Noise stability of functions with low influences: Invariance and optimality

Abstract

In this paper we study functions with low influences on product probability spaces. These are functions f W 1 n ! ‫ޒ‬ that have EOEVar i OEf small compared to VarOEf for each i . The analysis of boolean functions f W f 1; 1g n ! f 1; 1g with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics.We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known nonlinear invariance principles. It has the advantage that its proof is simple and that its error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability"-type problems.In particular, as applications of the invariance principle we prove two conjectures: Khot, Kindler, Mossel, and O'Donnell's "Majority Is Stablest" conjecture from theoretical computer science, which was the original motivation for this work, and Kalai and Friedgut's "It Ain't Over Till It's Over" conjecture from social choice theory.

On aggregation for heavy-tailed classes

“…By an appropriate version of the Berry-Esseen inequality for independent copies of Z ∈ L q for q > 2 [21], if ℓ ≥ c 3 (q, L) then…”

Standard simplices and pluralities are not the most noise stable

“…This partition is always given by three 120 degree sectors, so if the a i are not all 1/3 then it is a shifted standard simplex partition. This, along with the standard simplex conjecture of [24] may suggest that standard simplex partitions are always optimal. In our main result we prove that this is not the case:…”

“…The relevant question in this setting involves lowinfluence partitions. Using a non-linear invariance principle it was shown that the majority function maximizes noise stability [24] as conjectured both in the context of hardness of approximation [17] and in voting [16]. Since plurality is the natural generalization of majority, it was conjectured in [17] that plurality is the most stable low-influence partition into k ≥ 3 parts of {1, .…”

“…The answer to the noise stability question is analogous: half-spaces maximize the Gaussian noise stability among all sets of a given measure [3,23,12]. Using an invariance principle, this implies that the majority functions maximize noise stability among all low-influence functions on the discrete cube [24]. Informally, a common feature of all of the results above is that the geometric nature of the optimal partition remains the same no matter what the prescribed measure is (a half-space, a cap etc.).…”

“…Discrete Noise Stability. Using the invariance principle [24], [15,Theorem 1.10], it is by now standard to deduce discrete analogues of our main theorem. For simplicity we formulate one special case for partitions into 3 parts with 0 < ρ < 1 in Corollary 2.9 below, which only requires the Central Limit Theorem.…”