40 research outputs found
The intersection of two halfspaces has high threshold degree
The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least
degree of a real polynomial p such that f(x)=sgn p(x). We construct two
halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an
exponential improvement on previous lower bounds. This solves an open problem
due to Klivans (2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central unresolved challenge
in computational learning. We also prove that the intersection of two majority
functions has threshold degree Omega(log n), which is tight and settles a
conjecture of O'Donnell and Servedio (2003).
Our proof consists of two parts. First, we show that for any nonconstant
Boolean functions f and g, the intersection f(x)^g(y) has threshold degree O(d)
if and only if ||f-F||_infty + ||g-G||_infty < 1 for some rational functions F,
G of degree O(d). Second, we settle the least degree required for approximating
a halfspace and a majority function to any given accuracy by rational
functions.
Our technique further allows us to make progress on Aaronson's challenge
(2008) and contribute strong direct product theorems for polynomial
representations of composed Boolean functions of the form F(f_1,...,f_n). In
particular, we give an improved lower bound on the approximate degree of the
AND-OR tree.Comment: Full version of the FOCS'09 pape
Algorithmic Polynomials
The approximate degree of a Boolean function is
the minimum degree of a real polynomial that approximates pointwise within
. Upper bounds on approximate degree have a variety of applications in
learning theory, differential privacy, and algorithm design in general. Nearly
all known upper bounds on approximate degree arise in an existential manner
from bounds on quantum query complexity. We develop a first-principles,
classical approach to the polynomial approximation of Boolean functions. We use
it to give the first constructive upper bounds on the approximate degree of
several fundamental problems:
- for the -element
distinctness problem;
- for the -subset sum problem;
- for any -DNF or -CNF formula;
- for the surjectivity problem.
In all cases, we obtain explicit, closed-form approximating polynomials that
are unrelated to the quantum arguments from previous work. Our first three
results match the bounds from quantum query complexity. Our fourth result
improves polynomially on the quantum query complexity of the
problem and refutes the conjecture by several experts that surjectivity has
approximate degree . In particular, we exhibit the first natural
problem with a polynomial gap between approximate degree and quantum query
complexity
Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0
The threshold degree of a Boolean function is
the minimum degree of a real polynomial that represents in sign:
A related notion is sign-rank, defined for a
Boolean matrix as the minimum rank of a real matrix with
. Determining the maximum threshold degree
and sign-rank achievable by constant-depth circuits () is a
well-known and extensively studied open problem, with complexity-theoretic and
algorithmic applications.
We give an essentially optimal solution to this problem. For any
we construct an circuit in variables that has
threshold degree and sign-rank
improving on the previous best lower bounds of
and , respectively. Our
results subsume all previous lower bounds on the threshold degree and sign-rank
of circuits of any given depth, with a strict improvement
starting at depth . As a corollary, we also obtain near-optimal bounds on
the discrepancy, threshold weight, and threshold density of ,
strictly subsuming previous work on these quantities. Our work gives some of
the strongest lower bounds to date on the communication complexity of
.Comment: 99 page
Synchronization Strings: Explicit Constructions, Local Decoding, and Applications
This paper gives new results for synchronization strings, a powerful
combinatorial object that allows to efficiently deal with insertions and
deletions in various communication settings:
We give a deterministic, linear time synchronization string
construction, improving over an time randomized construction.
Independently of this work, a deterministic time
construction was just put on arXiv by Cheng, Li, and Wu. We also give a
deterministic linear time construction of an infinite synchronization string,
which was not known to be computable before. Both constructions are highly
explicit, i.e., the symbol can be computed in time.
This paper also introduces a generalized notion we call
long-distance synchronization strings that allow for local and very fast
decoding. In particular, only time and access to logarithmically
many symbols is required to decode any index.
We give several applications for these results:
For any we provide an insdel correcting
code with rate which can correct any fraction
of insdel errors in time. This near linear computational
efficiency is surprising given that we do not even know how to compute the
(edit) distance between the decoding input and output in sub-quadratic time. We
show that such codes can not only efficiently recover from fraction of
insdel errors but, similar to [Schulman, Zuckerman; TransInf'99], also from any
fraction of block transpositions and replications.
We show that highly explicitness and local decoding allow for
infinite channel simulations with exponentially smaller memory and decoding
time requirements. These simulations can be used to give the first near linear
time interactive coding scheme for insdel errors