# Deep Limit Model-free Prediction in Regression

arXiv: https://arxiv.org/abs/2408.09532 · HTML: https://mlsanity.com/paper/2408.09532/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 18 Aug 2024.

## Builds on (most cited first, 17 of 17)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Bag of Tricks for Adversarial Training](https://mlsanity.com/paper/2010.00467/) (2020)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [Wasserstein Generative Learning of Conditional Distribution](https://mlsanity.com/paper/2112.10039/) (2021)
- [Conditional Variational Autoencoder for Neural Machine Translation](https://mlsanity.com/paper/1812.04405/) (2018)
- [Scalable Subsampling Inference for Deep Neural Networks](https://mlsanity.com/paper/2405.08276/) (2024)
- [Model-free Bootstrap and Conformal Prediction in Regression: Conditionality, Conjecture Testing, and Pertinent Prediction Intervals](https://mlsanity.com/paper/2109.12156/) (2021)
- [Model-free time-aggregated predictions for econometric datasets](https://mlsanity.com/paper/2101.02273/) (2021)

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

- [Calibration Prediction Interval for Non-parametric Regression and Neural Networks](https://mlsanity.com/paper/2509.02735/) (2025)
- [Prediction Inference of Time Series with Standard ReLU Deep Neural Networks](https://mlsanity.com/paper/2608.15362/) (2026)

## Abstract (arXiv)

In this paper, we provide a novel Model-free approach based on Deep Neural Network (DNN) to accomplish point prediction and prediction interval under a general regression setting. Usually, people rely on parametric or non-parametric models to bridge dependent and independent variables (Y and X). However, this classical method relies heavily on the correct model specification. Even for the non-parametric approach, some additive form is often assumed. A newly proposed Model-free prediction principle sheds light on a prediction procedure without any model assumption. Previous work regarding this principle has shown better performance than other standard alternatives. Recently, DNN, one of the machine learning methods, has received increasing attention due to its great performance in practice. Guided by the Model-free prediction idea, we attempt to apply a fully connected forward DNN to map X and some appropriate reference random variable Z to Y. The targeted DNN is trained by minimizing a specially designed loss function so that the randomness of Y conditional on X is outsourced to Z through the trained DNN. Our method is more stable and accurate compared to other DNN-based counterparts, especially for optimal point predictions. With a specific prediction procedure, our prediction interval can capture the estimation variability so that it can render a better coverage rate for finite sample cases. The superior performance of our method is verified by simulation and empirical studies.