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：我们研究了KL散度下离散分布估计问题，并给出了拉普拉斯估计量的集中界。我们证明，当$n \ge k$时，偏离均值的尺度为$\sqrt{k}/n$，优于先前最优结果$k/n$。我们还建立了匹配的下界，表明我们的界在多对数因子内是紧的。