
\section{Introduction}
\label{sec:intro}
In the fast-evolving landscape of Deep Learning, ensuring the robustness and reliability of Neural Networks (NNs) is paramount, particularly for critical decision-making applications. This work introduces a simple approach for estimating the local robustness of trained Neural Networks against uncertainties, with a focus on their performance in the vicinity of clean inputs.
We propose a method that combines adversarial attacks with the Importance Sampling (IS) technique.

Adversarial attacks, traditionally aimed at uncovering NN vulnerabilities, are repurposed in our methodology as a strategic guide for the IS process. The point of this approach is to identify the most error-prone regions in the input space, thus directing the sampling process contrary to the commonly used Crude Monte Carlo method.

A key contribution of this research is the comparative analysis of our method with classical techniques from the field of Statistical Reliability Engineering \citep{stat_rel}. These techniques include the First Order Reliability Method (FORM), Second Order Reliability Method (SORM), and Line Sampling \citep{line_sampling}, which have not been extensively applied to DNNs in very high-dimensional spaces, a gap our study aims to fill.

In addition, we compare this IS estimator to classical rare event simulation algorithms.
 These include Cross-entropy-based Adaptive Importance Sampling
(CE-AIS) \citep{rubinstein_kroese} and Adaptive Multilevel Splitting (AMS) \citep{beck_mls} methods. We show that the proposed method is more efficient and faster than these techniques for various architectures and datasets.

However, this novel estimator is not without limitations. Its effectiveness is inherently tied to the efficiency of adversarial attacks; it can only be as good as the adversarial attacks it relies on. Moreover, the occurrence of weight degeneracy in extremely high-dimensional data, such as ImageNet data where $d=150528$, restricts the applicability of this method. These constraints highlight the need for a continuum of solutions from fast methods, like the one proposed here, to more advanced but slower methods for complex settings.
%continued refinement and adaptation of the methodology, especially in dealing with large-scale, complex data structures.

This paper delves into the intricacies of integrating adversarial attack strategies within the IS framework, addressing both the algorithmic challenges and the theoretical aspects. We focus on adapting these strategies for high-dimensional reliability analysis in NNs, confronting computational and conceptual hurdles.
We validate our approach through empirical studies and experiments on a variety of deep learning models using the computer vision datasets MNIST and CIFAR10. These evaluations demonstrate the method's efficacy in rapidly estimating NN probabilistic robustness.

%We conclude this paper by discussing the broader impact of our findings, their potential applications across various domains, and directions for future research. By leveraging adversarial phenomena within neural networks, this work contributes a novel perspective to reliability assessment in complex machine learning models.