# ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees

arXiv: https://arxiv.org/abs/2605.06265 · HTML: https://mlsanity.com/paper/2605.06265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 50 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Distributional Reinforcement Learning with Quantile Regression](https://mlsanity.com/paper/1710.10044/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [The alignment property of SGD noise and how it helps select flat minima: A stability analysis](https://mlsanity.com/paper/2207.02628/) (2022)
- [Understanding Generalization through Visualizations](https://mlsanity.com/paper/1906.03291/) (2019)
- [Smooth Loss Functions for Deep Top-k Classification](https://mlsanity.com/paper/1802.07595/) (2018)
- [Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems](https://mlsanity.com/paper/1808.08798/) (2018)
- [Quantile regression with deep ReLU Networks: Estimators and minimax rates](https://mlsanity.com/paper/2010.08236/) (2020)
- [Smoothed Quantile Regression with Large-Scale Inference](https://mlsanity.com/paper/2012.05187/) (2020)
- [CRPS Learning](https://mlsanity.com/paper/2102.00968/) (2021)
- [Smoothing quantile regressions](https://mlsanity.com/paper/1905.08535/) (2019)
- [High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization](https://mlsanity.com/paper/2109.05640/) (2021)
- [A Cram'er Distance perspective on Quantile Regression based Distributional Reinforcement Learning](https://mlsanity.com/paper/2110.00535/) (2021)
- [Deep Distributional Learning with Non-crossing Quantile Network](https://mlsanity.com/paper/2504.08215/) (2025)
- [Smoothed estimating equations for instrumental variables quantile regression](https://mlsanity.com/paper/1609.09033/) (2016)
- [Quantile Propagation for Wasserstein-Approximate Gaussian Processes](https://mlsanity.com/paper/1912.10200/) (2019)

## Abstract (arXiv)

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 convolution-smoothed quantile ReLU neural networks, 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.