  We introduce a new model of membership query (MQ) learning, where the
learning algorithm is restricted to query points that are \emph{close} to
random examples drawn from the underlying distribution. The learning model is
intermediate between the PAC model (Valiant, 1984) and the PAC+MQ model (where
the queries are allowed to be arbitrary points).
  Membership query algorithms are not popular among machine learning
practitioners. Apart from the obvious difficulty of adaptively querying
labelers, it has also been observed that querying \emph{unnatural} points leads
to increased noise from human labelers (Lang and Baum, 1992). This motivates
our study of learning algorithms that make queries that are close to examples
generated from the data distribution.
  We restrict our attention to functions defined on the $n$-dimensional Boolean
hypercube and say that a membership query is local if its Hamming distance from
some example in the (random) training data is at most $O(\log(n))$. We show the
following results in this model:
  (i) The class of sparse polynomials (with coefficients in R) over $\{0,1\}^n$
is polynomial time learnable under a large class of \emph{locally smooth}
distributions using $O(\log(n))$-local queries. This class also includes the
class of $O(\log(n))$-depth decision trees.
  (ii) The class of polynomial-sized decision trees is polynomial time
learnable under product distributions using $O(\log(n))$-local queries.
  (iii) The class of polynomial size DNF formulas is learnable under the
uniform distribution using $O(\log(n))$-local queries in time
$n^{O(\log(\log(n)))}$.
  (iv) In addition we prove a number of results relating the proposed model to
the traditional PAC model and the PAC+MQ model.
