Nearly Minimax Discrete Distribution Estimation in Kullback-Leibler Divergence with High Probability

TL;DR

This work analyzes discrete distribution estimation under Kullback–Leibler divergence with high-probability guarantees on the minimax risk. It introduces an online-to-batch based estimator with suffix averaging that achieves near-tight KL bounds, and establishes a lower bound showing the tail of the rate must include a doubly logarithmic factor in the domain size, up to constants. The authors develop a novel weak hypothesis-testing reduction to obtain the ln(K) ln(1/δ)/n lower bound and complement it with a K/n lower bound via a Yang–Barron framework, yielding a tight sandwich up to a ln ln K factor. The results clarify that KL minimax estimation is inherently harder in the tails than squared TV or Hellinger, highlight the role of δ-dependence in smoothing, and connect to concurrent work that corroborates the bounds while outlining potential extensions to more complex unbounded-loss settings.

Abstract

We consider the fundamental problem of estimating a discrete distribution on a domain of size~$K$ with high probability in Kullback-Leibler divergence. We provide upper and lower bounds on the minimax estimation rate, which show that the optimal rate is between $\big(K + \ln(K)\ln(1/δ)\big) /n$ and $\big(K\ln\ln(K) + \ln(K)\ln(1/δ)\big) /n$ at error probability $δ$ and sample size $n$, which pins down the rate up to the doubly logarithmic factor $\ln \ln K$ that multiplies $K$. Our upper bound uses techniques from online learning to construct a novel estimator via online-to-batch conversion. Perhaps surprisingly, the tail behavior of the minimax rate is worse than for the squared total variation and squared Hellinger distance, for which it is $\big(K + \ln(1/δ)\big) /n$, i.e.\ without the $\ln K$ multiplying $\ln (1/δ)$. As a consequence, we cannot obtain a fully tight lower bound from the usual reduction to these smaller distances. Moreover, we show that this lower bound cannot be achieved by the standard lower bound approach based on a reduction to hypothesis testing, and instead we need to introduce a new reduction to what we call weak hypothesis testing. We investigate the source of the gap with other divergences further in refined results, which show that the total variation rate is achievable for Kullback-Leibler divergence after all (in fact by he maximum likelihood estimator) if we rule out outcome probabilities smaller than $O(\ln(K/δ) / n)$, which is a vanishing set as $n$ increases for fixed $K$ and~$δ$. This explains why minimax Kullback-Leibler estimation is more difficult than asymptotic estimation.

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