Coresets are one of the central methods to facilitate the analysis of large data.We continue a recent line of research applying the theory of coresets to logistic regression. First, we show the negative result that no strongly sublinear sized coresets exist for logistic regression. To deal with intractable worst-case instances we introduce a complexity measure μ(X), which quantiAes the hardness of compressing a data set for logistic regression. μ(X) has an intuitive statistical interpretation that may be of independent interest. For data sets with bounded μ(X)-complexity, we show that a novel sensitivity sampling scheme produces the Arst provably sublinear (1 ± ϵ)-coreset.