# An Analytic Framework for Robust Training of Artificial Neural Networks

arXiv: https://arxiv.org/abs/2205.13502 · HTML: https://mlsanity.com/paper/2205.13502/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 25 references that are in the map.
- First posted on arXiv 26 May 2022.

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

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [Transferability in Machine Learning: from Phenomena to Black-Box Attacks using Adversarial Samples](https://mlsanity.com/paper/1605.07277/) (2016)
- [Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients](https://mlsanity.com/paper/1711.09404/) (2017)
- [Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation](https://mlsanity.com/paper/1705.08475/) (2017)
- [A Boundary Tilting Persepective on the Phenomenon of Adversarial Examples](https://mlsanity.com/paper/1608.07690/) (2016)
- [The Dimpled Manifold Model of Adversarial Examples in Machine Learning](https://mlsanity.com/paper/2106.10151/) (2021)
- [Explaining Adversarial Vulnerability with a Data Sparsity Hypothesis](https://mlsanity.com/paper/2103.00778/) (2021)
- [Towards Explaining Adversarial Examples Phenomenon in Artificial Neural Networks](https://mlsanity.com/paper/2107.10599/) (2021)

## Abstract (arXiv)

The reliability of a learning model is key to the successful deployment of machine learning in various industries. Creating a robust model, particularly one unaffected by adversarial attacks, requires a comprehensive understanding of the adversarial examples phenomenon. However, it is difficult to describe the phenomenon due to the complicated nature of the problems in machine learning. Consequently, many studies investigate the phenomenon by proposing a simplified model of how adversarial examples occur and validate it by predicting some aspect of the phenomenon. While these studies cover many different characteristics of the adversarial examples, they have not reached a holistic approach to the geometric and analytic modeling of the phenomenon. This paper propose a formal framework to study the phenomenon in learning theory and make use of complex analysis and holomorphicity to offer a robust learning rule for artificial neural networks. With the help of complex analysis, we can effortlessly move between geometric and analytic perspectives of the phenomenon and offer further insights on the phenomenon by revealing its connection with harmonic functions. Using our model, we can explain some of the most intriguing characteristics of adversarial examples, including transferability of adversarial examples, and pave the way for novel approaches to mitigate the effects of the phenomenon.