用ReLU网络实现更可靠的置信区间与方差估计，突破传统假设限制。
Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks
- 基于残差的方差估计框架，适用于异方差和同方差场景。
- 首次给出ReLU网络在均值与方差估计上的非渐近界，涵盖亚指数噪声。
- 提出带理论保证的鲁棒自助法，提升深度学习不确定性量化可靠性。
本文研究使用带有ReLU激活函数的密集网络进行非参数回归中的条件方差估计与置信区间构建问题。提出一种基于残差的方差估计框架，推导了在异方差和同方差设置下方差估计的非渐近界。放宽了传统的次高斯噪声假设，使所提边界可处理亚指数噪声及更广范围噪声。针对ReLU神经网络估计器，首次给出了其条件均值与方差估计的非渐近界。进一步设计了一种基于ReLU网络的鲁棒自助法（Efron, 1992），用于构建真实均值的置信区间，并提供了覆盖率的理论保证，显著推进了深度学习环境下不确定性量化的进展。
原文摘要 · Abstract (English)
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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