  We study the problem of estimating the mean of a distribution in high
dimensions when either the samples are adversarially corrupted or the
distribution is heavy-tailed. Recent developments in robust statistics have
established efficient and (near) optimal procedures for both settings. However,
the algorithms developed on each side tend to be sophisticated and do not
directly transfer to the other, with many of them having ad-hoc or complicated
analyses.
  In this paper, we provide a meta-problem and a duality theorem that lead to a
new unified view on robust and heavy-tailed mean estimation in high dimensions.
We show that the meta-problem can be solved either by a variant of the Filter
algorithm from the recent literature on robust estimation or by the quantum
entropy scoring scheme (QUE), due to Dong, Hopkins and Li (NeurIPS '19). By
leveraging our duality theorem, these results translate into simple and
efficient algorithms for both robust and heavy-tailed settings. Furthermore,
the QUE-based procedure has run-time that matches the fastest known algorithms
on both fronts.
  Our analysis of Filter is through the classic regret bound of the
multiplicative weights update method. This connection allows us to avoid the
technical complications in previous works and improve upon the run-time
analysis of a gradient-descent-based algorithm for robust mean estimation by
Cheng, Diakonikolas, Ge and Soltanolkotabi (ICML '20).
