  We propose a novel nonparametric online predictor for discrete labels
conditioned on multivariate continuous features. The predictor is based on a
feature space discretization induced by a full-fledged k-d tree with randomly
picked directions and a recursive Bayesian distribution, which allows to
automatically learn the most relevant feature scales characterizing the
conditional distribution. We prove its pointwise universality, i.e., it
achieves a normalized log loss performance asymptotically as good as the true
conditional entropy of the labels given the features. The time complexity to
process the $n$-th sample point is $O(\log n)$ in probability with respect to
the distribution generating the data points, whereas other exact nonparametric
methods require to process all past observations. Experiments on challenging
datasets show the computational and statistical efficiency of our algorithm in
comparison to standard and state-of-the-art methods.
