claimroutine

Noise factorizes into smoothed function payoff

Noisy payoff equals payoff of smoothed

T_{1-\lambda}F

on clean inputs

reduces to the smoothed function for invariance

Claim·§A.2 The finite ordering transfer·inside lemLow-influence functions cannot beat opt_q(J)

Statement (paper)

With

H=T_{1-\lambda}F

, independence of the noise between positions and multilinearity of

P_0^{\mathrm{ml}}

give

\mathbb E_{c,Z,\widetilde Z}P_0^{\mathrm{ml}}\bigl(F(\widetilde Z_1),\ldots,F(\widetilde Z_k)\bigr)=\mathbb E_c\mathbb E_{Z\sim\mu_c^{\otimes R}}P_0^{\mathrm{ml}}\bigl(H(Z_1),\ldots,H(Z_k)\bigr).

In plain English

Writing

H=T_{1-\lambda}F

, the expected payoff

P_0^{\mathrm{ml}}

of the noisy outputs

F(\widetilde Z_1),\ldots,F(\widetilde Z_k)

equals the expected payoff of the smoothed outputs

H(Z_1),\ldots,H(Z_k)

evaluated on the unperturbed strings

Z\sim\mu_c^{\otimes R}

, averaged over

c

Why it's here

Each

\widetilde Z_i

is resampled independently of the others given

Z_i

, and

P_0^{\mathrm{ml}}

is multilinear in its

k

arguments. So the expectation over noise can be taken separately in each slot, replacing

F(\widetilde Z_i)

by its conditional mean

H(Z_i)

. This turns the problem into one about the smoothed function

H

, whose influences are controlled by hypothesis, which is what the later invariance step needs.

Classified routine:

Standard step in Raghavendra-style dictatorship-test analyses: conditional independence of the per-coordinate noise plus multilinearity lets the noise expectation pass inside each slot, replacing F by its Bonami-Beckner smoothing T_{1-λ}F.

Uses

Nothing else in the paper

Used by

claimInvariance hypotheses hold for smoothed functions

Sources

PaperOpen

Section: §A.2 The finite ordering transfer
Page: —
Label: —
Line: appendices/finite-consequences.tex:545

Leannot aligned

Phase 2 aligns this node to a Lean declaration.