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Failing to distinguish between a sheepdog and a skyscraper should be worseand penalized more than failing to distinguish between a sheepdog and a poodle;after all, sheepdogs and poodles are both breeds of dogs. However, existingmetrics of failure (so-called "loss" or "win") used in textual or visualclassification/recognition via neural networks seldom leverage a-prioriinformation, such as a sheepdog being more similar to a poodle than to askyscraper. We define a metric that, inter alia, can penalize failure todistinguish between a sheepdog and a skyscraper more than failure todistinguish between a sheepdog and a poodle. Unlike previously employedpossibilities, this metric is based on an ultrametric tree associated with anygiven tree organization into a semantically meaningful hierarchy of aclassifier's classes. An ultrametric tree is a tree with a so-calledultrametric distance metric such that all leaves are at the same distance fromthe root. Unfortunately, extensive numerical experiments indicate that thestandard practice of training neural networks via stochastic gradient descentwith random starting points often drives down the hierarchical loss nearly asmuch when minimizing the standard cross-entropy loss as when trying to minimizethe hierarchical loss directly. Thus, this hierarchical loss is unreliable asan objective for plain, randomly started stochastic gradient descent tominimize; the main value of the hierarchical loss may be merely as a meaningfulmetric of success of a classifier.

A hierarchical loss and its problems when classifying non-hierarchically