  Multi-distribution learning is a natural generalization of PAC learning to
settings with multiple data distributions. There remains a significant gap
between the known upper and lower bounds for PAC-learnable classes. In
particular, though we understand the sample complexity of learning a VC
dimension d class on $k$ distributions to be $O(\epsilon^{-2} \ln(k)(d + k) +
\min\{\epsilon^{-1} dk, \epsilon^{-4} \ln(k) d\})$, the best lower bound is
$\Omega(\epsilon^{-2}(d + k \ln(k)))$. We discuss recent progress on this
problem and some hurdles that are fundamental to the use of game dynamics in
statistical learning.
