We study the problem of classifying data in a given taxonomy of labels, where the tax- onomy is specified as a tree forest. We assume that every data instance is labelled with a (possibly empty) set of class labels called multilabel, with the only requirement that mul- tilabels including some node i in the taxonony must also include all ancestors of i. Thus, each multilabel corresponds to the union of one or more paths in the forest, where each path must start from a root but it can terminate on an internal node (rather than a leaf). Learning algorithms for hierarchical classification have been investigated in, e.g., [8, 9, 10, 11, 12, 14, 15, 17, 20]. However, the scenario where labelling includes multiple and partial paths has received very little attention. The analysis in [5], which is mainly theoretical, shows in the multiple and partial path case a 0/1-loss bound for a hierarchical learning algorithm based on regularized least-squares estimates. In this work we extend [5] in several ways. First, we introduce a new hierarchical loss func- tion, the H-loss, which is better suited than the 0/1-loss to analyze hierarchical classification tasks, and we derive the corresponding Bayes-optimal classifier under the parametric data model introduced in [5].

We study the problem of hierarchical classification when labels corresponding topartial and/or multiple paths in the underlying taxonomy are allowed. We introduce a new hierarchical loss function, the H-loss, implementing thesimple intuition that additional mistakes in the subtree of a mistaken class should not be charged for. Based on a probabilistic data model introduced in earlier work, we derive the Bayes-optimal classifier for the H-loss. We then empirically compare two incremental approximations ofthe Bayes-optimal classifier with a flat SVM classifier and with classifiers obtained by using hierarchical versions of the Perceptron and SVM algorithms. The experiments show that our simplest incremental approximationof the Bayes-optimal classifier performs, after just one training epoch, nearly as well as the hierarchical SVM classifier (which performs best). For the same incremental algorithm we also derive an H-loss bound showing, when data are generated by our probabilistic data model, exponentially fast convergence to the H-loss of the hierarchical classifier based on the true model parameters.

We study the problem of hierarchical classification when labels corresponding to partial and/or multiple paths in the underlying taxonomy are allowed. We introduce a new hierarchical loss function, the H-loss, implementing the simple intuition that additional mistakes in the subtree of a mistaken class should not be charged for. Based on a probabilistic data model introduced in earlier work, we derive the Bayes-optimal classifier for the H-loss. We then empirically compare two incremental approximations of the Bayes-optimal classifier with a flat SVM classifier and with classifiers obtained by using hierarchical versions of the Perceptron and SVM algorithms. The experiments show that our simplest incremental approximation of the Bayes-optimal classifier performs, after just one training epoch, nearly as well as the hierarchical SVM classifier (which performs best). For the same incremental algorithm we also derive an H-loss bound showing, when data are generated by our probabilistic data model, exponentially fast convergence to the H-loss of the hierarchical classifier based on the true model parameters.

We study the problem of hierarchical classification when labels corresponding to partial and/or multiple paths in the underlying taxonomy are allowed. We introduce a new hierarchical loss function, the H-loss, implementing the simple intuition that additional mistakes in the subtree of a mistaken class should not be charged for. Based on a probabilistic data model introduced in earlier work, we derive the Bayes-optimal classifier for the H-loss. We then empirically compare two incremental approximations of the Bayes-optimal classifier with a flat SVM classifier and with classifiers obtained by using hierarchical versions of the Perceptron and SVM algorithms. The experiments show that our simplest incremental approximation of the Bayes-optimal classifier performs, after just one training epoch, nearly as well as the hierarchical SVM classifier (which performs best). For the same incremental algorithm we also derive an H-loss bound showing, when data are generated by our probabilistic data model, exponentially fast convergence to the H-loss of the hierarchical classifier based on the true model parameters.