Are the two approaches to \ell_2 regularized logistic regression in scikit-learn equivalent?

Are these two approaches to ell_2 regularized logistic regression in scikit-learn equivalent?

From my understanding, scikit-learn has a regularizaed logistic regression module based on LibLinear (called LogisticRegression) and a second one based on stochastic gradient descent (called SGDClassifier).

The LibLinear-based solver solves the following problem:

$$ min_{w} C sum_{i} log{(1+ e^{y^i X_i^T w)} + frac{1}{2} w^Tw$$

while the SGD solver minimizes $$ min_{w} sum_{i} log{(1+ e^{y^i X_i^T w)} + frac{rho}{2} w^Tw$$,

where we are assuming that the labels $y_i in {-1, +1}$.

If the above is correct regarding which problems each technique is solving, then they should be equivalent if $C gets frac{1}{alpha}$.

However, running both of these solvers on even very small, low-dimensional classification problems, I don't get the same weights $hat{w}$. Nor does it appear that the objective functions are minimized.

I wrote a Python link:script that compares the solutions returned by the LibLinear approach, the SGD approach, and my own implementation in Cvxpy.

Here are the results for the three techniques:

1. CVXPY

   • Coefficients=(array([-0.39929211]), array([[ 0.02260313]]))

   • Loss=59.0909466078

2. LogisticRegression

   • Coefficients=(array([-0.03588511]), array([[ 0.03284387]]))

   • Loss=67.3132269213

3. SGDClassifier

   • Coefficients=(array([-0.17192813]), array([[ 0.00436165]]))

   • Loss=62.4807117467

Now, I understand that for gradient methods, they can get within a neighborhood of the global minimum fairly quickly but require many iterations to converge to the global minimum.

I also understand that there may be additional parameter settings for SGD that lead to smaller losses. Still, I am a bit surprised that neither the LibLinear approach nor the SGD approach work well out of the box on what is essentially a tiny problem (there are only 2 parameters to learn and 100 training points!)

So I suspect either a programming error on my part, or something else important that I am missing.