Concentration Bounds for Discrete Distribution Estimation in KL Divergence

We study the problem of discrete distribution estimation in KL divergence and provide concentration bounds for the Laplace estimator. We show that the deviation from mean scales as $\sqrt{k}\mathrm{/}n$ when $n\ge k$, improving upon the best prior result of $k\mathrm{/}n$. We also establish a matching lower bound that shows that our bounds are tight up to polylogarithmic factors.