title: Multiclass Boosting and the Cost of Weak Learning
abstract: 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 the number of classes $k$, for both the booster and weak learner.
  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 a 
  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.
authors: Nataly Brukhim Elad Hazan Shay Moran Indraneel Mukherjee Robert E. Schapire
hIndex: Robert E. Schapire 77