Learning discrete distributions with infinite support

Doron Cohen; Aryeh Kontorovich; Geoffrey Wolfer

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arXiv · 2004.12680

Learning discrete distributions with infinite support

Doron Cohen·Aryeh Kontorovich·Geoffrey Wolfer

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Abstract

We present a novel approach to estimating discrete distributions with (potentially) infinite support in the total variation metric. In a departure from the established paradigm, we make no structural assumptions whatsoever on the sampling distribution. In such a setting, distribution-free risk bounds are impossible, and the best one could hope for is a fully empirical data-dependent bound. We derive precisely such bounds, and demonstrate that these are, in a well-defined sense, the best possible. Our main discovery is that the half-norm of the empirical distribution provides tight upper and lower estimates on the empirical risk. Furthermore, this quantity decays at a nearly optimal rate as a function of the true distribution. The optimality follows from a minimax result, of possible independent interest. Additional structural results are provided, including an exact Rademacher complexity calculation and apparently a first connection between the total variation risk and the missing mass.

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Doron Cohen, Aryeh Kontorovich, Geoffrey Wolfer. 2020-04-27. Learning discrete distributions with infinite support. https://arxiv.org/abs/2004.12680

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