is logistic regression large margin classifier?

Question

As I understand large margin effect in SVM:

For example let's look at this image:

In SVM optimization objective by regularization term we trying to find a set of parameters, where the norm of (parameters vector) theta is small. So we must find vector theta which is small and projections of positive examples (p) on this vector large (to compensate small Theta vector for inner product). In the same time large p gives us large margin. In this image we find ideal theta, and big p with it (and large margin):

My question:

Why logistic regression is not large margin classifier? In LR we minimize Theta vector in regularization term in the same way. Maybe I did not understand something, if so - correct me.

I've used images and theory from Coursera ml class.

Answer 1

Logistic Regression is a large margin loss. Lecun mentions this in one or more of his papers on energy-based learning.

To see that LR does induce a margin, it is easier to look at the softmax loss (which is equivalent to LR).

There are two terms in the softmax loss: L(z)=z_{true} - log(\sum_i \exp(z_i))

which means that the distance of an example from its true decision boundary needs to beat the log sum of the distances from all of the decision boundaries.

Because the softmax function is a probability distribution, the largest the log softmax can be is 0, so the log softmax returns a negative value (i.e. a penalty) that approaches 0 as the probability of the true class under the softmax function approaches 1.