Manuscript Psy03-C@1 · Psy03-C-v1.tex · theorem · d3380b90

Theorem: Finite-horizon Gaussian margin certificate

Noise, parameter, and control certificates › The two temporal noise allowances · lines 799–844 · proof included or linked

Psy03-C@1#thm:stress, Psy03-C@1#eq:margincertificate

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Let all relevant noise coordinates be bounded by $r_{δ}$ on an event of probability at least $1 - δ$ . For example, if $N$ Gaussian coordinates are used, take

r_{δ} = 2 lo g (2 N / δ) .

Suppose $R, L_{x}, L_{h}$ are valid for the compared trajectories on that event, and put $c = a L_{h}$ . Then the ordinary logit allowance is

ϵ_{O} = \norm V_{op} a L_{x} (g_{e} ξ + τ) d r_{δ} S_{T} (c),

and the biased allowance is

ϵ_{B} = \norm V_{op} a [L_{x} g_{e} ξ d r_{δ} + σ H r_{δ} + \abs β b_{p} H] S_{T} (c) .

For the corresponding clean true-label margin $m_{0}$ ,

Q_{perturbed} \geq P {m_{0} > 2 ϵ} - δ .

The lower bound may be truncated at zero.

Proof. Apply (eq:noiseeta) on the common noise event and then Theorem thm:recurrence. The margin satisfies

\abs m (z, y) - m (z^{'}, y) \leq 2 \norm z - z^{'}_{\infty} \leq 2 \norm z - z^{'}_{2} .

A clean margin exceeding $2 ϵ$ remains positive. Subtract the exceptional-event probability.

Displayed formulas in this passage

unlabeled · line 804
$r_{δ} = 2 lo g (2 N / δ) .$
unlabeled · line 810
$ϵ_{O} = \norm V_{op} a L_{x} (g_{e} ξ + τ) d r_{δ} S_{T} (c),$
unlabeled · line 816
$ϵ_{B} = \norm V_{op} a [L_{x} g_{e} ξ d r_{δ} + σ H r_{δ} + \abs β b_{p} H] S_{T} (c) .$
Psy03-C@1#eq:margincertificate · line 826
$Q_{perturbed} \geq P {m_{0} > 2 ϵ} - δ .$
unlabeled · line 838
$\abs m (z, y) - m (z^{'}, y) \leq 2 \norm z - z^{'}_{\infty} \leq 2 \norm z - z^{'}_{2} .$

Theorem: Finite-horizon Gaussian margin certificate

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