Bounded Independence Fools Degree-2 Threshold Functions

TL;DR

This paper demonstrates that degree-2 threshold functions can be effectively fooled by bounded k-wise independent distributions, providing new derandomization tools and extending previous results with a novel multivariate FT-mollification technique.

Contribution

It proves that k-wise independence suffices to approximate degree-2 threshold functions, introduces multivariate FT-mollification, and extends results to intersections of such functions.

Findings

Bounded independence epsilon-fools degree-2 threshold functions.

Explicit generators with seed length log(n)*poly(1/epsilon) are constructed.

The approach extends to intersections of degree-2 threshold functions and derandomizes hyperplane rounding.

Abstract

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.