Theoretical Analysis of Relative Errors in Gradient Computations for Adversarial Attacks with CE Loss

TL;DR

This paper provides a theoretical analysis of floating-point errors in gradient computations for adversarial attacks with CE loss, revealing error patterns and proposing a new loss function to improve attack accuracy.

Contribution

It offers the first comprehensive theoretical study of numerical errors in gradient-based adversarial attacks and introduces T-MIFPE, a loss function that minimizes floating-point errors.

Findings

T-MIFPE outperforms existing loss functions in attack success rates.

Floating-point errors significantly affect gradient accuracy in adversarial attacks.

Theoretical insights reveal key error patterns and contributors like underflow and rounding.

Abstract

Gradient-based adversarial attacks using the Cross-Entropy (CE) loss often suffer from overestimation due to relative errors in gradient computation induced by floating-point arithmetic. This paper provides a rigorous theoretical analysis of these errors, conducting the first comprehensive study of floating-point computation errors in gradient-based attacks across four distinct scenarios: (i) unsuccessful untargeted attacks, (ii) successful untargeted attacks, (iii) unsuccessful targeted attacks, and (iv) successful targeted attacks. We establish theoretical foundations characterizing the behavior of relative numerical errors under different attack conditions, revealing previously unknown patterns in gradient computation instability, and identify floating-point underflow and rounding as key contributors. Building on this insight, we propose the Theoretical MIFPE (T-MIFPE) loss function, which incorporates an optimal scaling factor $T = t^*$ to minimize the impact of floating-point errors, thereby enhancing the accuracy of gradient computation in adversarial attacks. Extensive experiments on the MNIST, CIFAR-10, and CIFAR-100 datasets demonstrate that T-MIFPE outperforms existing loss functions, including CE, C\&W, DLR, and MIFPE, in terms of attack potency and robustness evaluation accuracy.