# Instance Optimal Learning

We consider the following basic learning task: given independent draws from an unknown distribution over a discrete support, output an approximation of the distribution that is as accurate as possible in $\ell_1$ distance (i.e. total variation or statistical distance). Perhaps surprisingly, it is often possible to "de-noise" the empirical distribution of the samples to return an approximation of the true distribution that is significantly more accurate than the empirical distribution, without relying on any prior assumptions on the distribution. We present an instance optimal learning algorithm which optimally performs this de-noising for every distribution for which such a de-noising is possible. More formally, given $n$ independent draws from a distribution $p$, our algorithm returns a labelled vector whose expected distance from $p$ is equal to the minimum possible expected error that could be obtained by any algorithm that knows the true unlabeled vector of probabilities of distribution $p$ and simply needs to assign labels, up to an additive subconstant term that is independent of $p$ and goes to zero as $n$ gets large. One conceptual implication of this result is that for large samples, Bayesian assumptions on the "shape" or bounds on the tail probabilities of a distribution over discrete support are not helpful for the task of learning the distribution. As a consequence of our techniques, we also show that given a set of $n$ samples from an arbitrary distribution, one can accurately estimate the expected number of distinct elements that will be observed in a sample of any size up to $n \log n$. This sort of extrapolation is practically relevant, particularly to domains such as genomics where it is important to understand how much more might be discovered given larger sample sizes, and we are optimistic that our approach is practically viable.

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