Extreme multi-label classification (XMC) refers to supervised multi-label learning involving hundreds of thousands or even millions of labels. In this paper, we develop a suite of algorithms, called Bonsai, which generalizes the notion of label representation in XMC, and partitions the labels in the representation space to learn shallow trees. We show three concrete realizations of this label representation space including: (i) the input space which is spanned by the input features, (ii) the output space spanned by label vectors based on their co-occurrence with other labels, and (iii) the joint space by combining the input and output representations. Furthermore, the constraint-free multi-way partitions learnt iteratively in these spaces lead to shallow trees. By combining the effect of shallow trees and generalized label representation, Bonsai achieves the best of both worlds-fast training which is comparable to state-of-the-art tree-based methods in XMC, and much better prediction accuracy, particularly on tail-labels. On a benchmark Amazon-3M dataset with 3 million labels, Bonsai outperforms a state-of-the-art one-vs-rest method in terms of prediction accuracy, while being approximately 200 times faster to train. The code for Bonsai is available at https ://githu b.com/xmc-aalto /bonsa i.

The goal in extreme multi-label classification is to learn a classifier which can assign a small subset of relevant labels to an instance from an extremely large set of target labels. Datasets in extreme classification exhibit a long tail of labels which have small number of positive training instances. In this work, we pose the learning task in extreme classification with large number of tail-labels as learning in the presence of adversarial perturbations. This view motivates a robust optimization framework and equivalence to a corresponding regularized objective.Under the proposed robustness framework, we demonstrate efficacy of Hamming loss function for taillabel detection in extreme classification. The equivalent regularized objective, in combination with proximal gradient based optimization, performs better than state-ofthe-art methods on propensity scored versions of preci-sion@k and nDCG@k(upto 20% relative improvement over PFastreXML -a leading tree-based approach and 60% relative improvement over SLEEC -a leading labelembedding approach). Furthermore, we also highlight the sub-optimality of a sparse solver in a widely used package for large-scale linear classification, which is interesting in its own right. We also investigate the spectral properties of label graphs for providing novel insights towards understanding the conditions governing the performance of Hamming loss based one-vs-rest scheme vis-à-vis label embedding methods.

Deep neural networks (DNN) are the de-facto solution behind many intelligent applications of today, ranging from machine translation to autonomous driving. DNNs are accurate but resource-intensive, especially for embedded devices such as mobile phones and smart objects in the Internet of Things. To overcome the related resource constraints, DNN inference is generally offloaded to the edge or to the cloud. This is accomplished by partitioning the DNN and distributing computations at the two different ends. However, most of existing solutions simply split the DNN into two parts, one running locally or at the edge, and the other one in the cloud. In contrast, this article proposes a technique to divide a DNN in multiple partitions that can be processed locally by end devices or offloaded to one or multiple powerful nodes, such as in fog networks. The proposed scheme includes both an adaptive DNN partitioning scheme and a distributed algorithm to offload computations based on a matching game approach. Results obtained by using a selfdriving car dataset and several DNN benchmarks show that the proposed solution significantly reduces the total latency for DNN inference compared to other distributed approaches and is 2.6 to 4.2 times faster than the state of the art. Index Terms-DNN inference, task partitioning, task offloading, distributed algorithm, matching game.

In many of the large-scale physical and social complex systems phenomena fat-tailed distributions occur, for which different generating mechanisms have been proposed. In this paper, we study models of generating power law distributions in the evolution of large-scale taxonomies such as Open Directory Project, which consist of websites assigned to one of tens of thousands of categories. The categories in such taxonomies are arranged in tree or DAG structured configurations having parent-child relations among them. We first quantitatively analyse the formation process of such taxonomies, which leads to power law distribution as the stationary distributions. In the context of designing classifiers for large-scale taxonomies, which automatically assign unseen documents to leaf-level categories, we highlight how the fat-tailed nature of these distributions can be leveraged to analytically study the space complexity of such classifiers. Empirical evaluation of the space complexity on publicly available datasets demonstrates the applicability of our approach.