  Multitask learning and related areas such as multi-source domain adaptation
address modern settings where datasets from $N$ related distributions $\{P_t\}$
are to be combined towards improving performance on any single such
distribution ${\cal D}$. A perplexing fact remains in the evolving theory on
the subject: while we would hope for performance bounds that account for the
contribution from multiple tasks, the vast majority of analyses result in
bounds that improve at best in the number $n$ of samples per task, but most
often do not improve in $N$. As such, it might seem at first that the
distributional settings or aggregation procedures considered in such analyses
might be somehow unfavorable; however, as we show, the picture happens to be
more nuanced, with interestingly hard regimes that might appear otherwise
favorable.
  In particular, we consider a seemingly favorable classification scenario
where all tasks $P_t$ share a common optimal classifier $h^*,$ and which can be
shown to admit a broad range of regimes with improved oracle rates in terms of
$N$ and $n$. Some of our main results are as follows:
  $\bullet$ We show that, even though such regimes admit minimax rates
accounting for both $n$ and $N$, no adaptive algorithm exists; that is, without
access to distributional information, no algorithm can guarantee rates that
improve with large $N$ for $n$ fixed.
  $\bullet$ With a bit of additional information, namely, a ranking of tasks
$\{P_t\}$ according to their distance to a target ${\cal D}$, a simple
rank-based procedure can achieve near optimal aggregations of tasks' datasets,
despite a search space exponential in $N$. Interestingly, the optimal
aggregation might exclude certain tasks, even though they all share the same
$h^*$.
