  We study universal consistency and convergence rates of simple
nearest-neighbor prototype rules for the problem of multiclass classification
in metric paces. We first show that a novel data-dependent partitioning rule,
named Proto-NN, is universally consistent in any metric space that admits a
universally consistent rule. Proto-NN is a significant simplification of
OptiNet, a recently proposed compression-based algorithm that, to date, was the
only algorithm known to be universally consistent in such a general setting.
Practically, Proto-NN is simpler to implement and enjoys reduced computational
complexity.
  We then proceed to study convergence rates of the excess error probability.
We first obtain rates for the standard $k$-NN rule under a margin condition and
a new generalized-Lipschitz condition. The latter is an extension of a recently
proposed modified-Lipschitz condition from $\mathbb R^d$ to metric spaces.
Similarly to the modified-Lipschitz condition, the new condition avoids any
boundness assumptions on the data distribution. While obtaining rates for
Proto-NN is left open, we show that a second prototype rule that hybridizes
between $k$-NN and Proto-NN achieves the same rates as $k$-NN while enjoying
similar computational advantages as Proto-NN. However, as $k$-NN, this hybrid
rule is not consistent in general.
