Recently I have been reading some interesting machine learning research papers about classification.

### Multi-class support vector machines

The support vector machine or SVM is a machine learning algorithm for binary classification. It is based on the idea of learning a decision boundary which maximizes the margin, which is a geometric concept (this is a discriminative rather than probabilistic model / loss function).

During my PhD and postdoc years I studied the SVM and proposed an extension for learning ranking and comparison functions. In this context the data consist of pairs of items such as chess players; each player has a feature vector and each match has a label which can take one of three values (win, lose, draw).

Although naively this can be viewed as a multi-class problem (three possible label values), I showed that it can be converted to a binary problem, with labels that correspond to “somebody wins” and “draw” (aka significant rank difference between items or not). In this paper we showed that this framework allows learning a ranking function (which predicts a real-valued rank for any item/player) and a comparison function (which predicts a win/lose/draw label given a pair of items/players).

Other extensions to the binary SVM include various formulations for supporting multi-class. One of the most well known approaches is the Weston-Watkins SVM. Each component of the loss function is a hinge loss of the difference between predicted values (between the target class for each example, and another class). The loss is then summed over all observations, and all other classes.

The new paper An exact solver for the Weston-Watkins SVM subproblem proposes a new algorithm for solving the dual problem using block coordinate descent. This basically means holding all optimization variables constant, and only optimizing some subset (here the subset is the dual variables involved with a particular labeled example). The subproblem to solve is a quadratic program (quadratic objective function to minimize, subject to linear constraints). There is a very interesting and complicated derivation of the exact solver, and a proof of linear convergence. The experimental section compares with a previous iterative subproblem solver, and shows substantial speedups when there are a large number of classes.

The same authors had a paper at NeurIPS last year, Weston-Watkins Hinge Loss and Ordered Partitions, which explores the same model from a more theoretical angle. Other authors had already showed that the Crammer-Singer SVM is “calibrated” for the discrete abstention loss. In this context calibration refers to the ability of the learning algorithm to find the Bayes optimal decision boundary in the large sample limit. Similarly, the Lee-Lin-Wahba SVM, and other hinge losses mentioned in this paper, were previously shown to be calibrated for the 0-1 loss. This new paper shows that the WW-SVM is calibrated for a new discrete loss, which they call the ordered partition loss. This paper has a great related work section.

### AUC optimization

There are many papers which discuss ways to optimize the Area Under the ROC Curve (AUC), which is often used to evaluate learned binary classification models. Most of the papers involve optimizing a convex surrogate of the Mann-Whitney-Wilcoxon statistic, which involves a double sum, over all negative and positive examples. Computing this loss function and using it for learning is therefore quadratic in the number of labeled examples, which means it is not feasible to scale to large data.

Some possible solutions are discussed in Robust Deep AUC Maximization: A New Surrogate Loss and Empirical Studies on Medical Image Classification. One of the interesting ideas discussed therein (but not original) is taking the dual of the problem, in order to obtain a loss function which is linear rather than quadratic. The main idea they suggest in this paper is to replace the square loss with the squared hinge loss, which makes sense in the context of classification (as explained intuitively in their Figure 1). However the idea of using the squared hinge loss as a surrogate is not new; in fact it dates back (at least) to Yan 2003.

Another paper by the same group involves extending these ideas to the setting of huge data distributed over several servers, Federated Deep AUC Maximization for Heterogeneous Data with a Constant Communication Complexity. The interesting result in this paper is that AUC can be optimized in a federated algorithm in a reasonable amount of time, despite the fact that the AUC is not separable on examples/observations.