Multiclass Boosting and the Cost of Weak Learning

Boosting is an algorithmic approach which is based on the idea of combining weak and moderately inaccurate hypotheses to a strong and accurate one. In this work we study multiclass boosting with a possibly large number of classes or categories. Multiclass boosting can be formulated in various ways. Here, we focus on an especially natural formulation in which the weak hypotheses are assumed to belong to an ``easy-to-learn'' base class, and the weak learner is an agnostic PAC learner for that class with respect to the standard classification loss. This is in contrast with other, more complicated losses as have often been considered in the past. The goal of the overall boosting algorithm is then to learn a combination of weak hypotheses by repeatedly calling the weak learner. We study the resources required for boosting, especially how they depend on $k$, the number of classes. For the booster, these include the number of examples needed for learning and the number of oracle calls to the weak learner. For the weak learner, we measure the required accuracy, that is, how close the returned hypothesis must be to the best in the base class, or alternatively, the number of unweighted samples the boosting algorithm feeds the weak learner in each round. We find that the boosting algorithm itself only requires $O(\log k)$ samples, as we show by analyzing a variant of AdaBoost for our setting. In stark contrast, assuming typical limits on the number of weak-learner calls, we prove that the number of samples required by any weak learner is at least polynomial in $k$, exponentially more than the number of samples needed by the booster. Alternatively, we prove that the weak learner's accuracy parameter must be smaller than an inverse polynomial in $k$, showing that the returned weak hypotheses must be nearly the best in their class when $k$ is large. We also prove a trade-off between number of oracle calls and the resources required of the weak learner, meaning that the fewer calls to the weak learner the more that is demanded on each call.