C2Margin & kernel methods

Margin and kernel methods ask a sharper question than ordinary classifiers: not just where to draw the boundary, but how to draw it with the widest possible buffer between classes. Support vector machines formalize that intuition, and the kernel trick then lets you carry it into rich, even infinite-dimensional feature spaces without ever computing the features explicitly. The payoff is strong, principled models that work well even when data is scarce.

You will start with the perceptron and its convergence proof, build the hard-margin SVM as a quadratic program, relax it with slack variables and the hinge loss, and pass to the dual to see support vectors emerge from the KKT conditions. Then you will replace inner products with Mercer kernels, construct the RBF and polynomial workhorses, and finish with Gaussian process regression, which treats the unknown function itself as a distribution and returns calibrated uncertainty for free.

The margin idea reaches far beyond SVMs (it underlies one-class novelty detection and much of statistical learning theory), and Gaussian processes are the backbone of Bayesian optimization and principled uncertainty estimates throughout ML.