The complexity of approximating entropy

We consider the problem of approximating the entropy of a discrete distribution under several models. If the distribution is given explicitly as an array where the i-th location is the probability of the i-th element, then linear time is both necessary and sufficient for approximating the entropy. We consider a model in which the algorithm is given access only to independent samples from the distribution. Here, we show that a γ-multiplicative approximation to the entropy can be obtained in O (n^(1+η)/γ(2) poly(log n)) time for distributions with entropy Ω(γ/η), where n is the size of the domain of the distribution and η is an arbitrarily small positive constant. We show that one cannot get a multiplicative approximation to the entropy in general in this model. Even for the class of distributions to which our upper bound applies, we obtain a lower bound of Ω (n^{max(1/(2γ2),2/(5γ2-2))}). We next consider a hybrid model in which both the explicit distribution as well as independent samples are available. Here, significantly more efficient algorithms can be achieved: a γ-multiplicative approximation to the entropy can be obtained in O (γ² log² n/h²(γ-1)²) time for distributions with entropy Ω(h); we show a lower bound of Ω (log n/h(γ²-1)). Finally, we consider two special families of distributions: those for which the probability of an element decreases monotonically in the label of the element, and those that are uniform over a subset of the domain. In each case, we give more efficient algorithms for approximating the entropy.