# New Approximation Results and Optimal Estimation for Fully Connected Deep Neural Networks

arXiv: https://arxiv.org/abs/2512.09853 · HTML: https://mlsanity.com/paper/2512.09853/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Optimal Nonparametric Inference via Deep Neural Network](https://mlsanity.com/paper/1902.01687/) (2019)
- [Statistical Properties of Deep Neural Networks with Dependent Data](https://mlsanity.com/paper/2410.11113/) (2024)
- [Causal inference through multi-stage learning and doubly robust deep neural networks](https://mlsanity.com/paper/2407.08560/) (2024)

## Abstract (arXiv)

citetfarrell2021deep establish non-asymptotic high-probability bounds for general deep feedforward neural network (with rectified linear unit activation function) estimators, with citet[Theorem 1]farrell2021deep achieving a suboptimal convergence rate for fully connected feedforward networks. The authors suggest that improved approximation of fully connected networks could yield sharper versions of citet[Theorem 1]farrell2021deep without altering the theoretical framework. By deriving approximation bounds specifically for a narrower fully connected deep neural network, this note demonstrates that citet[Theorem 1]farrell2021deep can be improved to achieve an optimal rate (up to a logarithmic factor). Furthermore, this note briefly shows that deep neural network estimators can mitigate the curse of dimensionality for functions with compositional structure and functions defined on manifolds.