Strange definition of RSS for elastic net in scikit-learn (Python) I've been reading the manual about GLM, specifically the one on the elastic net in this link, and they come up with the following function to minimize:
$$\frac{1}{2n}||Xw-y||^2_{2}+\alpha \rho ||w||_1+\frac{\alpha(1-\rho)}{2}||w||_2^2,\ \ \ \ \ \ \ \ \ \ \ (1)$$ 
where $n$ is the number of samples and $w$ are the coefficients of the regression. However, obviating the division by $2n$ (which I find unnecessary), I'm confused by the second part. In Zou & Hastie (2005) the RSS (Residual Sum of Squares) of the elastic net is defined as:
$$RSS=||Xw-y||^2_{2}+\lambda \left(\alpha ||w||_1+(1-\alpha)||w||_2^2\right),\ \ \ (2)$$ 
and I really don't see the similarity between criterions (1) and (2) (like, the relations between the $\rho$ in eq. (1) and $\alpha$ of eq. (2), and the relation between $\alpha$ in eq.(1) and $\lambda$ in eq. (2)): am I missing something here?
 A: The term $\frac{1}{2n}$ is there for convenience to obtain a first and second derivative of $RSS$ that looks nicer.
Regarding the different formulations, I don't think it makes a big difference how you define the weights of the L1 and L2 penalty, and indeed different formulations exist. You can see that both are equal. Let's start with equation (1):
$$
\alpha \rho ||w||_1 + \frac{\alpha(1 - \rho)}{2} ||w||_2^2
= \alpha \rho ||w||_1 + \frac{\alpha - \alpha \rho}{2} ||w||_2^2
$$
And equation (2) becomes:
$$
\lambda ( \alpha ||w||_1 + (1 - \alpha)||w||_2^2)
= \lambda \alpha ||w||_1 + \lambda (1 - \alpha)||w||_2^2
= \lambda \alpha ||w||_1 + (\lambda - \lambda \alpha)||w||_2^2
$$
In essence, $\alpha$ of (1) is the same as $\lambda$ in (2) and $\rho$ of (1) is the same as $\alpha$ in (2). The $\frac{1}{2}$ part of (1) was added to obtain a nicer looking derivative, because
$$
\frac{d}{dw_j} \left( \frac{\alpha(1 - \rho)}{2} ||w||_2^2 \right)
= \alpha(1 - \rho) w_j .
$$
In fact they also use it in another paper by Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1-22.
