  For many learning problems one may not have access to fine grained label
information; e.g., an image can be labeled as husky, dog, or even animal
depending on the expertise of the annotator. In this work, we formalize these
settings and study the problem of learning from such coarse data. Instead of
observing the actual labels from a set $\mathcal{Z}$, we observe coarse labels
corresponding to a partition of $\mathcal{Z}$ (or a mixture of partitions).
  Our main algorithmic result is that essentially any problem learnable from
fine grained labels can also be learned efficiently when the coarse data are
sufficiently informative. We obtain our result through a generic reduction for
answering Statistical Queries (SQ) over fine grained labels given only coarse
labels. The number of coarse labels required depends polynomially on the
information distortion due to coarsening and the number of fine labels
$|\mathcal{Z}|$.
  We also investigate the case of (infinitely many) real valued labels focusing
on a central problem in censored and truncated statistics: Gaussian mean
estimation from coarse data. We provide an efficient algorithm when the sets in
the partition are convex and establish that the problem is NP-hard even for
very simple non-convex sets.
