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

Source: [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](<https://www.sota2.com/research/paper/69fd6e1f5497b3b670c894ba>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-05-09T05:00:12.752Z

Authors: Tianpai Luo, Fangwei Wu, Weichi Wu

Published: 2026-05-07T13:40:52.000Z

## Abstract

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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Quantile Regression on Scenario 2 n=1000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-2-n-1000.md>): Quantile Regression; Scenario 2 n=1000; MSE (τ=0.05): 0.4515.

- [Quantile Regression on Scenario 2 (n=5000)](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-2-n-5000.md>): Quantile Regression; Scenario 2 (n=5000); MSE (τ=0.05): 0.143.

- [Quantile Regression on Scenario 2 n=10000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-2-n-10000.md>): Quantile Regression; Scenario 2 n=10000; MSE (τ=0.05): 0.1008.

- [Quantile Regression on Scenario 3 n=1000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-3-n-1000.md>): Quantile Regression; Scenario 3 n=1000; MSE (τ=0.05): 1.9425.

- [Quantile Regression on Scenario 3 n=5000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-3-n-5000.md>): Quantile Regression; Scenario 3 n=5000; MSE (τ=0.05): 0.8839.

- [Quantile Regression on Scenario 3 n=10000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-3-n-10000.md>): Quantile Regression; Scenario 3 n=10000; MSE (tau=0.05): 0.6307.

- [BMI Prediction on Kaggle BMI Male](<https://www.sota2.com/research/sota/bmi-prediction-on-kaggle-bmi-male.md>): BMI Prediction; Kaggle BMI Male; Pinball Loss (τ=0.05): 0.5214.

- [BMI Prediction on Kaggle BMI (Female)](<https://www.sota2.com/research/sota/bmi-prediction-on-kaggle-bmi-female.md>): BMI Prediction; Kaggle BMI (Female); Pinball Loss (τ=0.05): 0.5202.

- [Joint Quantile Regression on Scenario 2 Model A](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-2-model-a.md>): Joint Quantile Regression; Scenario 2 Model A; MSE (τ=0.05): 0.0377.

- [Quantile Regression on Scenario 1](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-1.md>): Quantile Regression; Scenario 1; MAE (τ=0.05): 0.1678.

- [Joint Quantile Regression on Scenario 1 Model A (n=5000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-1-model-a-n-5000.md>): Joint Quantile Regression; Scenario 1 Model A (n=5000); MSE (τ=0.05): 0.0471.

- [Joint Quantile Regression on Scenario 2 Model A (n=1000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-2-model-a-n-1000.md>): Joint Quantile Regression; Scenario 2 Model A (n=1000); MSE (τ=0.05): 0.1669.

- [Joint Quantile Regression on Scenario 2 Model B (n=10000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-2-model-b-n-10000.md>): Joint Quantile Regression; Scenario 2 Model B (n=10000); MSE (τ=0.05): 0.0466.

- [Joint Quantile Regression on Scenario 3 Model A](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-a.md>): Joint Quantile Regression; Scenario 3 Model A; MSE (τ=0.05): 0.8659.

- [Joint Quantile Regression on Scenario 3 Model A (n=5000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-a-n-5000.md>): Joint Quantile Regression; Scenario 3 Model A (n=5000); MSE (τ=0.05): 0.3208.

- [Joint Quantile Regression on Scenario 3 Model A (n=10000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-a-n-10000.md>): Joint Quantile Regression; Scenario 3 Model A (n=10000); MSE (τ=0.05): 0.2297.

- [Joint Quantile Regression on Scenario 3 Model B (n=10000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-b-n-10000.md>): Joint Quantile Regression; Scenario 3 Model B (n=10000); MSE ($	au=0.05$): 0.1916.

- [Quantile Regression on Scenario 2](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-2.md>): Quantile Regression; Scenario 2; MAE (τ=0.05): 0.2112.

- [Quantile Regression on Scenario 3](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-3.md>): Quantile Regression; Scenario 3; MAE (τ=0.05): 0.3422.

- [Quantile Regression on Scenario 1 n=1000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-1-n-1000.md>): Quantile Regression; Scenario 1 n=1000; MSE (τ=0.05): 0.3356.

- [Quantile Regression on Scenario 1 (n=5000)](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-1-n-5000.md>): Quantile Regression; Scenario 1 (n=5000); MSE (Quantile 0.05): 0.1006.

- [Quantile Regression on Scenario 1 n=10000](<https://www.sota2.com/research/sota/quantile-regression-on-scenario-1-n-10000.md>): Quantile Regression; Scenario 1 n=10000; MSE (τ=0.05): 0.0627.

- [Joint Quantile Regression on Scenario 1 Model A](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-1-model-a.md>): Joint Quantile Regression; Scenario 1 Model A; MSE (τ=0.05): 0.0338.

- [Joint Quantile Regression on Scenario 1 Model B (n=1000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-1-model-b-n-1000.md>): Joint Quantile Regression; Scenario 1 Model B (n=1000); MSE (τ=0.05): 0.2105.

- [Joint Quantile Regression on Scenario 1 Model B (n=5000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-1-model-b-n-5000.md>): Joint Quantile Regression; Scenario 1 Model B (n=5000); MSE (τ=0.05): 0.0864.

- [Joint Quantile Regression on Scenario 1 Model B](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-1-model-b.md>): Joint Quantile Regression; Scenario 1 Model B; MSE (τ=0.05): 0.0674.

- [Joint Quantile Regression on Scenario 2 Model B (n=1000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-2-model-b-n-1000.md>): Joint Quantile Regression; Scenario 2 Model B (n=1000); MSE (τ=0.05): 0.1449.

- [Joint Quantile Regression on Scenario 2 Model B (n=5000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-2-model-b-n-5000.md>): Joint Quantile Regression; Scenario 2 Model B (n=5000); MSE (τ=0.05): 0.0728.

- [Joint Quantile Regression on Scenario 3 Model B (n=1000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-b-n-1000.md>): Joint Quantile Regression; Scenario 3 Model B (n=1000); MSE (τ=0.05): 0.7214.

- [Joint Quantile Regression on Scenario 3 Model B (n=5000)](<https://www.sota2.com/research/sota/joint-quantile-regression-on-scenario-3-model-b-n-5000.md>): Joint Quantile Regression; Scenario 3 Model B (n=5000); MSE (τ=0.05): 0.3078.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2605.06265>)

[PDF](<https://arxiv.org/pdf/2605.06265.pdf>)