多分布学习在标签噪声下无法实现快速收敛，样本复杂度随分布数增长。
Is Multi-Distribution Learning as Easy as PAC Learning: Sharp Rates with Bounded Label Noise
- 构建结构化假设检验框架，量化多源学习中近优性验证的统计代价。
- 在有界噪声下，收敛速率退化为 k/ε²，远慢于单任务学习的 1/ε。
- 首次揭示多源学习中随机噪声与Massart噪声的统计本质差异。
为理解异构数据源学习的统计复杂性，本文研究多分布学习问题：给定 k 个数据源，目标是利用共享结构为每个源输出分类器以降低样本复杂度。在有界标签噪声设定下，我们探究单任务学习中可实现的快速 1/ε 收敛率是否能在多源场景中保持且对 k 的依赖最小。令人惊讶的是，答案是否定的：即使在恒定噪声水平下，跨 k 个分布学习仍会引发与 k/ε² 相关的慢速收敛，除非对每个分布单独学习。关键技术贡献是一个结构化假设检验框架，用于刻画在有界噪声下验证近最优性的统计成本——该成本在多分布设置中不可避免。最后，我们证明当以各分布的最优贝叶斯误差为基准时，样本复杂度产生关于 k 的乘法惩罚。这一结果建立了随机分类噪声与 Massart 噪声之间的统计分离，凸显了多源学习独有的根本性障碍。
原文摘要 · Abstract (English)
Towards understanding the statistical complexity of learning from heterogeneous sources, we study the problem of multi-distribution learning. Given $k$ data sources, the goal is to output a classifier for each source by exploiting shared structure to reduce sample complexity. We focus on the bounded label noise setting to determine whether the fast $1/ε$ rates achievable in single-task learning extend to this regime with minimal dependence on $k$. Surprisingly, we show that this is not the case. We demonstrate that learning across $k$ distributions inherently incurs slow rates scaling with $k/ε^2$, even under constant noise levels, unless each distribution is learned separately. A key technical contribution is a structured hypothesis-testing framework that captures the statistical cost of certifying near-optimality under bounded noise-a cost we show is unavoidable in the multi-distribution setting. Finally, we prove that when competing with the stronger benchmark of each distribution's optimal Bayes error, the sample complexity incurs a \textit{multiplicative} penalty in $k$. This establishes a \textit{statistical} separation between random classification noise and Massart noise, highlighting a fundamental barrier unique to learning from multiple sources.
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