8,369 research outputs found
Learning DNF Expressions from Fourier Spectrum
Since its introduction by Valiant in 1984, PAC learning of DNF expressions
remains one of the central problems in learning theory. We consider this
problem in the setting where the underlying distribution is uniform, or more
generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed
that in this setting a DNF expression can be efficiently approximated from its
"heavy" low-degree Fourier coefficients alone. This is in contrast to previous
approaches where boosting was used and thus Fourier coefficients of the target
function modified by various distributions were needed. This property is
crucial for learning of DNF expressions over smoothed product distributions, a
learning model introduced by Kalai et al. (2009) and inspired by the seminal
smoothed analysis model of Spielman and Teng (2001).
We introduce a new approach to learning (or approximating) a polynomial
threshold functions which is based on creating a function with range [-1,1]
that approximately agrees with the unknown function on low-degree Fourier
coefficients. We then describe conditions under which this is sufficient for
learning polynomial threshold functions. Our approach yields a new, simple
algorithm for approximating any polynomial-size DNF expression from its "heavy"
low-degree Fourier coefficients alone. Our algorithm greatly simplifies the
proof of learnability of DNF expressions over smoothed product distributions.
We also describe an application of our algorithm to learning monotone DNF
expressions over product distributions. Building on the work of Servedio
(2001), we give an algorithm that runs in time \poly((s \cdot
\log{(s/\eps)})^{\log{(s/\eps)}}, n), where is the size of the target DNF
expression and \eps is the accuracy. This improves on \poly((s \cdot
\log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n) bound of Servedio
(2001).Comment: Appears in Conference on Learning Theory (COLT) 201
Learning Coverage Functions and Private Release of Marginals
We study the problem of approximating and learning coverage functions. A
function is a coverage function, if
there exists a universe with non-negative weights for each
and subsets of such that . Alternatively, coverage functions can be described
as non-negative linear combinations of monotone disjunctions. They are a
natural subclass of submodular functions and arise in a number of applications.
We give an algorithm that for any , given random and uniform
examples of an unknown coverage function , finds a function that
approximates within factor on all but -fraction of the
points in time . This is the first fully-polynomial
algorithm for learning an interesting class of functions in the demanding PMAC
model of Balcan and Harvey (2011). Our algorithms are based on several new
structural properties of coverage functions. Using the results in (Feldman and
Kothari, 2014), we also show that coverage functions are learnable agnostically
with excess -error over all product and symmetric
distributions in time . In contrast, we show that,
without assumptions on the distribution, learning coverage functions is at
least as hard as learning polynomial-size disjoint DNF formulas, a class of
functions for which the best known algorithm runs in time
(Klivans and Servedio, 2004).
As an application of our learning results, we give simple
differentially-private algorithms for releasing monotone conjunction counting
queries with low average error. In particular, for any , we obtain
private release of -way marginals with average error in time
Discrete Spectra of Semirelativistic Hamiltonians
We review various attempts to localize the discrete spectra of
semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r)
(w.l.o.g. in three spatial dimensions) as entering, for instance, in the
spinless Salpeter equation. Every Hamiltonian in this class of operators
consists of the relativistic kinetic energy \beta \sqrt{m^2 + p^2} (where \beta
> 0 allows for the possibility of more than one particles of mass m) and a
spherically symmetric attractive potential V(r), r = |x|. In general, accurate
eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of
a numerical approximation procedure. Our main emphasis, however, is on the
derivation of rigorous semi-analytical expressions for both upper and lower
bounds to the energy levels of such operators. We compare the bounds obtained
within different approaches and present relationships existing between the
bounds.Comment: 21 pages, 3 figure
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