Functional and Geometric Margins, and the Optimal Margin Classifier

The perceptron stops the instant it finds any hyperplane separating two classes, with no notion of quality beyond "zero training error." Intuitively, though, a boundary that barely squeezes between the classes should feel less trustworthy than one with generous breathing room on both sides. This note makes that intuition precise via the margin, and shows that maximising it is a clean, convex optimisation problem — the starting point for everything else in this topic.