  When many labels are possible, choosing a single one can lead to low
precision. A common alternative, referred to as top-$K$ classification, is to
choose some number $K$ (commonly around 5) and to return the $K$ labels with
the highest scores. Unfortunately, for unambiguous cases, $K>1$ is too many
and, for very ambiguous cases, $K \leq 5$ (for example) can be too small. An
alternative sensible strategy is to use an adaptive approach in which the
number of labels returned varies as a function of the computed ambiguity, but
must average to some particular $K$ over all the samples. We denote this
alternative average-$K$ classification. This paper formally characterizes the
ambiguity profile when average-$K$ classification can achieve a lower error
rate than a fixed top-$K$ classification. Moreover, it provides natural
estimation procedures for both the fixed-size and the adaptive classifier and
proves their consistency. Finally, it reports experiments on real-world image
data sets revealing the benefit of average-$K$ classification over top-$K$ in
practice. Overall, when the ambiguity is known precisely, average-$K$ is never
worse than top-$K$, and, in our experiments, when it is estimated, this also
holds.
