# Theoretical guarantees for neural estimators in parametric statistics

arXiv: https://arxiv.org/abs/2506.18508 · HTML: https://mlsanity.com/paper/2506.18508/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 23 Jun 2025.

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

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [A Walk with SGD](https://mlsanity.com/paper/1802.08770/) (2018)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
- [Neural Networks for Parameter Estimation in Intractable Models](https://mlsanity.com/paper/2107.14346/) (2021)
- [Fast covariance parameter estimation of spatial Gaussian process models using neural networks](https://mlsanity.com/paper/2012.15339/) (2020)
- [Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE](https://mlsanity.com/paper/2012.06691/) (2020)
- [Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods](https://mlsanity.com/paper/2305.04634/) (2023)
- [Exact simulation of max-stable processes](https://mlsanity.com/paper/1506.04430/) (2015)
- [Statistical Inference for H"usler-Reiss Graphical Models Through Matrix Completions](https://mlsanity.com/paper/2210.14292/) (2022)
- [Progression: an extrapolation principle for regression](https://mlsanity.com/paper/2410.23246/) (2024)
- [Bayesian inference for multivariate extreme value distributions](https://mlsanity.com/paper/1611.05602/) (2016)
- [On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions](https://mlsanity.com/paper/1410.6733/) (2014)

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

- [Mitigating Representation Gaps in Amortized Bayesian Inference with Auxiliary Supervision](https://mlsanity.com/paper/2609.39525/) (2026)
- [Learning Decision-Stump Thresholds in Context: Dynamics of Softmax Attention](https://mlsanity.com/paper/2610.07074/) (2026)

## Abstract (arXiv)

Neural estimators are simulation-based estimators for the parameters of a family of statistical models, which build a direct mapping from the sample to the parameter vector. They benefit from the versatility of available network architectures and efficient training methods developed in the field of deep learning. Neural estimators are amortized in the sense that, once trained, they can be applied to any new data set with almost no computational cost. While many papers have shown very good performance of these methods in simulation studies and real-world applications, so far no statistical guarantees are available to support these observations theoretically. In this work, we study the risk of neural estimators by decomposing it into several terms that can be analyzed separately. We formulate easy-to-check assumptions ensuring that each term converges to zero, and we verify them for popular applications of neural estimators. Our results provide a general recipe to derive theoretical guarantees also for broader classes of architectures and estimation problems.