# Learning Discrete Distributions from Untrusted Batches

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.

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