# Error Analysis of Neural-Network-Based Engression

arXiv: https://arxiv.org/abs/2607.27723 · HTML: https://mlsanity.com/paper/2607.27723/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 38 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [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)
- [Do Bayesian Neural Networks Need To Be Fully Stochastic?](https://mlsanity.com/paper/2211.06291/) (2022)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [An error analysis of generative adversarial networks for learning distributions](https://mlsanity.com/paper/2105.13010/) (2021)
- [Wasserstein Generative Regression](https://mlsanity.com/paper/2306.15163/) (2023)
- [Theoretical Analysis of Engression and Reverse Markov Engression](https://mlsanity.com/paper/2606.01002/) (2026)
- [Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk](https://mlsanity.com/paper/2205.04360/) (2022)
- [Neural Generative Distributional Regression](https://mlsanity.com/paper/2605.02062/) (2026)
- [Semi-Supervised Learning on Graphs using Graph Neural Networks](https://mlsanity.com/paper/2602.17115/) (2026)

## Built on by (most cited first, 4 of 4)

- [Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression](https://mlsanity.com/paper/2609.23789/) (2026)
- [Sufficiently Reduced Distributional Regression](https://mlsanity.com/paper/2609.29291/) (2026)
- [Optimal Transport Dropout for Structured Predictive Uncertainty](https://mlsanity.com/paper/2609.33377/) (2026)
- [Generalized Engression Models](https://mlsanity.com/paper/2610.01823/) (2026)

## Abstract (arXiv)

Engression (Shen and Meinshausen, 2024) learns a conditional distribution by fitting a generative model Y = f(X,varepsilon) under the energy score, a strictly proper scoring rule. We provide a theoretical error analysis of engression implemented with deep neural networks. We decompose the excess risk into three components: the approximation error, the stochastic error, and the Monte Carlo error. Based on this decomposition, we establish convergence rates under the assumption that the target conditional generator admits a compositional smoothness structure.