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：我们研究了以 KL 差差数计算的离散分布估计问题,并为 Laplace 估测器提供了浓度界限。 我们发现,当当 $\ ge k$ / n$ 时, 平均比值偏差为$\ sqrt{k}/ n$, 高于美元/ n$ 的先前最佳结果。 我们还建立了一个匹配的较低比值界限, 显示我们的界限接近于多元系数 。