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Two-variable Dual Coordinate Descent Methods for Linear SVM with/without the Bias Term

Coordinate descent (CD) methods have been a stateof-the-art technique for training large-scale linear SVM. The most used setting is to solve the dual problem of an SVM formulation without the bias term (or an SVM formulation by embedding the bias term in the weight vector). The reason of omitting the bias term is that dual SVM no longer has a linear constraint and the CD procedure of updating one variable at a time is very simple. However, some have criticized the decision of not considering the bias term. To understand the role of the bias term in the design of CD methods for linear SVM, we give a thorough study on twovariable CD. First, if the bias term is not considered, we develop a two-variable CD that is competitive with the commonly used one-variable CD and is superior for difficult problems. The procedure is simple and has theoretical linear-rate convergence. Second, we investigate two-variable CD for linear SVM with the bias term. Analysis shows that CD is much less efficient for such a setting. Therefore, we conclude that in using CD for linear SVM, the bias term should not be considered.