# Optimal Nonparametric Inference via Deep Neural Network

arXiv: https://arxiv.org/abs/1902.01687 · HTML: https://mlsanity.com/paper/1902.01687/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2019.

## Built on by (most cited first, 9 of 9)

- [Calibrating multi-dimensional complex ODE from noisy data via deep neural networks](https://mlsanity.com/paper/2106.03591/) (2021)
- [Deep Feature Screening: Feature Selection for Ultra High-Dimensional Data via Deep Neural Networks](https://mlsanity.com/paper/2204.01682/) (2022)
- [Deep Neural Network Classifier for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.08592/) (2022)
- [Solar Flare Index Prediction Using SDO/HMI Vector Magnetic Data Products with Statistical and Machine Learning Methods](https://mlsanity.com/paper/2209.13779/) (2022)
- [Robust Deep Neural Network Estimation for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.09604/) (2022)
- [Semiparametric Regression for Spatial Data via Deep Learning](https://mlsanity.com/paper/2301.03747/) (2023)
- [Nonparametric regression with modified ReLU networks](https://mlsanity.com/paper/2207.08306/) (2022)
- [New Approximation Results and Optimal Estimation for Fully Connected Deep Neural Networks](https://mlsanity.com/paper/2512.09853/) (2025)
- [Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples](https://mlsanity.com/paper/2605.08559/) (2026)

## Abstract (arXiv)

Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance in nonparametric estimation, whereas the results are suboptimal due to a redundant logarithmic sacrifice. In this paper, we show that such log-factors are not necessary. We derive upper bounds for the L2 minimax risk in nonparametric estimation. Sufficient conditions on network architectures are provided such that the upper bounds become optimal (without log-sacrifice). Our proof relies on an explicitly constructed network estimator based on tensor product B-splines. We also derive asymptotic distributions for the constructed network and a relating hypothesis testing procedure. The testing procedure is further proven as minimax optimal under suitable network architectures.