Dimension-free empirical entropy estimationDownload PDF

21 May 2021, 20:43 (modified: 16 Oct 2021, 06:43)NeurIPS 2021 PosterReaders: Everyone
Keywords: entropy, estimation, dimension-free
TL;DR: We give finite-sample risk bounds for estimating entropy over infinite alphabets.
Abstract: We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} --- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest.
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