354 research outputs found
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
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
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
Pseudorandom Generators for Width-3 Branching Programs
We construct pseudorandom generators of seed length that -fool ordered read-once branching programs
(ROBPs) of width and length . For unordered ROBPs, we construct
pseudorandom generators with seed length . This is the first improvement for pseudorandom
generators fooling width ROBPs since the work of Nisan [Combinatorica,
1992].
Our constructions are based on the `iterated milder restrictions' approach of
Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson
framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the
last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered
case, we combine iterated milder restrictions with the generator of
Chattopadhyay et al. [CCC, 2018].
Two conceptual ideas that play an important role in our analysis are: (1) A
relabeling technique allowing us to analyze a relabeled version of the given
branching program, which turns out to be much easier. (2) Treating the number
of colliding layers in a branching program as a progress measure and showing
that it reduces significantly under pseudorandom restrictions.
In addition, we achieve nearly optimal seed-length
for the classes of: (1) read-once polynomials on
variables, (2) locally-monotone ROBPs of length and width
(generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length
having a layer of width in every consecutive
layers.Comment: 51 page
Better Pseudorandom Generators from Milder Pseudorandom Restrictions
We present an iterative approach to constructing pseudorandom generators,
based on the repeated application of mild pseudorandom restrictions. We use
this template to construct pseudorandom generators for combinatorial rectangles
and read-once CNFs and a hitting set generator for width-3 branching programs,
all of which achieve near-optimal seed-length even in the low-error regime: We
get seed-length O(log (n/epsilon)) for error epsilon. Previously, only
constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for
these classes with polynomially small error.
The (pseudo)random restrictions we use are milder than those typically used
for proving circuit lower bounds in that we only set a constant fraction of the
bits at a time. While such restrictions do not simplify the functions
drastically, we show that they can be derandomized using small-bias spaces.Comment: To appear in FOCS 201
Fourier Growth of Structured ??-Polynomials and Applications
We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F?-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013].
- We show that any read-? degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? (k ? d)^{O(k)}.
- We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F?-polynomials
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
- …