Factorized AdaBoost.MH Matches Original AdaBoost Convergence Rate.

AdaBoost.MH is a well-established algorithm for multi-class classification, known for its strong convergence guarantees. A more structured and often practically superior variant, Factorized AdaBoost.MH, simplifies the base classifiers by sharing a single binary classifier across all classes, with label dependence handled by a vote vector. Previously, there was uncertainty regarding whether this factorization introduced a dimension-dependent slowdown in its convergence rate compared to the original AdaBoost.MH. This research provides a rigorous mathematical proof that Factorized AdaBoost.MH indeed achieves the same boosting-type convergence rate as its non-factorized counterpart, up to a universal constant factor. The paper sharpens the combinatorial analysis of the minimax quantity governing the factorized edge, demonstrating that it remains uniformly bounded over varying numbers of training examples and classes. This finding removes the previously suggested additional dependence on $n$ or $K$ (number of examples or classes), confirming that the factorized variant is as theoretically efficient as the original, while often offering practical advantages.