Manuscript Psy01-A@1 · Psy01-A.tex · theorem · 94477fa3

Theorem: Strict trade-off transfer

Stress certificates and a conditional trade-off › Transfer to a ReLU checkpoint · lines 1091–1130 · proof included or linked

Psy01-A@1#thm:transfer

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For two specified ReLU checkpoints $θ_{0}, θ_{1}$ , suppose a declared comparison model gives observables $A_{i}^{cmp}, S_{i}^{cmp}$ , and valid error certificates give

\abs A_{θ_{i}} - A_{i}^{cmp} \leq ϵ_{i}^{A}, \abs S_{θ_{i}} (σ) - S_{i}^{cmp} \leq ϵ_{i}^{S} .

A_{1}^{cmp} - A_{0}^{cmp} > ϵ_{0}^{A} + ϵ_{1}^{A}

and

S_{0}^{cmp} - S_{1}^{cmp} > ϵ_{0}^{S} + ϵ_{1}^{S},

then

A_{θ_{1}} > A_{θ_{0}}, S_{θ_{1}} (σ) < S_{θ_{0}} (σ) .

Proof. Subtract the two accuracy inequalities:

A_{θ_{1}} - A_{θ_{0}} \geq A_{1}^{cmp} - A_{0}^{cmp} - ϵ_{0}^{A} - ϵ_{1}^{A} > 0.

Apply the same argument to $S_{θ_{0}} - S_{θ_{1}}$ .

Displayed formulas in this passage

unlabeled · line 1097
$\abs A_{θ_{i}} - A_{i}^{cmp} \leq ϵ_{i}^{A}, \abs S_{θ_{i}} (σ) - S_{i}^{cmp} \leq ϵ_{i}^{S} .$
unlabeled · line 1104
$A_{1}^{cmp} - A_{0}^{cmp} > ϵ_{0}^{A} + ϵ_{1}^{A}$
unlabeled · line 1109
$S_{0}^{cmp} - S_{1}^{cmp} > ϵ_{0}^{S} + ϵ_{1}^{S},$
unlabeled · line 1114
$A_{θ_{1}} > A_{θ_{0}}, S_{θ_{1}} (σ) < S_{θ_{0}} (σ) .$
unlabeled · line 1123
$A_{θ_{1}} - A_{θ_{0}} \geq A_{1}^{cmp} - A_{0}^{cmp} - ϵ_{0}^{A} - ϵ_{1}^{A} > 0.$

Nearby in this section

section: Transfer to a ReLU checkpoint