  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].
