Theorem: Global stress lower bound

For any input and Gaussian realization,

Let $H=n_1+n_2+n_3=640$, let $0<\delta <1$, and define

Then

The right side may be truncated below at zero.

Proof. Apply the $1$-Lipschitz property of ReLU recursively. The noise introduced at hidden layer $r$ is propagated through the downstream operator norms, giving (eq:stresspathbound).

For a standard scalar Gaussian $G$, $\mathbb{P}(\text{abs}G>r)\le 2e^{-r^2/2}$. A union bound over the $H$ hidden-noise coordinates shows that all coordinates have magnitude at most $R_{\delta }$ with probability at least $1-\delta$. On that event,

If the clean true-label margin exceeds $2\sigma R_{\delta }{C}_{\theta }$, equation (eq:marginlip) preserves correct classification. Subtracting the probability of the exceptional noise event proves (eq:globalstress).