# Why is the regularization term multiplied by the error term in the cost function of SVM?

The cost function of the Optimal Margin Classifier(non-kernelized SVM) is given as : $$J(\mathbf{\vec w}, b) = \frac{1}{2}\|\mathbf{\vec w}\|_{2}^{2} + C \sum_{i=1}^{n}\max(0, 1-y ^{(i)}(\mathbf{\vec w}\cdot \mathbf{\vec x}^{(i)} + b))$$

Why do we multiply the regularization parameter $$C$$ with the error term instead of the regularization term $$\frac{1}{2}\|\mathbf{\vec{w}}\|_2^2$$ ?

The general form of a cost function is typically this : $$J_\lambda(\boldsymbol \theta) = J(\boldsymbol \theta) + \lambda R(\boldsymbol \theta)$$ where $$R(\boldsymbol \theta)$$ is the regularization term. As you can see, the regularization parameter $$\lambda$$ is always multiplied by the regularization term. So why is the cost function of SVM so peculiar?

They're equivalent; just re-express $$\lambda = \frac{1}{C}$$ and multiply both sides by $$\lambda$$. Obviously, we assume $$C > 0$$.
$$J(\mathbf{\vec w}, b) = \frac{1}{2}\|\mathbf{\vec w}\|_{2}^{2} + C \sum_{i=1}^{n}\max(0, 1-y ^{(i)}(\mathbf{\vec w}\cdot \mathbf{\vec x}^{(i)} + b)) \\ \frac{1}{C} J(\mathbf{\vec w}, b) = \frac{1}{2C}\|\mathbf{\vec w}\|_{2}^{2} + \sum_{i=1}^{n}\max(0, 1-y ^{(i)}(\mathbf{\vec w}\cdot \mathbf{\vec x}^{(i)} + b))$$