Annealed correlation and quenched noise sensitivity in biased deep neural networks

Johan Jonasson

In earlier work by Jonasson, Steif and Zetterqvist (2023), fully connected deep neural networks with layers of width $n$, i.i.d.\ standard Gaussian weights, sign activations and no bias terms were shown to be annealed and quenched noise sensitive with respect to any joint probability distribution $\mathbb{Q}^{(n)}$ on the input $\boldsymbolω$ and its perturbation $\boldsymbolη$ satisfying a mild positive-correlation condition and $\mathbb{Q}^{(n)}(\boldsymbolη \in \{\boldsymbolω,-\boldsymbolω\}) \to 0$. Here we add independent Gaussian biases of the natural order and show that quenched noise sensitivity is retained over a broad range of depths, indeed under the slightly weaker assumption that $\mathbb{Q}^{(n)}(\boldsymbolω = \boldsymbolη) \to 0$. Thus, subject to the positive-correlation condition, quenched noise sensitivity holds under the minimal requirement that the perturbed input is asymptotically unlikely to coincide with the original input. At the same time, the annealed covariance remains bounded away from zero and is entirely accounted for by fluctuations in the network-dependent output means.