On the Impossibility of Learning the Missing Mass
This formalizes a known limitation in statistics and machine learning, showing incremental theoretical insight into why predicting rare events requires heavy-tailed distributions.
The paper tackles the problem of learning the probability of rare events, known as the 'missing mass', from i.i.d. samples of a discrete distribution, and proves that it is impossible to learn this without additional structural assumptions, specifically showing it is not distribution-free PAC-learnable in relative error.
This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impossibility result can then be stated as: the missing mass is not distribution-free PAC-learnable in relative error. The proof is semi-constructive and relies on a coupling argument using a dithered geometric distribution. This result formalizes the folklore that in order to predict rare events, one necessarily needs distributions with "heavy tails".