  Complex classification performance metrics such as the F${}_\beta$-measure
and Jaccard index are often used, in order to handle class-imbalanced cases
such as information retrieval and image segmentation. These performance metrics
are not decomposable, that is, they cannot be expressed in a per-example
manner, which hinders a straightforward application of M-estimation widely used
in supervised learning. In this paper, we consider linear-fractional metrics,
which are a family of classification performance metrics that encompasses many
standard ones such as the F${}_\beta$-measure and Jaccard index, and propose
methods to directly maximize performances under those metrics. A clue to tackle
their direct optimization is a calibrated surrogate utility, which is a
tractable lower bound of the true utility function representing a given metric.
We characterize sufficient conditions which make the surrogate maximization
coincide with the maximization of the true utility. Simulation results on
benchmark datasets validate the effectiveness of our calibrated surrogate
maximization especially if the sample sizes are extremely small.
