Despite the widespread use of boosting in structured prediction, a general theoretical understanding of aggregation beyond scalar losses remains incomplete. We study vector-valued and conditional density prediction under general divergences and identify stability conditions under which aggregation amplifies weak guarantees into strong ones. We formalize this stability property as \emph{$(α,β)$-boostability}. We show that geometric median aggregation achieves $(α,β)$-boostability for a broad class of divergences, with tradeoffs that depend on the underlying geometry. For vector-valued prediction and conditional density estimation, we characterize boostability under common divergences ($\ell_1$, $\ell_2$, $\TV$, and $\Hel$) with geometric median, revealing a sharp distinction between dimension-dependent and dimension-free regimes. We further show that while KL divergence is not directly boostable via geometric median aggregation, it can be handled indirectly through boostability under Hellinger distance. Building on these structural results, we propose a generic boosting framework \textsc{GeoMedBoost} based on exponential reweighting and geometric-median aggregation. Under a weak learner condition and $(α,β)$-boostability, we obtain exponential decay of the empirical divergence exceedance error. Our framework recovers classical algorithms such as \textsc{MedBoost}, \textsc{AdaBoost}, and \textsc{SAMME} as special cases, and provides a unified geometric view of boosting for structured prediction.

翻译：尽管Boosting在结构化预测中广泛应用，但超越标量损失的聚合理论理解仍不完整。我们研究一般散度下的向量值与条件密度预测，并识别出能够将弱保证放大为强保证的稳定性条件。我们将此稳定性形式化为\emph{$(α,β)$-可提升性}。我们证明几何中位数聚合能为广泛类别的散度实现$(α,β)$-可提升性，其权衡取决于底层几何结构。针对向量值预测与条件密度估计，我们刻画了在常见散度（$\ell_1$、$\ell_2$、$\TV$与$\Hel$）下使用几何中位数的可提升性，揭示了维度依赖与维度无关机制之间的显著区别。我们进一步证明，虽然KL散度无法直接通过几何中位数聚合实现可提升性，但可通过Hellinger距离下的可提升性间接处理。基于这些结构结果，我们提出一个基于指数重加权与几何中位数聚合的通用Boosting框架\textsc{GeoMedBoost}。在弱学习器条件与$(α,β)$-可提升性假设下，我们获得了经验散度超限误差的指数衰减。我们的框架将经典算法如\textsc{MedBoost}、\textsc{AdaBoost}和\textsc{SAMME}作为特例进行统一，为结构化预测的Boosting提供了统一的几何视角。