  We propose a new framework for algorithmic stability in the context of
multiclass classification. In practice, classification algorithms often operate
by first assigning a continuous score (for instance, an estimated probability)
to each possible label, then taking the maximizer -- i.e., selecting the class
that has the highest score. A drawback of this type of approach is that it is
inherently unstable, meaning that it is very sensitive to slight perturbations
of the training data, since taking the maximizer is discontinuous. Motivated by
this challenge, we propose a pipeline for constructing stable classifiers from
data, using bagging (i.e., resampling and averaging) to produce stable
continuous scores, and then using a stable relaxation of argmax, which we call
the "inflated argmax," to convert these scores to a set of candidate labels.
The resulting stability guarantee places no distributional assumptions on the
data, does not depend on the number of classes or dimensionality of the
covariates, and holds for any base classifier. Using a common benchmark data
set, we demonstrate that the inflated argmax provides necessary protection
against unstable classifiers, without loss of accuracy.
