# A general framework for deep learning

arXiv: https://arxiv.org/abs/2512.23425 · HTML: https://mlsanity.com/paper/2512.23425/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## Builds on (most cited first, 16 of 16)

- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Factor Augmented Sparse Throughput Deep ReLU Neural Networks for High Dimensional Regression](https://mlsanity.com/paper/2210.02002/) (2022)
- [Smooth function approximation by deep neural networks with general activation functions](https://mlsanity.com/paper/1906.06903/) (2019)
- [How do noise tails impact on deep ReLU networks?](https://mlsanity.com/paper/2203.10418/) (2022)
- [Quantile regression with deep ReLU Networks: Estimators and minimax rates](https://mlsanity.com/paper/2010.08236/) (2020)
- [Nonconvex sparse regularization for deep neural networks and its optimality](https://mlsanity.com/paper/2003.11769/) (2020)
- [Classification with Deep Neural Networks and Logistic Loss](https://mlsanity.com/paper/2307.16792/) (2023)
- [On the rate of convergence of a deep recurrent neural network estimate in a regression problem with dependent data](https://mlsanity.com/paper/2011.00328/) (2020)
- [Adaptive deep learning for nonlinear time series models](https://mlsanity.com/paper/2207.02546/) (2022)
- [Excess risk bound for deep learning under weak dependence](https://mlsanity.com/paper/2302.07503/) (2023)
- [Sparse-penalized deep neural networks estimator under weak dependence](https://mlsanity.com/paper/2303.01406/) (2023)
- [Learning theory estimates with observations from general stationary stochastic processes](https://mlsanity.com/paper/1605.02887/) (2016)
- [Robust deep learning from weakly dependent data](https://mlsanity.com/paper/2405.05081/) (2024)
- [Minimax optimality of deep neural networks on dependent data via PAC-Bayes bounds](https://mlsanity.com/paper/2410.21702/) (2024)
- [Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality](https://mlsanity.com/paper/2406.08321/) (2024)

## Built on by (most cited first, 2 of 2)

- [Deep regression learning from dependent observations with minimum error entropy principle](https://mlsanity.com/paper/2603.11138/) (2026)
- [Adaptive deep nonparametric regression from dependent data under covariate shift](https://mlsanity.com/paper/2607.20309/) (2026)

## Abstract (arXiv)

This paper develops a general approach for deep learning for a setting that includes nonparametric regression and classification. We perform a framework from data that fulfills a generalized Bernstein-type inequality, including independent, φ-mixing, strongly mixing and C-mixing observations. Two estimators are proposed: a non-penalized deep neural network estimator (NPDNN) and a sparse-penalized deep neural network estimator (SPDNN). For each of these estimators, bounds of the expected excess risk on the class of H"older smooth functions and composition H"older functions are established. Applications to independent data, as well as to φ-mixing, strongly mixing, C-mixing processes are considered. For each of these examples, the upper bounds of the expected excess risk of the proposed NPDNN and SPDNN predictors are derived. It is shown that both the NPDNN and SPDNN estimators are minimax optimal (up to a logarithmic factor) in many classical settings.