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 $\sqrt{k}/n$ when $n \ge k$, improving upon the best prior result of $k/n$. We also establish a matching lower bound that shows that our bounds are tight up to polylogarithmic factors.

翻译：我们研究离散分布在KL散度下的估计问题，并给出了拉普拉斯估计量的集中界限。我们证明，当$n \ge k$时，均值偏差的量级为$\sqrt{k}/n$，改进了此前最优的$k/n$结果。我们还建立了一个匹配的下界，表明我们的界限在多项式对数因子范围内是紧致的。