Consistent algorithms for multi-label classification with macro-at-$k$ metrics
This addresses a challenging optimization problem in multi-label classification for applications with long-tail labels, though it is incremental as it builds on existing frameworks.
The paper tackles the optimization of macro-at-k metrics in multi-label classification, where predictions are constrained to exactly k labels per instance, by proving the existence of an optimal classifier and proposing a statistically consistent algorithm based on the Frank-Wolfe method, with empirical results showing competitive performance.
We consider the optimization of complex performance metrics in multi-label classification under the population utility framework. We mainly focus on metrics linearly decomposable into a sum of binary classification utilities applied separately to each label with an additional requirement of exactly $k$ labels predicted for each instance. These "macro-at-$k$" metrics possess desired properties for extreme classification problems with long tail labels. Unfortunately, the at-$k$ constraint couples the otherwise independent binary classification tasks, leading to a much more challenging optimization problem than standard macro-averages. We provide a statistical framework to study this problem, prove the existence and the form of the optimal classifier, and propose a statistically consistent and practical learning algorithm based on the Frank-Wolfe method. Interestingly, our main results concern even more general metrics being non-linear functions of label-wise confusion matrices. Empirical results provide evidence for the competitive performance of the proposed approach.