62 research outputs found
Agnostic Learning of Disjunctions on Symmetric Distributions
We consider the problem of approximating and learning disjunctions (or
equivalently, conjunctions) on symmetric distributions over .
Symmetric distributions are distributions whose PDF is invariant under any
permutation of the variables. We give a simple proof that for every symmetric
distribution , there exists a set of
functions , such that for every disjunction , there is function
, expressible as a linear combination of functions in , such
that -approximates in distance on or
. This directly
gives an agnostic learning algorithm for disjunctions on symmetric
distributions that runs in time . The best known
previous bound is and follows from approximation of the
more general class of halfspaces (Wimmer, 2010). We also show that there exists
a symmetric distribution , such that the minimum degree of a
polynomial that -approximates the disjunction of all variables is
distance on is . Therefore the
learning result above cannot be achieved via -regression with a
polynomial basis used in most other agnostic learning algorithms.
Our technique also gives a simple proof that for any product distribution
and every disjunction , there exists a polynomial of
degree such that -approximates in
distance on . This was first proved by Blais et al.
(2008) via a more involved argument
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
Quantum entanglement, sum of squares, and the log rank conjecture
For every , we give an
-time algorithm for the vs
\emph{Best Separable State (BSS)} problem of distinguishing, given
an matrix corresponding to a quantum measurement,
between the case that there is a separable (i.e., non-entangled) state
that accepts with probability , and the case that every
separable state is accepted with probability at most .
Equivalently, our algorithm takes the description of a subspace (where can be either the real or
complex field) and distinguishes between the case that contains a
rank one matrix, and the case that every rank one matrix is at least
far (in distance) from .
To the best of our knowledge, this is the first improvement over the
brute-force -time algorithm for this problem. Our algorithm is based
on the \emph{sum-of-squares} hierarchy and its analysis is inspired by Lovett's
proof (STOC '14, JACM '16) that the communication complexity of every rank-
Boolean matrix is bounded by .Comment: 23 pages + 1 title-page + 1 table-of-content
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