Different formulations for SVM with slack variables (primal)

Question

I have seen two different ways to formulate the SVM optimization but I was not sure what the difference was between them or if there was any difference.

First formulation:

$$min \frac{1}{2}||\theta||^2 + C \sum^n_{i=1}\xi_i $$ $$\text{s.t. }\ \ y^{(i)}(\theta \cdot x^{(i)} +\theta_0) \geq 1 - \xi_i$$

Second formulation:

$$min \frac{\lambda}{2}||\theta||^2 + \frac{1}{n} \sum^n_{i=1}\xi_i $$ $$\text{s.t. }\ \ y^{(i)}(\theta \cdot x^{(i)} +\theta_0) \geq 1 - \xi_i$$

Are they the same formulation?

The thing that is confusing me is that I think what they mean that:

$$min \frac{1}{2}||\theta||^2 + \frac{1}{\lambda} \sum^n_{i=1}\xi_i $$

where:

$$C = \frac{1}{\lambda}$$

Is the same as:

$$min \frac{\lambda}{2}||\theta||^2 + \sum^n_{i=1}\xi_i $$

Am I right that there should not be and 1/n?

Or does it make a difference?

Answer 1

I doesn't make the difference given $C, \lambda >0$ because in both cases minimum will be achieved at same $\theta, \xi_i$. In original paper on soft-margin SVM there were no $\frac{1}{n}$, although it still won't matter because that coefficient is non-negative.