Most Convolutional Networks Suffer from Small Adversarial Perturbations

The existence of adversarial examples is relatively understood for random fully connected neural networks, but much less so for convolutional neural networks (CNNs). The recent work [Daniely, 2025] establishes that adversarial examples can be found in CNNs, in some non-optimal distance from the input. We extend over this work and prove that adversarial examples in random CNNs with input dimension $d$ can be found already in $\ell_2$-distance of order $\lVert x \rVert /\sqrt{d}$ from the input $x$, which is essentially the nearest possible. We also show that such adversarial small perturbations can be found using a single step of gradient descent. To derive our results we use Fourier decomposition to efficiently bound the singular values of a random linear convolutional operator, which is the main ingredient of a CNN layer. This bound might be of independent interest.