On the Statistical Complexity of Sample Amplification

Given $n$ i.i.d. samples drawn from an unknown distribution $P$, when is it possible to produce a larger set of $n+m$ samples which cannot be distinguished from $n+m$ i.i.d. samples drawn from $P$? (Axelrod et al. 2019) formalized this question as the sample amplification problem, and gave optimal amplification procedures for discrete distributions and Gaussian location models. However, these procedures and associated lower bounds are tailored to the specific distribution classes, and a general statistical understanding of sample amplification is still largely missing. In this work, we place the sample amplification problem on a firm statistical foundation by deriving generally applicable amplification procedures, lower bound techniques and connections to existing statistical notions. Our techniques apply to a large class of distributions including the exponential family, and establish a rigorous connection between sample amplification and distribution learning.