  We consider the problem of multi-class classification, where a stream of
adversarially chosen queries arrive and must be assigned a label online. Unlike
traditional bounds which seek to minimize the misclassification rate, we
minimize the total distance from each query to the region corresponding to its
correct label. When the true labels are determined via a nearest neighbor
partition -- i.e. the label of a point is given by which of $k$ centers it is
closest to in Euclidean distance -- we show that one can achieve a loss that is
independent of the total number of queries. We complement this result by
showing that learning general convex sets requires an almost linear loss per
query. Our results build off of regret guarantees for the geometric problem of
contextual search. In addition, we develop a novel reduction technique from
multiclass classification to binary classification which may be of independent
interest.
