K-nn active learning under local smoothness condition

There is a large body of work on convergence rates either in passive or active learning. Here we outline some of the results that have been obtained, more specifically in a nonparametric setting under assumptions about the smoothness and the margin noise. We also discuss the relative merits of these underlying assumptions by putting active learning in perspective with recent work on passive learning. We provide a novel active learning algorithm with a rate of convergence better than in passive learning, using a particular smoothness assumption customized for $k$-nearest neighbors. This smoothness assumption provides a dependence on the marginal distribution of the instance space unlike other recent algorithms. Our algorithm thus 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.