## Elastic net in Scikit-Learn vs. Keras

Logistic regression with elastic net regularization is available in sklearn and keras. To compare these two approaches, we must be able to set the same hyperparameters for both learning algorithms. However, how to do is not immediately obvious.

The objective function and variable names differ between the original paper and these libraries. This blog post describes and reconciles these differences. It is assumed the reader understands the purpose of elastic net and the concepts behind regularization.

The elastic net, first proposed by Zou and Hastie (J. R. Statist. Soc. B. 2005), adds L1 and L2 penalties of lasso and ridge regression methods to the objective function $L(\lambda_{1}, \lambda_{2}, \beta)$:

$\vert y - X\beta \vert^{2} + \lambda_{2} \vert \beta \vert^{2} + \lambda_{1} \vert \beta \vert_{1}$

The model coefficients $\hat{\beta}$ minimize this objective function:

$\underset{\beta}{\operatorname{argmin}} \{L(\lambda_{1}, \lambda_{2}, \beta)\}$

Elastic net with $\lambda_{2}=0$ is simply ridge regression. Likewise, elastic net with $\lambda_{1}=0$ is simply lasso.

In sklearn, per the documentation for elastic net, the objective function $L$ to minimize is different:

$\frac{1}{2n} \| y - Xw \| ^{2}_{2} + \alpha \rho \|w\|_1 + \frac{1}{2} \alpha ( 1 - \rho ) \|w\|^{2}_{2} \\$

Note l1_ratio is denoted as $\rho$ here.

There are several differences between (1) and (3):

1. Model coefficients are denoted by $w$, not $\beta$.
2. Norms are denoted with double instead of single lines.
3. Terms with $L2$ norms are multiplied by $\frac{1}{2}$ or $\frac{1}{2n}$. This is a mathematical convenience that cancels with the $2$ that arises from taking the derivative. Adding $k$ to any term being minimized does not change the solution because $\underset{\beta}{\operatorname{argmin}} ( L ) = \underset{\beta}{\operatorname{argmin}} ( kL )$ for $k \in \mathbb{R}$.
4. The lasso term (L1 penalty) comes first, whereas in the paper it comes after the ridge regression term (L2 penalty).
5. $\alpha \rho$ is used instead of $\lambda_{1}$
6. $\alpha (1-\rho)$ is used instead of $\lambda_{2}$

To make variables in the sklearn documentation match those in the original Zhou and Hastie paper, I set $a=\lambda_{1}$ and $b=\lambda_{2}$ and use the latter notation below:

$\alpha = \lambda_{1} + \lambda_{2}$ $\rho = \frac{\lambda_{1}}{\lambda_{2} + \lambda_{2}}$

These equations, written in Python, will set elastic net hyperparameters $\alpha$ and $\rho$ for elastic net in sklearn as functions of $\lambda_{1}$ and $\lambda_{2}$:

alpha = lambda1 + lambda2
l1_ratio = lambda1 / (lambda1 + lambda2)


This enables the use of $\lambda_{1}$ and $\lambda_{2}$ for elastic net in either sklearn or keras:

from sklearn.linear_model import ElasticNet
alpha = args.l1 + args.l2
l1_ratio = args.l1 / (args.l1 + args.l2)
model = ElasticNet(alpha=alpha, l1_ratio=l1_ratio)
model.fit(X, y)


The keras documentation for elastic net is minimal. No equation for the objective function is given. L1 L2 regularization is not even referred to as elastic net. However, the implementation is straightforward; simply use the l1_l2 regularizer function and set the parameters l1 and l2, which are equivalent to $\lambda_{1}$ and $\lambda_{2}$, respectively):

tf.keras.regularizers.l1_l2(l1=args.l1, l2=args.l2)