Oracle efficient truncated statistics

Abstract

We study the problem of learning from truncated samples: instead of observing samples from some underlying population $p^\ast$, we observe only the examples that fall in some survival set $S \subset \mathbb{R}^d$ whose probability mass (measured with respect to $p^\ast$) is at least $\alpha$. Assuming membership oracle access to the truncation set $S$, prior works obtained algorithms for the case where $p^\ast$ is Gaussian or more generally an exponential family with strongly convex likelihood --- albeit with a super-polynomial dependency on the (inverse) survival mass $1/\alpha$ both in terms of runtime and in number of oracle calls to the set $S$. In this work we design a new learning method with runtime and query complexity polynomial in $1/\alpha$. Our result significantly improves over the prior works by focusing on efficiently solving the underlying optimization problem using a general purpose optimization algorithm with minimal assumptions.