Improper Multiclass Boosting

We study the setting of multiclass boosting with a possibly large number of classes. A recent work by Brukhim, Hazan, Moran, and Schapire, 2021, proved a hardness result for a large class of natural boosting algorithms we call proper. These algorithms output predictors that correspond to a plurality-vote aggregation of weak hypotheses. In particular, they showed that proper boosting algorithms must incur a large cost that scales with the number of classes. In this work we propose an efficient improper multiclass boosting algorithm that circumvents this hardness result. A key component of our algorithm is based on the technique of list learning. In list learning, instead of predicting a single outcome for a given unseen input, the goal is to provide a short menu of predictions. The resulting boosting algorithm has sample and oracle complexity bounds that are entirely independent of the number of classes. A corollary of the above is that plurality-vote over a learnable class is also learnable. We complement this result by showing that other simple aggregations over hypotheses from a learnable class do not preserve learnability, unlike in the binary setting.