Sharp Matrix Empirical Bernstein Inequalities

We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by the first-order $1/\sqrt{n}$ term asymptotically matches the matrix Bernstein inequality exactly, including constants, the latter requiring knowledge of the variance. Our first inequality holds for the sample mean of independent matrices, and our second inequality holds for a mean estimator under martingale dependence at stopping times.