Manuscript Psy03-B@1 · Psy03-B-v1.tex · corollary · 8ca3caf9

Corollary: Biased-noise path bound

Biased-noise and full-state perturbation certificates › Common-protection challenge paths · lines 967–1003 · proof included or linked

Psy03-B@1#cor:path, Psy03-B@1#eq:path

Compare the same network and protection state at $(ξ, β, σ)$ and $(0, 0, 0)$ under a common input and Gaussian coupling. Then

\norm z_{ξ, β, σ} - z_{0, 0, 0}_{2} \leq ξ q^{3} (ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}) \norm G_{x}_{2} + r = 1 \sum^{3} q^{4 - r} P_{r} [σ \norm G_{r}_{2} + \abs β b_{bias} (p) n_{r}] .

The protection offset cancels in this comparison. The result does not require $q \leq 1$ .

Proof. Both admitted activations are $1$ -Lipschitz. Starting from $d_{0} = ξ \norm G_{x}_{2}$ , the hidden differences obey

d_{ℓ} \leq q \norm B_{ℓ}_{op} d_{ℓ - 1} + q [σ \norm G_{ℓ}_{2} + \abs β b_{bias} (p) n_{ℓ}] .

Iterate and multiply by the head norm. This is the layerwise interface in Psy03-A lem:path, with gain and bias explicitly included CheungPsy03A.

Displayed formulas in this passage

Psy03-B@1#eq:path · line 973
$\norm z_{ξ, β, σ} - z_{0, 0, 0}_{2} \leq ξ q^{3} (ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}) \norm G_{x}_{2} + r = 1 \sum^{3} q^{4 - r} P_{r} [σ \norm G_{r}_{2} + \abs β b_{bias} (p) n_{r}] .$
unlabeled · line 992
$d_{ℓ} \leq q \norm B_{ℓ}_{op} d_{ℓ - 1} + q [σ \norm G_{ℓ}_{2} + \abs β b_{bias} (p) n_{ℓ}] .$