# Uncertainty Quantification for Sparse Deep Learning

arXiv: https://arxiv.org/abs/2002.11815 · HTML: https://mlsanity.com/paper/2002.11815/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

## Built on by (most cited first, 13 of 13)

- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
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- [A Kernel-Expanded Stochastic Neural Network](https://mlsanity.com/paper/2201.05319/) (2022)
- [Sparse Deep Learning: A New Framework Immune to Local Traps and Miscalibration](https://mlsanity.com/paper/2110.00653/) (2021)
- [An Uncertainty-aware Loss Function for Training Neural Networks with Calibrated Predictions](https://mlsanity.com/paper/2110.03260/) (2021)
- [Seven challenges for harmonizing explainability requirements](https://mlsanity.com/paper/2108.05390/) (2021)
- [Merging Two Cultures: Deep and Statistical Learning](https://mlsanity.com/paper/2110.11561/) (2021)
- [Towards a Unified Framework for Uncertainty-aware Nonlinear Variable Selection with Theoretical Guarantees](https://mlsanity.com/paper/2204.07293/) (2022)
- [Bayesian Neural Networks for Macroeconomic Analysis](https://mlsanity.com/paper/2211.04752/) (2022)
- [Sparse Deep Learning for Time Series Data: Theory and Applications](https://mlsanity.com/paper/2310.03243/) (2023)
- [Approximate Bayesian Computation via Classification](https://mlsanity.com/paper/2111.11507/) (2021)
- [Density Regression and Uncertainty Quantification with Bayesian Deep Noise Neural Networks](https://mlsanity.com/paper/2206.05643/) (2022)
- [Uncertainty Quantification for Large-Scale Deep Networks via Post-StoNet Modeling](https://mlsanity.com/paper/2508.01217/) (2025)

## Abstract (arXiv)

Deep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of available theoretical results, the literature has been largely silent about uncertainty quantification for deep learning. This paper takes a step forward in this important direction by taking a Bayesian point of view. We study Gaussian approximability of certain aspects of posterior distributions of sparse deep ReLU architectures in non-parametric regression. Building on tools from Bayesian non-parametrics, we provide semi-parametric Bernstein-von Mises theorems for linear and quadratic functionals, which guarantee that implied Bayesian credible regions have valid frequentist coverage. Our results provide new theoretical justifications for (Bayesian) deep learning with ReLU activation functions, highlighting their inferential potential.