9 research outputs found
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
Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits
We study correlation bounds and pseudorandom generators for depth-two
circuits that consist of a -gate (computing an arbitrary
symmetric function) or -gate (computing an arbitrary linear
threshold function) that is fed by gates. Such circuits were
considered in early influential work on unconditional derandomization of Luby,
Veli\v{c}kovi\'c, and Wigderson [LVW93], who gave the first non-trivial PRG
with seed length that -fools
these circuits.
In this work we obtain the first strict improvement of [LVW93]'s seed length:
we construct a PRG that -fools size-
circuits over
with seed length
an exponential (and near-optimal) improvement of the -dependence
of [LVW93]. The above PRG is actually a special case of a more general PRG
which we establish for constant-depth circuits containing multiple
or gates, including as a special case
circuits. These more
general results strengthen previous results of Viola [Vio06] and essentially
strengthen more recent results of Lovett and Srinivasan [LS11].
Our improved PRGs follow from improved correlation bounds, which are
transformed into PRGs via the Nisan--Wigderson "hardness versus randomness"
paradigm [NW94]. The key to our improved correlation bounds is the use of a
recent powerful \emph{multi-switching} lemma due to H{\aa}stad [H{\aa}s14]
Dimension Reduction for Polynomials over Gaussian Space and Applications
We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest
Noise stability is computable and approximately low-dimensional
Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of R[superscript n] for n ≥ 1 to k parts with given Gaussian measures μ[superscript 1], . . . , μ[superscript k]. We call a partition ϵ-optimal, if its noise stability is optimal up to an additive ϵ. In this paper, we give an explicit, computable function n(ϵ) such that an ϵ-optimal partition exists in R[superscript n(ϵ)]. This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work. Keywords: Gaussian noise stability; Plurality is stablest; Ornstein Uhlenbeck operatorNational Science Foundation (U.S.) (Award CCF 1320105)United States. Office of Naval Research (Grant N00014-16-1-2227
Towards derandomising Markov chain Monte Carlo
We present a new framework to derandomise certain Markov chain Monte Carlo
(MCMC) algorithms.
As in MCMC, we first reduce counting problems to sampling from a sequence of
marginal distributions.
For the latter task,
we introduce a method called coupling towards the past that can, in
logarithmic time,
evaluate one or a constant number of variables from a stationary Markov chain
state.
Since there are at most logarithmic random choices, this leads to very simple
derandomisation.
We provide two applications of this framework, namely efficient deterministic
approximate counting algorithms for hypergraph independent sets and hypergraph
colourings,
under local lemma type conditions matching, up to lower order factors, their
state-of-the-art randomised counterparts.Comment: 57 page