  We study the collaborative PAC learning problem recently proposed in Blum et
al.~\cite{BHPQ17}, in which we have $k$ players and they want to learn a target
function collaboratively, such that the learned function approximates the
target function well on all players' distributions simultaneously. The quality
of the collaborative learning algorithm is measured by the ratio between the
sample complexity of the algorithm and that of the learning algorithm for a
single distribution (called the overhead). We obtain a collaborative learning
algorithm with overhead $O(\ln k)$, improving the one with overhead $O(\ln^2
k)$ in \cite{BHPQ17}. We also show that an $\Omega(\ln k)$ overhead is
inevitable when $k$ is polynomial bounded by the VC dimension of the hypothesis
class. Finally, our experimental study has demonstrated the superiority of our
algorithm compared with the one in Blum et al. on real-world datasets.
