SVM - what is a functional margin?

Question

A geometric margin is simply the euclidean distance between a certain x (data point) to the hyperlane.

What is the intuitive explanation to what a function

Answer 1

Check Andrew Ng's Lecture Notes from Lecture 3 on SVMs (notation changed to make it easier to type without mathjax/TeX on this site):

"Let’s formalize the notions of the functional and geometric margins . Given a training example (x_i, y_i) we define the functional margin of (w, b) with respect to the training example

gamma_i = y_i( (w^T)x_i + b )

Note that if y_i > 0 then for the functional margin to be large (i.e., for our prediction to be confident and correct), we need (w^T)x + b to be a large positive number. Conversely, if y_i < 0, then for the functional margin to be large, we need (w^T)x + b to be a large negative number. Moreover, if

y_i( (w^T)x_i + b) > 0

then our prediction on this example is correct. (Check this yourself.) Hence, a large functional margin represents a confident and a correct prediction."

Page 3 from the Lecture 3 PDF linked at the materials page linked above.