Multi-distribution learning generalizes the classic PAC learning to handle data coming from multiple distributions. Given a set of $k$ data distributions and a hypothesis class of VC dimension $d$, the goal is to learn a hypothesis that minimizes the maximum population loss over $k$ distributions, up to $\epsilon$ additive error. In this paper, we settle the sample complexity of multi-distribution learning by giving an algorithm of sample complexity $\widetilde{O}((d+k)\epsilon^{-2}) \cdot (k/\epsilon)^{o(1)}$. This matches the lower bound up to sub-polynomial factor and resolves the COLT 2023 open problem of Awasthi, Haghtalab and Zhao [AHZ23].

翻译：多分布学习将经典的PAC学习推广至处理来自多个分布的数据。给定一组$k$个数据分布和一个VC维数为$d$的假设类，目标是在$k$个分布上学习一个使最大总体损失最小化的假设，误差不超过$\epsilon$的加性误差。在本文中，我们通过给出一个样本复杂度为$\widetilde{O}((d+k)\epsilon^{-2}) \cdot (k/\epsilon)^{o(1)}$的算法，解决了多分布学习的样本复杂度问题。该结果与下界匹配仅相差次多项式因子，并解决了Awasthi、Haghtalab和Zhao [AHZ23]在COLT 2023上提出的开放问题。