Concentration inequalities for the sample mean, like those due to Bernstein and Hoeffding, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem (CLT) provides asymptotic confidence intervals with optimal width, but these are invalid for all sample sizes. To resolve this tension, we develop new computable concentration inequalities with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. These bounds enable the construction of efficient confidence intervals with correct coverage for any sample size. We derive our inequalities by tightly bounding the Hellinger distance, Stein discrepancy, and Wasserstein distance to a Gaussian, and, as a byproduct, we obtain the first explicit bounds for the Hellinger CLT.
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