When finding the maximum margin separator in the primal form we have the quadratic program:
$$min\frac{1}{2}||\theta||^2$$ $$\text{ subject to: } y^{(t)}(\theta \cdot x^{(t)} + \theta_0) \geq 1, \ t=1,...,n$$
Saying basically to find the maximum margin separator. The margin size will be:
$$\frac{1}{||\theta||}$$
does the size of the margin change whether we change the constant of the constraints?
i.e. if we have:
$$\text{ subject to: } y^{(t)}(\theta \cdot x^{(t)} + \theta_0) \geq k, \ t=1,...,n$$
instead of 1?
If it does not matter, why doesn't this matter? How is it an equivalent formulation regardless of the exact constant for the constraint?
The constraints are homogeneous in the triple $(θ,θ_0,1)$ resp. $(θ,θ_0,k)$, so yes, multiplying this triple with a constant will still have the same separating plane as the solution, however, the margin size will have to be computed as $\frac{k}{\|θ\|}$.
That is why there exists a second formulation where the separation number $k$ is maximized while the hyperplane normal $θ$ stays normed, $\|θ\|=1$.