论文标题
在高维度的有限字母上的离散熵的最佳均匀浓度不平等
An Optimal Uniform Concentration Inequality for Discrete Entropies on Finite Alphabets in the High-dimensional Setting
论文作者
论文摘要
我们证明了指数衰减的浓度不等式，以绑定有限字母内离散随机变量的对数可能性之间差的尾巴概率。我们得出的浓度结合在所有参数值上均匀保存。新结果提高了Zhao（2020）结果的收敛速率，从$（k^2 \ log k）/n = o（1）$（\ log k）^2/n = o（1）$，其中$ n $是样本大小，$ k $是Alphabet的大小。我们进一步证明了费率$ $（\ log k）^2/n = o（1）$是最佳的。结果将结果扩展到分组随机变量的误指定的对数可能性。我们在信息理论中提供了新结果的应用。
We prove an exponential decay concentration inequality to bound the tail probability of the difference between the log-likelihood of discrete random variables on a finite alphabet and the negative entropy. The concentration bound we derive holds uniformly over all parameter values. The new result improves the convergence rate in an earlier result of Zhao (2020), from $(K^2\log K)/n=o(1)$ to $ (\log K)^2/n=o(1)$, where $n$ is the sample size and $K$ is the size of the alphabet. We further prove that the rate $(\log K)^2/n=o(1)$ is optimal. The results are extended to misspecified log-likelihoods for grouped random variables. We give applications of the new result in information theory.
