# why is SVM cost function the norm of the hyperplane parameters, and not the mean square error?

working through Dr. Ng's coursera course

basically, you want to minimize the difference squared between your guess and the actual value

cost function for SVM: this or this

he says the cost function is now the norm of the vector representing the hyperparameters: sqrt(theta_1^2 + theta_2^2... )

Why is that the case?

Let's imagine we have two parallel seperating hyperplanes defined by the vector $\theta$ and with no data points in between, their distance is then given by: $$m= \frac{2}{\|\theta\|}$$ To maximize this distance, the two hyperplanes must pass through data points of the opposing classes. In this case we can define another hyperplane in the middle (with same distance to both hyperplanes and the same vector $\theta$) that now has the margin $m$.
As you can see from the formula: the smaller $\|\theta\|$ the greater the margin $m$.
So maximizing the margin $m$ is equivalent to minimizing $\|\theta\|$. Thus, a separating hyperplane is optimal if it minimizes $\|\theta\|$.