  Confidence sequences are confidence intervals that can be sequentially
tracked, and are valid at arbitrary data-dependent stopping times. This paper
presents confidence sequences for a univariate mean of an unknown distribution
with a known upper bound on the $p$-th central moment ($p$ > 1), but allowing
for (at most) $\epsilon$ fraction of arbitrary distribution corruption, as in
Huber's contamination model. We do this by designing new robust exponential
supermartingales, and show that the resulting confidence sequences attain the
optimal width achieved in the nonsequential setting. Perhaps surprisingly, the
constant margin between our sequential result and the lower bound is smaller
than even fixed-time robust confidence intervals based on the trimmed mean, for
example. Since confidence sequences are a common tool used within A/B/n testing
and bandits, these results open the door to sequential experimentation that is
robust to outliers and adversarial corruptions.
