  We study consistency properties of machine learning methods based on
minimizing convex surrogates. We extend the recent framework of Osokin et al.
(2017) for the quantitative analysis of consistency properties to the case of
inconsistent surrogates. Our key technical contribution consists in a new lower
bound on the calibration function for the quadratic surrogate, which is
non-trivial (not always zero) for inconsistent cases. The new bound allows to
quantify the level of inconsistency of the setting and shows how learning with
inconsistent surrogates can have guarantees on sample complexity and
optimization difficulty. We apply our theory to two concrete cases: multi-class
classification with the tree-structured loss and ranking with the mean average
precision loss. The results show the approximation-computation trade-offs
caused by inconsistent surrogates and their potential benefits.
