Analysis and Optimization of Loss Functions for Multiclass, Top-k, and Multilabel Classification.

Top-k error is currently a popular performance measure on large scale image classification benchmarks such as ImageNet and Places. Despite its wide acceptance, our understanding of this metric is limited as most of the previous research is focused on its special case, the top-1 error. In this work, we explore two directions that shed more light on the top-k error. First, we provide an in-depth analysis of established and recently proposed single-label multiclass methods along with a detailed account of efficient optimization algorithms for them. Our results indicate that the softmax loss and the smooth multiclass SVM are surprisingly competitive in top-k error uniformly across all k, which can be explained by our analysis of multiclass top-k calibration. Further improvements for a specific k are possible with a number of proposed top-k loss functions. Second, we use the top-k methods to explore the transition from multiclass to multilabel learning. In particular, we find that it is possible to obtain effective multilabel classifiers on Pascal VOC using a single label per image for training, while the gap between multiclass and multilabel methods on MS COCO is more significant. Finally, our contribution of efficient algorithms for training with the considered top-k and multilabel loss functions is of independent interest.

Learning using privileged information: SVM+ and weighted SVM.

Prior knowledge can be used to improve predictive performance of learning algorithms or reduce the amount of data required for training. The same goal is pursued within the learning using privileged information paradigm which was recently introduced by Vapnik et al. and is aimed at utilizing additional information available only at training time-a framework implemented by SVM+. We relate the privileged information to importance weighting and show that the prior knowledge expressible with privileged features can also be encoded by weights associated with every training example. We show that a weighted SVM can always replicate an SVM+ solution, while the converse is not true and we construct a counterexample highlighting the limitations of SVM+. Finally, we touch on the problem of choosing weights for weighted SVMs when privileged features are not available.