12,295 research outputs found
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
On the sum of the L1 influences of bounded functions
Let have degree as a multilinear
polynomial. It is well-known that the total influence of is at most .
Aaronson and Ambainis asked whether the total influence of can also
be bounded as a function of . Ba\v{c}kurs and Bavarian answered this
question in the affirmative, providing a bound of for general
functions and for homogeneous functions. We improve on their results
by providing a bound of for general functions and for
homogeneous functions. In addition, we prove a bound of for
monotone functions, and provide a matching example.Comment: 16 pages; accepted for publication in the Israel Journal of
Mathematic
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
On the sum-of-squares degree of symmetric quadratic functions
We study how well functions over the boolean hypercube of the form
can be approximated by sums of squares of low-degree
polynomials, obtaining good bounds for the case of approximation in
-norm as well as in -norm. We describe three
complexity-theoretic applications: (1) a proof that the recent breakthrough
lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite
extension complexity of the correlation and TSP polytopes cannot be improved
further by showing better sum-of-squares degree lower bounds on
-approximation of ; (2) a proof that Grigoriev's lower bound on
the degree of Positivstellensatz refutations for the knapsack problem is
optimal, answering an open question from his work; (3) bounds on the query
complexity of quantum algorithms whose expected output approximates such
functions.Comment: 33 pages. Second version fixes some typos and adds reference
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