-asymptotic bounds for both its conditional mean and variance estimation, representing the first result for variance estimation using ReLU networks. Furthermore, we develop a ReLU network based robust bootstrap procedure (Efron, 1992) for constructing confidence intervals for the true mean that comes with a theoretical guarantee on the coverage, providing a significant advancement in uncertainty quantification and the construction of reliable confidence intervals in deep learning settings.

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Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla, Yik Lun Kei, Zhi Zhang, Yanzhen Chen. 2024-12-29. Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks. https://arxiv.org/abs/2412.20355

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