  In statistical learning and analysis from shared data, which is increasingly
widely adopted in platforms such as federated learning and meta-learning, there
are two major concerns: privacy and robustness. Each participating individual
should be able to contribute without the fear of leaking one's sensitive
information. At the same time, the system should be robust in the presence of
malicious participants inserting corrupted data. Recent algorithmic advances in
learning from shared data focus on either one of these threats, leaving the
system vulnerable to the other. We bridge this gap for the canonical problem of
estimating the mean from i.i.d. samples. We introduce PRIME, which is the first
efficient algorithm that achieves both privacy and robustness for a wide range
of distributions. We further complement this result with a novel exponential
time algorithm that improves the sample complexity of PRIME, achieving a
near-optimal guarantee and matching a known lower bound for (non-robust)
private mean estimation. This proves that there is no extra statistical cost to
simultaneously guaranteeing privacy and robustness.
