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

The functional margin represents the correctness and confidence of the prediction if the magnitude of the vector(w^T) orthogonal to the hyperplane has a constant value all the time.

By correctness, the functional margin should always be positive, since if wx + b is negative, then y is -1 and if wx + b is positive, y is 1. If the functional margin is negative then the sample should be divided into the wrong group.

By confidence, the functional margin can change due to two reasons: 1) the sample(y_i and x_i) changes or 2) the vector(w^T) orthogonal to the hyperplane is scaled (by scaling w and b). If the vector(w^T) orthogonal to the hyperplane remains the same all the time, no matter how large its magnitude is, we can determine how confident the point is grouped into the right side. The larger that functional margin, the more confident we can say the point is classified correctly.

But if the functional margin is defined without keeping the magnitude of the vector(w^T) orthogonal to the hyperplane the same, then we define the geometric margin as mentioned above. The functional margin is normalized by the magnitude of w to get the geometric margin of a training example. In this constraint, the value of the geometric margin results only from the samples and not from the scaling of the vector(w^T) orthogonal to the hyperplane.

The geometric margin is invariant to the rescaling of the parameter, which is the only difference between geometric margin and functional margin.

EDIT:

The introduction of functional margin plays two roles: 1) intuit the maximization of geometric margin and 2) transform the geometric margin maximization issue to the minimization of the magnitude of the vector orthogonal to the hyperplane.

Since scaling the parameters w and b can result in nothing meaningful and the parameters are scaled in the same way as the functional margin, then if we can arbitrarily make the ||w|| to be 1(results in maximizing the geometric margin) we can also rescale the parameters to make them subject to the functional margin being 1(then minimize ||w||).