# Exact Upper and Lower Bounds for the Output Distribution of Neural Networks with Random Inputs

arXiv: https://arxiv.org/abs/2502.11672 · HTML: https://mlsanity.com/paper/2502.11672/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 17 Feb 2025.

## Builds on (most cited first, 20 of 20)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [A Survey of Autonomous Driving: Common Practices and Emerging Technologies](https://mlsanity.com/paper/1906.05113/) (2019)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Branch and Bound for Piecewise Linear Neural Network Verification](https://mlsanity.com/paper/1909.06588/) (2019)
- [Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Bound](https://mlsanity.com/paper/2205.00263/) (2022)
- [Marabou 2.0: A Versatile Formal Analyzer of Neural Networks](https://mlsanity.com/paper/2401.14461/) (2024)
- [Adversarial vulnerability for any classifier](https://mlsanity.com/paper/1802.08686/) (2018)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Google's Cloud Vision API Is Not Robust To Noise](https://mlsanity.com/paper/1704.05051/) (2017)
- [Unwrapping The Black Box of Deep ReLU Networks: Interpretability, Diagnostics, and Simplification](https://mlsanity.com/paper/2011.04041/) (2020)
- [Interval Universal Approximation for Neural Networks](https://mlsanity.com/paper/2007.06093/) (2020)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Automatic Perturbation Analysis for Scalable Certified Robustness and Beyond](https://mlsanity.com/paper/2002.12920/) (2020)
- [Uncertainty Propagation in Deep Neural Network Using Active Subspace](https://mlsanity.com/paper/1903.03989/) (2019)
- [Tailored Uncertainty Estimation for Deep Learning Systems](https://mlsanity.com/paper/2204.13963/) (2022)

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

- [Central limit theorems for the outputs of fully convolutional neural networks with time series input](https://mlsanity.com/paper/2603.29612/) (2026)
- [A Survey on the Verification of Reinforcement Learning Policies](https://mlsanity.com/paper/2607.16210/) (2026)

## Abstract (arXiv)

We derive exact upper and lower bounds for the cumulative distribution function (cdf) of the output of a neural network (NN) over its entire support subject to noisy (stochastic) inputs. The upper and lower bounds converge to the true cdf over its domain as the resolution increases. Our method applies to any feedforward NN using continuous monotonic piecewise twice continuously differentiable activation functions (e.g., ReLU, tanh and softmax) and convolutional NNs, which were beyond the scope of competing approaches. The novelty and instrumental tool of our approach is to bound general NNs with ReLU NNs. The ReLU NN-based bounds are then used to derive the upper and lower bounds of the cdf of the NN output. Experiments demonstrate that our method delivers guaranteed bounds of the predictive output distribution over its support, thus providing exact error guarantees, in contrast to competing approaches.