  There is growing interest in improving our algorithmic understanding of
fundamental statistical problems such as mean estimation, driven by the goal of
understanding the limits of what we can extract from valuable data. The state
of the art results for mean estimation in $\mathbb{R}$ are 1) the optimal
sub-Gaussian mean estimator by [LV22], with the tight sub-Gaussian constant for
all distributions with finite but unknown variance, and 2) the analysis of the
median-of-means algorithm by [BCL13] and a lower bound by [DLLO16],
characterizing the big-O optimal errors for distributions for which only a
$1+\alpha$ moment exists for $\alpha \in (0,1)$. Both results, however, are
optimal only in the worst case. We initiate the fine-grained study of the mean
estimation problem: Can algorithms leverage useful features of the input
distribution to beat the sub-Gaussian rate, without explicit knowledge of such
features?
  We resolve this question with an unexpectedly nuanced answer: "Yes in limited
regimes, but in general no". For any distribution $p$ with a finite mean, we
construct a distribution $q$ whose mean is well-separated from $p$'s, yet $p$
and $q$ are not distinguishable with high probability, and $q$ further
preserves $p$'s moments up to constants. The main consequence is that no
reasonable estimator can asymptotically achieve better than the sub-Gaussian
error rate for any distribution, matching the worst-case result of [LV22]. More
generally, we introduce a new definitional framework to analyze the
fine-grained optimality of algorithms, which we call "neighborhood optimality",
interpolating between the unattainably strong "instance optimality" and the
trivially weak "admissibility" definitions. Applying the new framework, we show
that median-of-means is neighborhood optimal, up to constant factors. It is
open to find a neighborhood-optimal estimator without constant factor
slackness.
