  In this paper, we develop connections between two seemingly disparate, but
central, models in robust statistics: Huber's epsilon-contamination model and
the heavy-tailed noise model. We provide conditions under which this connection
provides near-statistically-optimal estimators. Building on this connection, we
provide a simple variant of recent computationally-efficient algorithms for
mean estimation in Huber's model, which given our connection entails that the
same efficient sample-pruning based estimators is simultaneously robust to
heavy-tailed noise and Huber contamination. Furthermore, we complement our
efficient algorithms with statistically-optimal albeit computationally
intractable estimators, which are simultaneously optimally robust in both
models. We study the empirical performance of our proposed estimators on
synthetic datasets, and find that our methods convincingly outperform a variety
of practical baselines.
