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On the uncertainty principle of neural networks

Jun-Jie Zhang, Dong-Xiao Zhang, Jian-Nan Chen, Long-Gang Pang, Deyu Meng

TL;DR

This work proposes that neural networks are subject to an uncertainty relation, which manifests as a fundamental limitation in their ability to simultaneously achieve high accuracy and robustness against adversarial attacks, and reveals that the complementarity principle applies to neural networks.

Abstract

In this study, we explore the inherent trade-off between accuracy and robustness in neural networks, drawing an analogy to the uncertainty principle in quantum mechanics. We propose that neural networks are subject to an uncertainty relation, which manifests as a fundamental limitation in their ability to simultaneously achieve high accuracy and robustness against adversarial attacks. Through mathematical proofs and empirical evidence, we demonstrate that this trade-off is a natural consequence of the sharp boundaries formed between different class concepts during training. Our findings reveal that the complementarity principle, a cornerstone of quantum physics, applies to neural networks, imposing fundamental limits on their capabilities in simultaneous learning of conjugate features. Meanwhile, our work suggests that achieving human-level intelligence through a single network architecture or massive datasets alone may be inherently limited. Our work provides new insights into the theoretical foundations of neural network vulnerability and opens up avenues for designing more robust neural network architectures.

On the uncertainty principle of neural networks

TL;DR

This work proposes that neural networks are subject to an uncertainty relation, which manifests as a fundamental limitation in their ability to simultaneously achieve high accuracy and robustness against adversarial attacks, and reveals that the complementarity principle applies to neural networks.

Abstract

In this study, we explore the inherent trade-off between accuracy and robustness in neural networks, drawing an analogy to the uncertainty principle in quantum mechanics. We propose that neural networks are subject to an uncertainty relation, which manifests as a fundamental limitation in their ability to simultaneously achieve high accuracy and robustness against adversarial attacks. Through mathematical proofs and empirical evidence, we demonstrate that this trade-off is a natural consequence of the sharp boundaries formed between different class concepts during training. Our findings reveal that the complementarity principle, a cornerstone of quantum physics, applies to neural networks, imposing fundamental limits on their capabilities in simultaneous learning of conjugate features. Meanwhile, our work suggests that achieving human-level intelligence through a single network architecture or massive datasets alone may be inherently limited. Our work provides new insights into the theoretical foundations of neural network vulnerability and opens up avenues for designing more robust neural network architectures.
Paper Structure (7 sections, 1 theorem, 21 equations, 1 figure, 1 table)

This paper contains 7 sections, 1 theorem, 21 equations, 1 figure, 1 table.

Table of Contents

  1. Some background information
  2. Uncertainty principle for neural networks

Key Result

Theorem 1

The standard deviations $\sigma_{{p}_{i}}$ and $\sigma_{{x}_{i}}$ corresponding to the attack and pixel operators $\hat{p_{i}}$ and $\hat{x_{i}}$, respectively, are restricted by the relation:

Figures (1)

  • Figure 1: Ratio of high frequencies is roughly proportional to the test accuracies. Ratio of high frequencies is obtained via the formula $(|\text{FFT}[\Delta_{1}]|-|\text{FFT}[\Delta_{1}]_{\text{LF}}|)/|\text{FFT}[\Delta_{1}]|$. For Cifar-10, we turn the RGB images into gray and $|\text{FFT}[\Delta_{1}]_{\text{LF}}|$ takes the slices $[15:19,15:19]$. For MNIST, $|\text{FFT}[\Delta_{1}]_{\text{LF}}|$ takes the slices $[12:18,12:18]$. The networks are trained with various epoch numbers to gain the different test accuracies. The implementation can be found in Ref. DVN/FKFJZQ_2021.

Theorems & Definitions (2)

  • Theorem 1
  • proof