2,227 research outputs found
Bounded Independence Fools Degree-2 Threshold Functions
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.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was
wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem
8.1 to m^4, not m^
Recommended from our members
Bounded Independence Fools Degree-2 Threshold Functions
For an n-variate degree-2 real polynomial p, we prove that Is determined up to an additive as long as D is a k-wise Independent distribution over for . This gives a broad class of explicit pseudorandom generators against degree-2 boolean threshold functions, and answers an open question of Diakonikolas et al. (FOCS 2009).Engineering and Applied Science
A Polylogarithmic PRG for Degree Threshold Functions in the Gaussian Setting
We devise a new pseudorandom generator against degree 2 polynomial threshold
functions in the Gaussian setting. We manage to achieve error with
seed length polylogarithmic in and the dimension, and exponential
improvement over previous constructions
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
A Small PRG for Polynomial Threshold Functions of Gaussians
We develop a pseudo-random generator to fool degree- polynomial threshold
functions with respect to the Gaussian distribution. For any constant, we
construct a pseudo-random generator that fools such functions to within
and has seed length
A Pseudorandom Generator for Polynomial Threshold Functions of Gaussian with Subpolynomial Seed Length
We develop a pseudorandom generator that fools degree- polynomial
threshold functions in variables with respect to the Gaussian distribution
and has seed length
A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut
We give improved pseudorandom generators (PRGs) for Lipschitz functions of
low-degree polynomials over the hypercube. These are functions of the form
psi(P(x)), where P is a low-degree polynomial and psi is a function with small
Lipschitz constant. PRGs for smooth functions of low-degree polynomials have
received a lot of attention recently and play an important role in constructing
PRGs for the natural class of polynomial threshold functions. In spite of the
recent progress, no nontrivial PRGs were known for fooling Lipschitz functions
of degree O(log n) polynomials even for constant error rate. In this work, we
give the first such generator obtaining a seed-length of (log
n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps.
Previous generators had an exponential dependence on the degree.
We use our PRG to get better integrality gap instances for sparsest cut, a
fundamental problem in graph theory with many applications in graph
optimization. We give an instance of uniform sparsest cut for which a powerful
semi-definite relaxation (SDP) first introduced by Goemans and Linial and
studied in the seminal work of Arora, Rao and Vazirani has an integrality gap
of exp(\Omega((log log n)^{1/2})). Understanding the performance of the
Goemans-Linial SDP for uniform sparsest cut is an important open problem in
approximation algorithms and metric embeddings and our work gives a
near-exponential improvement over previous lower bounds which achieved a gap of
\Omega(log log n)
Moment-Matching Polynomials
We give a new framework for proving the existence of low-degree, polynomial
approximators for Boolean functions with respect to broad classes of
non-product distributions. Our proofs use techniques related to the classical
moment problem and deviate significantly from known Fourier-based methods,
which require the underlying distribution to have some product structure.
Our main application is the first polynomial-time algorithm for agnostically
learning any function of a constant number of halfspaces with respect to any
log-concave distribution (for any constant accuracy parameter). This result was
not known even for the case of learning the intersection of two halfspaces
without noise. Additionally, we show that in the "smoothed-analysis" setting,
the above results hold with respect to distributions that have sub-exponential
tails, a property satisfied by many natural and well-studied distributions in
machine learning.
Given that our algorithms can be implemented using Support Vector Machines
(SVMs) with a polynomial kernel, these results give a rigorous theoretical
explanation as to why many kernel methods work so well in practice
- …