Recent work of Acharya et al. (NeurIPS 2019) showed how to estimate the entropy of a distribution $\mathcal D$ over an alphabet of size $k$ up to $\pmε$ additive error by streaming over $(k/ε^3) \cdot \text{polylog}(1/ε)$ i.i.d. samples and using only $O(1)$ words of memory. In this work, we give a new constant memory scheme that reduces the sample complexity to $(k/ε^2)\cdot \text{polylog}(1/ε)$. We conjecture that this is optimal up to $\text{polylog}(1/ε)$ factors.
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