78 research outputs found
Deterministic Approximate Counting for Juntas of Degree-2 Polynomial Threshold Functions
Let be any Boolean function and
be any degree-2 polynomials over We give a \emph{deterministic}
algorithm which, given as input explicit descriptions of and
an accuracy parameter \eps>0, approximates \Pr_{x \sim
\{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] to within an additive
\pm \eps. For any constant \eps > 0 and the running time of our
algorithm is a fixed polynomial in . This is the first fixed polynomial-time
algorithm that can deterministically approximately count satisfying assignments
of a natural class of depth-3 Boolean circuits.
Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a
deterministic approximate counting algorithm for a single degree-2 polynomial
threshold function \sign(q(x)), corresponding to the case of our
result.
Our algorithm and analysis requires several novel technical ingredients that
go significantly beyond the tools required to handle the case in
\cite{DDS13:deg2count}. One of these is a new multidimensional central limit
theorem for degree-2 polynomials in Gaussian random variables which builds on
recent Malliavin-calculus-based results from probability theory. We use this
CLT as the basis of a new decomposition technique for -tuples of degree-2
Gaussian polynomials and thus obtain an efficient deterministic approximate
counting algorithm for the Gaussian distribution. Finally, a third new
ingredient is a "regularity lemma" for \emph{-tuples} of degree-
polynomial threshold functions. This generalizes both the regularity lemmas of
\cite{DSTW:10,HKM:09} and the regularity lemma of Gopalan et al \cite{GOWZ10}.
Our new regularity lemma lets us extend our deterministic approximate counting
results from the Gaussian to the Boolean domain
Efficient deterministic approximate counting for low-degree polynomial threshold functions
We give a deterministic algorithm for approximately counting satisfying
assignments of a degree- polynomial threshold function (PTF). Given a
degree- input polynomial over and a parameter
, our algorithm approximates to within an additive in time . (Any sort of efficient multiplicative approximation is
impossible even for randomized algorithms assuming .) Note that the
running time of our algorithm (as a function of , the number of
coefficients of a degree- PTF) is a \emph{fixed} polynomial. The fastest
previous algorithm for this problem (due to Kane), based on constructions of
unconditional pseudorandom generators for degree- PTFs, runs in time
for all .
The key novel contributions of this work are: A new multivariate central
limit theorem, proved using tools from Malliavin calculus and Stein's Method.
This new CLT shows that any collection of Gaussian polynomials with small
eigenvalues must have a joint distribution which is very close to a
multidimensional Gaussian distribution. A new decomposition of low-degree
multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up
to some small error) any such polynomial can be decomposed into a bounded
number of multilinear polynomials all of which have extremely small
eigenvalues. We use these new ingredients to give a deterministic algorithm for
a Gaussian-space version of the approximate counting problem, and then employ
standard techniques for working with low-degree PTFs (invariance principles and
regularity lemmas) to reduce the original approximate counting problem over the
Boolean hypercube to the Gaussian version
Deterministic search for CNF satisfying assignments in almost polynomial time
We consider the fundamental derandomization problem of deterministically
finding a satisfying assignment to a CNF formula that has many satisfying
assignments. We give a deterministic algorithm which, given an -variable
-clause CNF formula that has at least
satisfying assignments, runs in time for
and outputs a satisfying assignment of
. Prior to our work the fastest known algorithm for this problem was simply
to enumerate over all seeds of a pseudorandom generator for CNFs; using the
best known PRGs for CNFs [DETT10], this takes time
even for constant . Our approach is based on a new general
framework relating deterministic search and deterministic approximate counting,
which we believe may find further applications
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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