Complexity of Estimating Rényi Entropy of Markov Chains

Abstract

Estimating entropy of random processes is one of the fundamental problems of machine learning and property testing. It has numerous applications to anything from DNA testing and predictability of human behaviour to modeling neural activity and cryptography. We investigate the problem of Renyi entropy estimation for sources that form Markov chains.Kamath and Verd (ISIT’16) showed that good mixing properties are essential for that task. We prove that even with very good mixing time, estimation of entropy of order α > 1 requires Ω(K2−1/α) samples, where K is the size of the alphabet; particularly min-entropy requires Ω(K2) sample size and collision entropy requires Ω(K3/2) samples. Our results hold both in asymptotic and non-asymptotic regimes (under mild restrictions). The analysis is completed by the upper complexity bound of O(K2) for the standard plug-in estimator. This leads to an interesting open question how to improve upon a plugin estimator, which looks much more challenging than for IID sources (which tensorize nicely).We achieve the results by applying Le Cam’s method to two Markov chains which differ by an appropriately chosen sparse perturbation; the discrepancy between these chains is estimated with help of perturbation theory. Our techniques might be of independent interest.