# Statistically guided deep learning

arXiv: https://arxiv.org/abs/2504.08489 · HTML: https://mlsanity.com/paper/2504.08489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 42 references that are in the map.
- First posted on arXiv 11 Apr 2025.

## Builds on (most cited first, 26 of 26)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Convolutional Neural Networks for Sentence Classification](https://mlsanity.com/paper/1408.5882/) (2014)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [A Rigorous Framework for the Mean Field Limit of Multilayer Neural Networks](https://mlsanity.com/paper/2001.11443/) (2020)
- [Learning with Square Loss: Localization through Offset Rademacher Complexity](https://mlsanity.com/paper/1502.06134/) (2015)
- [Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime](https://mlsanity.com/paper/2006.12297/) (2020)
- [Gradient Descent Finds Global Minima for Generalizable Deep Neural Networks of Practical Sizes](https://mlsanity.com/paper/1908.02419/) (2019)
- [Generalization Bounds for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01487/) (2019)
- [Approximating smooth functions by deep neural networks with sigmoid activation function](https://mlsanity.com/paper/2010.04596/) (2020)
- [Deep Neural Networks for Nonparametric Interaction Models with Diverging Dimension](https://mlsanity.com/paper/2302.05851/) (2023)
- [Generalization Error Bounds for Deep Neural Networks Trained by SGD](https://mlsanity.com/paper/2206.03299/) (2022)
- [Analysis of the expected L2 error of an over-parametrized deep neural network estimate learned by gradient descent without regularization](https://mlsanity.com/paper/2311.14609/) (2023)

## Abstract (arXiv)

We present a theoretically well-founded deep learning algorithm for nonparametric regression. It uses over-parametrized deep neural networks with logistic activation function, which are fitted to the given data via gradient descent. We propose a special topology of these networks, a special random initialization of the weights, and a data-dependent choice of the learning rate and the number of gradient descent steps. We prove a theoretical bound on the expected L2 error of this estimate, and illustrate its finite sample size performance by applying it to simulated data. Our results show that a theoretical analysis of deep learning which takes into account simultaneously optimization, generalization and approximation can result in a new deep learning estimate which has an improved finite sample performance.