提出向量预测的广义提升框架，实现弱学习到强学习的指数级误差下降。
$(α,β)$-Stability for Boosting Vector-Valued Prediction
- 基于几何中位数与指数重加权，构建$(α,β)$-稳定性理论
- 在ℓ₁、ℓ₂、TV等距离下实现经验误差指数衰减
- 适用于密度估计和结构化预测，适合理论研究者
尽管提升方法在结构化预测中广泛应用，但对超出标量预测的聚合机制仍缺乏普遍理论。本文研究目标分歧下的向量值预测，识别出一种几何稳定性，使聚合可将弱保证强化为强保证。通过几何中位数定义$(α,β)$-稳定性，并建立基于指数重加权与几何中位数聚合的提升框架。针对无约束向量预测，分析了ℓ₁和ℓ₂距离下的稳定性；针对有限概率向量上的密度估计，分析了总变差（TV）、Hellinger和KL散度。在此基础上，提出通用提升框架geomedboost。在弱学习器条件与$(α,β)$-稳定性下，获得经验分歧误差的指数衰减，进而通过泛化界推导出总体保证。
原文摘要 · Abstract (English)
Despite the widespread use of boosting in structured prediction, a general theoretical understanding of aggregation beyond scalar prediction remains incomplete. We study vector-valued prediction under a target divergence and identify a geometric stability property under which aggregation amplifies weak guarantees into strong ones. We formalize this property as $(α,β)$-stability by geometric median and show how it supports a boosting framework based on exponential reweighting and geometric-median aggregation. For vector-valued prediction, we characterize this stability property under several natural divergences: $\ell_1$ and $\ell_2$ distances for unconstrained vector-valued prediction, and TV, Hellinger, and KL for density estimation over finite probability vectors. Building on these results, we propose a generic boosting framework \geomedboost. Under a weak learner condition and $(α,β)$-stability, we obtain exponential decay of the empirical divergence error, which then yields population guarantees through a generalization bound.
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