ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees
This work provides a theoretically grounded and practically effective method for quantile regression with deep neural networks, addressing the non-smoothness bottleneck for distributional learning.
ConquerNet introduces convolution-smoothed quantile ReLU neural networks that achieve smooth optimization for quantile regression while preserving quantile structure, with minimax guarantees over Besov classes and superior performance over standard quantile neural networks across multiple quantile levels.
Quantile regression is a fundamental tool for distributional learning but poses significant optimization challenges for deep models due to the non-smoothness of the pinball loss. We propose ConquerNet, a class of \textbf{con}volution-smoothed \textbf{qu}antil\textbf{e} \textbf{R}eLU neural \textbf{net}works, which yield smooth objectives while preserving the underlying quantile structure. We establish general nonasymptotic risk bounds for ConquerNet under mild conditions, providing minimax guarantees over Besov function classes. In numerical studies, we demonstrate that the proposed approach outperforms standard quantile neural networks at multiple quantile levels, showing improved estimation accuracy and training efficiency across the board, with particularly pronounced advantages at high and low quantiles.