发表机构
School of Computer Science, Wuhan University（武汉大学计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了因子化AdaBoost.MH的收敛性分析，证明其收敛速率与AdaBoost.MH一致，消除了对样本量$n$或类别数$K$的额外依赖。
AI 中文摘要
AdaBoost.MH将多分类问题约简为一系列二分类子问题，在弱学习条件下享有经典的提升型收敛保证。更具结构化的变体因子化AdaBoost.MH使用形如$\boldsymbol{h}(x)=\boldsymbol{\beta}\boldsymbol{v}\boldsymbol{\bm{\rho}}(x)$的基分类器，其中单个二分类器$\boldsymbol{\bm{\rho}}$在所有类别间共享，标签依赖由投票向量$\boldsymbol{v}\in\{\pm1\}^K$承载。该因子化在算法层面颇具吸引力，且在实际应用中表现更优，但其收敛性取决于是否总能选择诱导二元权重质量足够大的投票向量。过往研究通过下界$\max\{1/n,1/\sqrt{2K}\}$解决了这一问题，该下界仍使算法相对于原始AdaBoost.MH分析存在维度相关的减速。本文中，我们对这一组合步骤进行了改进：对于控制因子化边的极小极大量$\boldsymbol{W}_{n,K}$，我们证明$\max\{1/n,C_K\}\le\boldsymbol{W}_{n,K}\le C_{\min\{n,K\}}$，其中$C_q$在$q=1$时为1，偶整数$q\ge2$时为$q/(3q-4)$，奇整数$q\ge2$时为$(q+1)/(3q-1)$。由于$C_q$随$q$增大趋近于1/3，我们的界表明$\boldsymbol{W}_{n,K}=\Theta(1)$在$n$和$K$上均匀成立。因此，因子化AdaBoost.MH在通用常数因子范围内达到与AdaBoost.MH相同的提升型收敛速率，消除了此前在提升轮数中对$n$或$K$的额外依赖。
英文摘要
{AdaBoost.MH} reduces multi-class classification to a collection of binary subproblems and enjoys the classical boosting-type convergence guarantee under a weak learning condition. A more structured variant, Factorized {AdaBoost.MH}, uses base classifiers of the form $\mathbf{h}(x)=α\mathbf{v} \bmφ(x)$, where a single binary classifier $\bmφ$ is shared across all classes and the label dependence is carried by a vote vector $\mathbf{v} \in\{\pm1\}^K$. This factorization is algorithmically attractive and achieves better performance in practice, but its convergence depends on whether one can always choose a vote vector with sufficiently large induced binary weight mass. Previous work resolved this question with a lower bound $\max\{1/n,1/\sqrt{2K}\}$, which still leaves a dimension-dependent slowdown relative to the original {AdaBoost.MH} analysis. In this paper, we sharpen this combinatorial step. For the minimax quantity $\mathfrak{W}_{n,K}$ governing the factorized edge, we prove $\mathfrak{W}_{n,K} = C_{\min\{n+1,K\}}$, where $C_q=1$ for $q=1$, $C_q=q/(3q-4)$ for even $q\ge2$, and $C_q=(q+1)/(3q-1)$ for odd $q\ge2$. Since $C_q\downarrow 1/3$, our bounds show that $\mathfrak{W}_{n,K}=Θ(1)$ uniformly over $n$ and $K$. Consequently, Factorized {AdaBoost.MH} achieves the same boosting-type convergence rate as {AdaBoost.MH} up to a universal constant factor, removing the previously suggested additional dependence on $n$ or $K$ in the number of boosting rounds.
