# Why does the formulation of the SVM problem has the bias (something we try to optimize) as a part of the constraint?

The common formulation of the SVM problem is

$$\min_{\theta, \theta_0}\frac{1}{2}||\theta||^2$$ $$\text{ subject to: } y^{(t)}(\theta \cdot x^{(t)} + \theta_0) \geq 1, \ t=1,...,n,$$

However, it seems to be unnatural that $$\theta_0$$ (something we try to optimize) is a part of the constraint, but it doesn't appear in the objective.

Is this natural?

Yes, because SVM is trying to maximize the margin between supporting planes. The margin is $$2\over||\theta||$$, which is the distance between the supporting hyperplanes. Distance between parallel hyperplanes is $$\frac{\text{bias difference}}{||\text{normal vector}||}$$, where normal vector is $$\theta$$ here. For classes $$1,-1$$ the bias terms are $$1-\theta_0$$ and $$-1-\theta_0$$, so their difference is $$2$$. Maximizing this term is equivalent to minimizing $$\frac{||\theta||^2}{2}$$.