Regret Analysis for Performance Metrics in Multi-Label Classification: The Case of Hamming and Subset Zero-One Loss

In multi-label classification (MLC), each instance is associated with a subset of labels instead of a single class, as in conventional classification, and this generalization enables the definition of a multitude of loss functions. Indeed, a large number of losses has already been proposed and is commonly applied as performance metrics in experimental studies. However, even though these loss functions are of a quite different nature, a concrete connection between the type of multi-label classifier used and the loss to be minimized is rarely established, implicitly giving the misleading impression that the same method can be optimal for different loss functions. In this paper, we elaborate on risk minimization and the connection between loss functions in MLC, both theoretically and empirically. In particular, we compare two important loss functions, namely the Hamming loss and the subset 0/1 loss. We perform a regret analysis, showing how poor a classifier intended to minimize the subset 0/1 loss can become in terms of Hamming loss and vice versa. The theoretical results are corroborated by experimental studies, and their implications for MLC methods are discussed in a broader context.