We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian sensitivity at noise rate $\epsilon$ is less than some quantity asymptotic to $\frac{d\sqrt{2\epsilon}}{\pi}$ and the Gaussian surface area is at most $\frac{d}{\sqrt{2\pi}}$. Furthermore these bounds are asymptotically tight as $\epsilon\to 0$ and $f$ the threshold function of a product of $d$ distinct homogeneous linear functions.
Deep Dive into The Gaussian Surface Area and Noise Sensitivity of Degree-$d$ Polynomials.
We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian sensitivity at noise rate $\epsilon$ is less than some quantity asymptotic to $\frac{d\sqrt{2\epsilon}}{\pi}$ and the Gaussian surface area is at most $\frac{d}{\sqrt{2\pi}}$. Furthermore these bounds are asymptotically tight as $\epsilon\to 0$ and $f$ the threshold function of a product of $d$ distinct homogeneous linear functions.
We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if f is a degree-d polynomial threshold function, then its Gaussian sensitivity at noise rate ǫ is less than some quantity asymptotic to d √ 2ǫ π and the Gaussian surface area is at most d √ 2π . Furthermore these bounds are asymptotically tight as ǫ → 0 and f the threshold function of a product of d distinct homogeneous linear functions.
The noise sensitivity and surface area are both of fundamental interest and useful in the analysis of agnostic learning algorithms (see [6]). In particular our results imply that the class of degree-d polynomial threshold functions is agnostically learnable under the n-dimensional Gaussian distribution in time n O(d 2 /ǫ 4 ) .
A number of other authors have attempted to prove bounds along these lines. [7] proves a bound on noise sensitivity in terms of surface area and we relate our bounds essentially by also proving the other direction of this inequality for boolean functions with smooth interface that switch signs a bounded number of times on any line through the origin. Our bounds are obtained via a simple computation in the case of d = 1. A bound of Õ(ǫ 1/(2d) ) noise sensitivity was recently proved by [3] and independently by [5] for multilinear polynomials.
There is also interest in related questions for points picked uniformly from vertices of the hypercube rather than with the Gaussian distribution. It is conjectured in [4] that the corresponding noise sensitivity in this case is also always O(d √ ǫ). The d = 1 case of this conjecture was proved by [8], improving upon a bound of O(ǫ 1/4 ) of [2]. It is noted in [3] that such a result would imply a similar bound for the Gaussian case. Hence our results can be thought of as a first step toward proving this conjecture.
Given a function f : R n → {-1, 1} we define the Gaussian noise sensitivity at noise rate ǫ as GNS ǫ (f
where X is an n-dimensional Gaussian random variable, and
This is closely related to the Gaussian surface area of f -1 (1). In particular we define the Gaussian surface area of a set A to be
Where the Gaussian volume of a region R is Pr(X ∈ R) for X a Gaussian random variable, and where A δ is the set of points x so that d(x, A) ≤ δ (under the Euclidean metric). We note that if A is an open region whose boundary is smooth away from codimension 2, that its Gaussian surface area is equal to ∂A φ(x)dσ.
Where φ(x) is the Gaussian density, and dσ is the surface measure on ∂A. Furthermore if A is such a region, then its Gaussian surface area is seen to be equal to
For f a boolean function, we define
The concepts of noise sensitivity and surface area are related to each other by noting that the noise sensitivity is roughly the probability that X is close enough to the boundary that wiggling it will push it over the boundary.
We focus on proving two main results. We define f to be a degree d polynomial threshold function if f (x) = sgn(p(x)) for some degree d polynomial p. We prove the following Theorems about such functions:
Furthermore this bound is asymptotically tight as ǫ → 0 for the threshold function of any product of distinct linear functions.
. Section 2 will be devoted to the proof of Theorem 1, Section 3 to the proof of Theorem 2, and Section 4 will provide some closing notes.
Proof of Theorem 1. We begin by letting θ = arcsin( √ 2ǫ -ǫ 2 ). We need to bound
We note that the value of p given in Equation 1 remains the same if X and Y are replaced by any X ′ and Y ′ that are i.i.d. Gaussian distributions. In particular we define
Note that X φ and X φ+π/2 are i.i.d. Gaussians. Using these distributions we find that for any φ that since
Therefore we have for any integer n that
(2) We define the random function F : R → {-1, 1} by
(F depends on X and Y as well as φ). We note that the left hand side of Equation 2 is at most the number of times that F (φ) changes signs on the interval [0, nθ]. Therefore we have that
We note that F (φ) is periodic in φ with period 2π. Therefore the number of times F changes sign on [0, nθ] is the number of times that F changes sign on [0, 2π) times nθ 2π + O (1) . Applying this to Equation 3, we get that
Taking a limit as n → ∞ yields
We now make use of the fact that f is a degree d polynomial threshold function. In particular we will show that for any X and Y that F changes signs at most 2d times on [0, 2π). We let f = sgn(g) for some degree d polynomial g. We note that the number of sign changes of F is equal to the number of zeroes of the function g(cos(φ)X + sin(φ)Y ) (unless this function is identically 0, which happens with probability 0 and can be ignored). It should be noted though that g(cos(φ)X + sin(φ)Y ) = 0 if and only if z = e iφ is a root of the degree-2d polynomial
Therefore the expectation in Equation 4is at most 2d. Therefore we have
as desired.
We also note the ways in
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