We consider multi-label prediction problems with large output spaces under the assumption of output sparsity – that the target vectors have small support. We develop a general theory for a variant of the popular ECOC (error correcting output code) scheme, based on ideas from compressed sensing for exploiting this sparsity. The method can be regarded as a simple reduction from multilabel regression problems to binary regression problems. It is shown that the number of subproblems need only be logarithmic in the total number of label values, making this approach radically more efficient than others. We also state and prove performance guarantees for this method, and test it empirically. 1.

A large body of research in supervised learning deals with the analysis of singlelabel data, where training examples are associated with a single label λ from a set of disjoint labels L. However, training examples in several application domains are often associated with a set of labels Y ⊆ L. Such data are called multi-label.

We propose a max-margin formulation for the multi-label classification problem where the goal is to tag a data point with a set of pre-specified labels. Given a set of L labels, a data point can be tagged with any of the 2 L possible subsets. The main challenge therefore lies in optimising over this exponentially large label space subject to label correlations. Existing solutions take either of two approaches. The first assumes, a priori, that there are no label correlations and independently trains a classifier for each label (as is done in the 1-vs-All heuristic). This reduces the problem complexity from exponential to linear and such methods can scale to large problems. The second approach explicitly models correlations by pairwise label interactions. However, the complexity remains exponential unless one assumes that label correlations are sparse. Furthermore, the learnt correlations reflect the training set biases. We take a middle approach that assumes labels are correlated but does not incorporate pairwise label terms in the prediction function. We show that the complexity can still be reduced from exponential to linear while modelling dense pairwise label correlations. By incorporating correlation priors we can overcome training set biases and improve prediction accuracy. We provide a principled interpretation of the 1-vs-All method and show

The fields of machine learning and mathematical programming are increasingly intertwined. Optimization problems lie at the heart of most machine learning approaches. The Special Topic on Machine Learning and Large Scale Optimization examines this interplay. Machine learning researchers have embraced the advances in mathematical programming allowing new types of models to be pursued. The special topic includes models using quadratic, linear, second-order cone, semidefinite, and semi-infinite programs. We observe that the qualities of good optimization algorithms from the machine learning and optimization perspectives can be quite different. Mathematical programming puts a premium on accuracy, speed, and robustness. Since generalization is the bottom line in machine learning and training is normally done off-line, accuracy and small speed improvements are of little concern in machine learning. Machine learning prefers simpler algorithms that work in reasonable computational time for specific classes of problems. Reducing machine learning problems to well-explored mathematical programming classes with robust general purpose optimization codes allows machine learning researchers to rapidly develop new techniques.

Abstract. In many applicative contexts in which textual documents are labelled with thematic categories, a distinction is made between the primary categories of a document, which represent the topics that are central to it, and and its secondary categories, which represent topics that the document only touches upon. We contend that this distinction, so far neglected in text categorization research, is important and deserves to be explicitly tackled. The contribution of this paper is three-fold. First, we propose an evaluation measure for this preferential text categorization task, whereby different kinds of misclassifications involving either primary or secondary categories have a different impact on effectiveness. Second, we establish several baseline results for this task on a well-known benchmark for patent classification in which the distinction between primary and secondary categories is present; these results are obtained by reformulating the preferential text categorization task in terms of well established classification problems, such as single and/or multi-label multiclass classification; state-of-the-art learning technology such as SVMs and kernel-based methods are used. Third, we improve on these results by using a recently proposed class of algorithms explicitly devised for learning from training data expressed in

Many real world applications employ multivariate performance measures and each example can belong to multiple classes. The currently most popular approaches train an SVM for each class, followed by ad hoc thresholding. Probabilistic models using Bayesian decision theory are also commonly adopted. In this paper, we propose aBayesian online multi-label classification framework (BOMC) which learns a probabilistic linear classifier. The likelihood is modeled by a graphical model similar to TrueSkillTM, and inference is based on Gaussian density filtering with expectation propagation. Using samples from the posterior, we label the testing data by maximizing the expected F1-score. Our experiments on Reuters1-v2 dataset show BOMC compares favorably to the state-of-the-art online learners in macroaveraged F1-score and training time. 1

Text categorization is a crucial and wellproven method for organizing the collection of large scale documents. In this paper, we propose a hierarchical multi-class text categorization method with global margin maximization. We not only maximize the margins among leaf categories, but also maximize the margins among their ancestors. Experiments show that the performance of our algorithm is competitive with the recently proposed hierarchical multi-class classification algorithms. 1

criterion (CLUHSIC) is a recent clustering algorithm that maximizes the dependence between cluster labels and data observations according to the Hilbert Schmidt independence criterion (HSIC). It is unique in that structure information on the cluster outputs can be easily utilized in the clustering process. However, while the choice of the loss function is known to be very important in supervised learning with structured outputs, we will show in this paper that CLUHSIC is implicitly using the often inappropriate zero-one loss. We propose an extension called CLUHSICAL (which stands for “Clustering using HSIC and loss”) which explicitly considers both the output dependency and loss function. Its optimization problem has the same form as CLUHSIC, except that its partition matrix is constructed in a different manner. Experimental results on a number of datasets with structured outputs show that CLUHSICAL often outperforms CLUHSIC in terms of both structured loss and clustering accuracy.

In this paper we present an algorithm to learn a multi-label classifier which attempts at directly optimising the F-score. The key novelty of our formulation is that we explicitly allow for assortative (submodular) pairwise label interactions, i.e., we can leverage the co-ocurrence of pairs of labels in order to improve the quality of prediction. Prediction in this model consists of minimising a particular submodular set function, what can be accomplished exactly and efficiently via graph-cuts. Learning however is substantially more involved and requires the solution of an intractable combinatorial optimisation problem. We present an approximate algorithm for this problem and prove that it is sound in the sense that it never predicts incorrect labels. We also present a nontrivial test of a sufficient condition for our algorithm to have found an optimal solution. We present experiments on benchmark multi-label datasets, which attest the value of the proposed technique. We also make available source code that enables the reproduction of our experiments. 1

Recently, hierarchical text classification has become an active research topic. The essential idea is that the descendant classes can share the information of the ancestor classes in a predefined taxonomy. In this paper, we claim that each class has several latent concepts and its subclasses share information with these d-ifferent concepts respectively. Then, we propose a variant Passive-Aggressive (PA) algorithm for hierarchical text classification with latent concepts. Experimental results show that the performance of our algorithm is competitive with the recently proposed hierarchical classification algorithms. 1