The F-measure is a widely used performance measure for multi-label
classification, where multiple labels can be active in an instance
simultaneously (e.g. in image tagging, multiple tags can be active in any
image). In particular, the F-measure explicitly balances recall (fraction of
active labels predicted to be active) and precision (fraction of labels
predicted to be active that are actually so), both of which are important in
evaluating the overall performance of a multi-label classifier. As with most
discrete prediction problems, however, directly optimizing the F-measure is
computationally hard. In this paper, we explore the question of designing
convex surrogate losses that are calibrated for the F-measure -- specifically,
that have the property that minimizing the surrogate loss yields (in the limit
of sufficient data) a Bayes optimal multi-label classifier for the F-measure.
We show that the F-measure for an $s$-label problem, when viewed as a $2^s
\times 2^s$ loss matrix, has rank at most $s^2+1$, and apply a result of
Ramaswamy et al. (2014) to design a family of convex calibrated surrogates for
the F-measure. The resulting surrogate risk minimization algorithms can be
viewed as decomposing the multi-label F-measure learning problem into $s^2+1$
binary class probability estimation problems. We also provide a quantitative
regret transfer bound for our surrogates, which allows any regret guarantees
for the binary problems to be transferred to regret guarantees for the overall
F-measure problem, and discuss a connection with the algorithm of Dembczynski
et al. (2013). Our experiments confirm our theoretical findings.

本文探讨设计针对 F-Measure 的凸罚函数问题，提出了 F-Measure 的凸罚函数方法，将多标签 F-Measure 学习问题分解为 $s^2+1$ 个二进制分类概率估计问题，并提供了差错传递边界的定量证明，与 Dembczynski 等人的算法有关，并进行了实验证实了论文理论结果。