Zhaoji Tang
arXiv 10 Dec 2025 · Econometrics
arXiv:2512.09853 · PDF · Extracted main text
\citet{farrell2021deep} 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.
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| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Farrell, Max H and Liang, Tengyuan and Misra, Sanjog (2021) Deep neural networks for estimation and inference | 1.000 | 36 | 4 | 100% |
| 2 | Yarotsky, Dmitry (2017) Error bounds for approximations with deep ReLU networks | 1.000 | 22 | 6 | 100% |
| 3 | Petersen, Philipp and Zech, Jakob (2024) Mathematical theory of deep learning | 1.000 | 8 | 3 | 100% |
| 4 | Stone, Charles J (1982) Optimal global rates of convergence for nonparametric regression | 0.928 | 4 | 3 | 100% |
| 5 | Liu, Ruiqi and Boukai, Ben and Shang, Zuofeng (2022) Optimal nonparametric inference via deep neural network | 0.874 | 5 | 2 | 100% |
| 6 | Schmidt-Hieber, Johannes (2020) Nonparametric regression using deep neural networks with ReLU activation function | 0.693 | 5 | 1 | 100% |
| 7 | Brown, Chad (2024) Statistical Properties of Deep Neural Networks with Dependent Data | 0.644 | 2 | 2 | 100% |
| 8 | Colangelo, Kyle and Lee, Ying-Ying (2025) Double debiased machine learning nonparametric inference with continuous treatments | 0.644 | 2 | 2 | 100% |
| 9 | Feng, Xingdong and Jiao, Yuling and Kang, Lican and Zhang, Baqun and… (2023) Over-parameterized deep nonparametric regression for dependent data with its applications to reinforcement learning | 0.644 | 2 | 2 | 100% |
| 10 | Jiao, Yuling and Kang, Lican and Liu, Jin and Lu, Xiliang and Yang,… (2025) Deep approximate policy iteration | 0.644 | 2 | 2 | 100% |
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