Manuscript Psy03-A@1 · Psy03-A.tex · theorem · b30eb6e9

Theorem: Accuracy and smooth-score stability

Stress, margins, and complete perturbation budgets › Coupled parameter comparisons · lines 781–812 · proof included or linked

Psy03-A@1#thm:accuracy, Psy03-A@1#eq:accuracy

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Let $D = \norm z - z^{'}_{\infty}$ under a specified common coupling. For every $t > 0$ ,

\abs Q - Q^{'} \leq P {\abs m (z, Y) \leq t} + \frac{2 E D}{t} .

If $D \leq η$ almost surely, the bound is instead

\abs Q - Q^{'} \leq P {\abs m (z, Y) \leq 2 η} .

For the margin ramp

ρ_{τ} (m) = min {1, max {0, m / τ}}, τ > 0,

\abs E ρ_{τ} (m (z, Y)) - E ρ_{τ} (m (z^{'}, Y)) \leq 2 E D / τ .

Proof. A change in correctness is possible only when $\abs m (z, Y) \leq 2 D$ ; ties are included in that event. Split at $t$ , then use Markov's inequality on $D$ . For the ramp, use its Lipschitz constant $1/ τ$ and (eq:marginlip).

Displayed formulas in this passage

Psy03-A@1#eq:accuracy · line 785
$\abs Q - Q^{'} \leq P {\abs m (z, Y) \leq t} + \frac{2 E D}{t} .$
unlabeled · line 792
$\abs Q - Q^{'} \leq P {\abs m (z, Y) \leq 2 η} .$
unlabeled · line 796
$ρ_{τ} (m) = min {1, max {0, m / τ}}, τ > 0,$
unlabeled · line 800
$\abs E ρ_{τ} (m (z, Y)) - E ρ_{τ} (m (z^{'}, Y)) \leq 2 E D / τ .$

Theorem: Accuracy and smooth-score stability

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