# Worst-Case Distance-Aware Error Bounds for Neural Networks

arXiv: https://arxiv.org/abs/2510.22021 · HTML: https://mlsanity.com/paper/2510.22021/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 55 of its 100 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 30 of 55)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [On Feature Collapse and Deep Kernel Learning for Single Forward Pass Uncertainty](https://mlsanity.com/paper/2102.11409/) (2021)
- [Structured and Efficient Variational Deep Learning with Matrix Gaussian Posteriors](https://mlsanity.com/paper/1603.04733/) (2016)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Interpretability and Explainability: A Machine Learning Zoo Mini-tour](https://mlsanity.com/paper/2012.01805/) (2020)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)

## Abstract (arXiv)

Safety-critical applications of machine learning require uncertainty estimates that support reliable worst-case analysis. Neural networks (NNs) provide expressive function approximation, while Gaussian processes (GPs) offer principled probabilistic uncertainty, but both face limitations in this setting: most modern neural architectures lack tractable worst-case error bounds, and Gaussian processes become computationally expensive at scale. For uncertainty to be interpretable, a central requirement is distance-awareness: that is, uncertainty increases with the distance between a test input and the nearest relevant training data. We present a general framework for worst-case distance-aware error bounds for NNs that combine dense layers with spline-based components. Our approach establishes error bounds that are both distance-aware, reflecting proximity of a test point to its nearest training data, and worst-case, providing deterministic guarantees under known Lipschitz constraints rather than probabilistic assumptions. Our algorithm, K-DAREK (Distance-Aware Error for Kurkova-Kolmogorov Networks), provides efficient and interpretable uncertainty quantification for NNs. K-DAREK is about four times faster and ten times more computationally efficient than an ensemble of KANs, 8.6 times more scalable than GP, eliminates up to 8.2% error-bound violation rate observed in DAREK, and reduces the average collision rate from 1.8% to 1.1% in the multi-agent safe control experiment. On high-dimensional real-world regression tasks (e.g., Real Estate Valuation), K-DAREK preserves distance-aware error bounds and achieves zero coverage violations, addressing the overgeneralization and inducing-point-coverage limitations exhibited by SNGP and DUE, respectively.