10. Support Vector Machines

We define the soft-margin support vector machine (SVM) directly in terms of its objective function (L2-regularized, hinge loss minimization over a linear hypothesis space). Using our knowledge of Lagrangian duality, we find a dual form of the SVM problem, apply the complementary slackness conditions, and derive some interesting insights into the connection between "support vectors" and margin. Read the "SVM Insights from Duality" in the Notes below for a high-level view of this mathematically dense lecture.
More...Notably absent from the lecture is the hard-margin SVM and its standard geometric derivation. Although the derivation is fun, since we start from the simple and visually appealing idea of maximizing the "geometric margin", the hard-margin SVM is rarely useful in practice, as it requires separable data, which precludes any datasets with repeated inputs and label noise. One fixes this by introducing "slack" variables, which leads to a formulation equivalent to the soft-margin SVM we present. Once we introduce slack variables, I've personally found the interpretation in terms of maximizing the margin to be much hazier, and I find understanding the SVM in terms of "just" a particular loss function and a particular regularization to be much more useful for understanding its properties. That said, Brett Bernstein gives a very nice development of the geometric approach to the SVM, which is linked in the References below. At the very least, it's a great exercise in basic linear algebra.
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