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

Theorem: Sufficient challenge-survival certificate

Biased-noise and full-state perturbation certificates › Common-protection challenge paths · lines 1021–1053 · proof included or linked

Psy03-B@1#thm:margin, Psy03-B@1#eq:survival

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Let $H = 2 + n_{1} + n_{2} + n_{3} = 1282$ , $0 < δ < 1$ , and

R_{δ} = 2 lo g (2 H / δ) .

Define

C_{δ} = ξ q^{3} (ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}) 2 R_{δ} + r = 1 \sum^{3} q^{4 - r} P_{r} n_{r} [σ R_{δ} + \abs β b_{bias} (p)] .

Then

Q (ϑ; ξ, β, σ) \geq P {m (z_{0, 0, 0} (X), Y) > 2 C_{δ}} - δ .

The right side may be truncated below at zero.

Proof. The scalar Gaussian bound $P (\abs G > u) \leq 2 e^{- u^{2} /2}$ and a union bound place all noise coordinates within $R_{δ}$ with probability at least $1 - δ$ . Equation (eq:path) is then at most $C_{δ}$ . The margin inequality preserves correct classification.

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unlabeled · line 1028
$C_{δ} = ξ q^{3} (ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}) 2 R_{δ} + r = 1 \sum^{3} q^{4 - r} P_{r} n_{r} [σ R_{δ} + \abs β b_{bias} (p)] .$
unlabeled · line 1024
$R_{δ} = 2 lo g (2 H / δ) .$
Psy03-B@1#eq:survival · line 1037
$Q (ϑ; ξ, β, σ) \geq P {m (z_{0, 0, 0} (X), Y) > 2 C_{δ}} - δ .$

Theorem: Sufficient challenge-survival certificate

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