WW hinge loss and so on.

Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Doǧan et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is minimally emblematic among discrete losses satisfying this property. Finally, we apply our theory to justify the empirical observation made by Doǧan et al that the WW-SVM can work well even under massive label noise, a challenging setting for multiclass SVMs.

Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Doǧan et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is minimally emblematic among discrete losses satisfying this property. Finally, we apply our theory to justify the empirical observation made by Doǧan et al that the WW-SVM can work well even under massive label noise, a challenging setting for multiclass SVMs.

Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Do\v{g}an et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is maximally informative among discrete losses satisfying this property. Finally, we apply our theory to justify the empirical observation made by Do\v{g}an et al. that the WW-SVM can work well even under massive label noise, a challenging setting for multiclass SVMs.

Assignment methods are at the heart of many algorithms for unsupervised learning and clustering - in particular, the well-known K-means and Expectation-Maximization (EM) algorithms. In this work, we study several different methods of assignment, including the "hard" assignments used by K-means and the ?soft' assignments used by EM. While it is known that K-means minimizes the distortion on the data and EM maximizes the likelihood, little is known about the systematic differences of behavior between the two algorithms. Here we shed light on these differences via an information-theoretic analysis. The cornerstone of our results is a simple decomposition of the expected distortion, showing that K-means (and its extension for inferring general parametric densities from unlabeled sample data) must implicitly manage a trade-off between how similar the data assigned to each cluster are, and how the data are balanced among the clusters. How well the data are balanced is measured by the entropy of the partition defined by the hard assignments. In addition to letting us predict and verify systematic differences between K-means and EM on specific examples, the decomposition allows us to give a rather general argument showing that K ?means will consistently find densities with less "overlap" than EM. We also study a third natural assignment method that we call posterior assignment, that is close in spirit to the soft assignments of EM, but leads to a surprisingly different algorithm.