  There is a large body of work on convergence rates either in passive or
active learning. Here we first outline some of the main results that have been
obtained, more specifically in a nonparametric setting under assumptions about
the smoothness of the regression function (or the boundary between classes) and
the margin noise. We discuss the relative merits of these underlying
assumptions by putting active learning in perspective with recent work on
passive learning. We design an active learning algorithm with a rate of
convergence better than in passive learning, using a particular smoothness
assumption customized for k-nearest neighbors. Unlike previous active learning
algorithms, we use a smoothness assumption that provides a dependence on the
marginal distribution of the instance space. Additionally, our algorithm avoids
the strong density assumption that supposes the existence of the density
function of the marginal distribution of the instance space and is therefore
more generally applicable.
