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

K-NN active learning under local smoothness assumption