51 research outputs found
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 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
Derandomized Graph Product Results using the Low Degree Long Code
In this paper, we address the question of whether the recent derandomization
results obtained by the use of the low-degree long code can be extended to
other product settings. We consider two settings: (1) the graph product results
of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is
stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and
Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of
approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of which exhibits the following property (shown for
by Alon et al.): independent sets close in size to the
maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur and
Shinkar shows that if there exist two sets of vertices and in
with very few edges with one endpoint in and another in
, then it must be the case that the two sets and share a single
influential coordinate. In our second result, we show that a similar "majority
is stablest" statement holds good for a considerably smaller subgraph of
. Furthermore using this result, we give a more efficient
reduction from Unique Games to the graph coloring problem, leading to improved
hardness of approximation results for coloring
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
Bounded Independence vs. Moduli
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m
Almost Optimal Pseudorandom Generators for Spherical Caps
Halfspaces or linear threshold functions are widely studied in complexity
theory, learning theory and algorithm design. In this work we study the natural
problem of constructing pseudorandom generators (PRGs) for halfspaces over the
sphere, aka spherical caps, which besides being interesting and basic geometric
objects, also arise frequently in the analysis of various randomized algorithms
(e.g., randomized rounding). We give an explicit PRG which fools spherical caps
within error and has an almost optimal seed-length of . For an inverse-polynomially
growing error , our generator has a seed-length optimal up to a
factor of . The most efficient PRG previously known (due
to Kane, 2012) requires a seed-length of in this
setting. We also obtain similar constructions to fool halfspaces with respect
to the Gaussian distribution.
Our construction and analysis are significantly different from previous works
on PRGs for halfspaces and build on the iterative dimension reduction ideas of
Kane et. al. (2011) and Celis et. al. (2013), the \emph{classical moment
problem} from probability theory and explicit constructions of \emph{orthogonal
designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion
in Lie groups.Comment: 28 Pages (including the title page
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Polynomial approximations to boolean functions have led to many positive
results in computer science. In particular, polynomial approximations to the
sign function underly algorithms for agnostically learning halfspaces, as well
as pseudorandom generators for halfspaces. In this work, we investigate the
limits of these techniques by proving inapproximability results for the sign
function.
Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput.
2008) shows that halfspaces can be learned with respect to log-concave
distributions on in the challenging agnostic learning model. The
power of this algorithm relies on the fact that under log-concave
distributions, halfspaces can be approximated arbitrarily well by low-degree
polynomials. We ask whether this technique can be extended beyond log-concave
distributions, and establish a negative result. We show that polynomials of any
degree cannot approximate the sign function to within arbitrarily low error for
a large class of non-log-concave distributions on the real line, including
those with densities proportional to .
Secondly, we investigate the derandomization of Chernoff-type concentration
inequalities. Chernoff-type tail bounds on sums of independent random variables
have pervasive applications in theoretical computer science. Schmidt et al.
(SIAM J. Discrete Math. 1995) showed that these inequalities can be established
for sums of random variables with only -wise independence,
for a tail probability of . We show that their results are tight up to
constant factors.
These results rely on techniques from weighted approximation theory, which
studies how well functions on the real line can be approximated by polynomials
under various distributions. We believe that these techniques will have further
applications in other areas of computer science.Comment: 22 page
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