  We consider the problem of learning a discrete distribution in the presence
of an $\epsilon$ fraction of malicious data sources. Specifically, we consider
the setting where there is some underlying distribution, $p$, and each data
source provides a batch of $\ge k$ samples, with the guarantee that at least a
$(1-\epsilon)$ fraction of the sources draw their samples from a distribution
with total variation distance at most $\eta$ from $p$. We make no assumptions
on the data provided by the remaining $\epsilon$ fraction of sources--this data
can even be chosen as an adversarial function of the $(1-\epsilon)$ fraction of
"good" batches. We provide two algorithms: one with runtime exponential in the
support size, $n$, but polynomial in $k$, $1/\epsilon$ and $1/\eta$ that takes
$O((n+k)/\epsilon^2)$ batches and recovers $p$ to error
$O(\eta+\epsilon/\sqrt{k})$. This recovery accuracy is information
theoretically optimal, to constant factors, even given an infinite number of
data sources. Our second algorithm applies to the $\eta = 0$ setting and also
achieves an $O(\epsilon/\sqrt{k})$ recover guarantee, though it runs in
$\mathrm{poly}((nk)^k)$ time. This second algorithm, which approximates a
certain tensor via a rank-1 tensor minimizing $\ell_1$ distance, is surprising
in light of the hardness of many low-rank tensor approximation problems, and
may be of independent interest.
