  Failing to distinguish between a sheepdog and a skyscraper should be worse
and penalized more than failing to distinguish between a sheepdog and a poodle;
after all, sheepdogs and poodles are both breeds of dogs. However, existing
metrics of failure (so-called "loss" or "win") used in textual or visual
classification/recognition via neural networks seldom leverage a-priori
information, such as a sheepdog being more similar to a poodle than to a
skyscraper. We define a metric that, inter alia, can penalize failure to
distinguish between a sheepdog and a skyscraper more than failure to
distinguish between a sheepdog and a poodle. Unlike previously employed
possibilities, this metric is based on an ultrametric tree associated with any
given tree organization into a semantically meaningful hierarchy of a
classifier's classes. An ultrametric tree is a tree with a so-called
ultrametric distance metric such that all leaves are at the same distance from
the root. Unfortunately, extensive numerical experiments indicate that the
standard practice of training neural networks via stochastic gradient descent
with random starting points often drives down the hierarchical loss nearly as
much when minimizing the standard cross-entropy loss as when trying to minimize
the hierarchical loss directly. Thus, this hierarchical loss is unreliable as
an objective for plain, randomly started stochastic gradient descent to
minimize; the main value of the hierarchical loss may be merely as a meaningful
metric of success of a classifier.
