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We study the fundamental problem of estimating the mean of a $d$-dimensionaldistribution with covariance $\Sigma \preccurlyeq \sigma^2 I_d$ given $n$samples. When $d = 1$, Catoni \cite{catoni} showed an estimator with error$(1+o(1)) \cdot \sigma \sqrt{\frac{2 \log \frac{1}{\delta}}{n}}$, withprobability $1 - \delta$, matching the Gaussian error rate. For $d>1$, anatural estimator outputs the center of the minimum enclosing ball ofone-dimensional confidence intervals to achieve a $1-\delta$ confidence radiusof $\sqrt{\frac{2 d}{d+1}} \cdot \sigma \left(\sqrt{\frac{d}{n}} +\sqrt{\frac{2 \log \frac{1}{\delta}}{n}}\right)$, incurring a$\sqrt{\frac{2d}{d+1}}$-factor loss over the Gaussian rate. When the$\sqrt{\frac{d}{n}}$ term dominates by a $\sqrt{\log \frac{1}{\delta}}$ factor,\cite{lee2022optimal-highdim} showed an improved estimator matching theGaussian rate. This raises a natural question: is the Gaussian rate achievablein general? Or is the $\sqrt{\frac{2 d}{d+1}}$ loss \emph{necessary} when the$\sqrt{\frac{2 \log \frac{1}{\delta}}{n}}$ term dominates?  We show that the answer to both these questions is \emph{no} -- we show that\emph{some} constant-factor loss over the Gaussian rate is necessary, butconstruct an estimator that improves over the above naive estimator by aconstant factor. We also consider robust estimation, where an adversary isallowed to corrupt an $\epsilon$-fraction of samples arbitrarily: in this case,we show that the above strategy of combining one-dimensional estimates andincurring the $\sqrt{\frac{2d}{d+1}}$-factor \emph{is} optimal in theinfinite-sample limit.

Beyond Catoni: Sharper Rates for Heavy-Tailed and Robust Mean Estimation