5 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