Implementation of the proximal operator

[Yu Umegaki]

In the source code, the proximal operator is defined as follows:

    def _prox(self, beta, thresh):

The second argument is the threshold of the proximal operator.

As this site, the threshold should be the product of the learning rate and the L1 regularization parameter.

However, this method is called with feeding the second argument with only the L1 regularization parameter(reg_lambda * alpha) as follows.

            # Apply proximal operator
            if self.fit_intercept:
                beta[1:] = self._prox(beta[1:], reg_lambda * alpha)
            else:
                beta = self._prox(beta, reg_lambda * alpha)

I think reg_lambda * alpha must be replaced with learning_rate * reg_lambda * alpha. Otherwise, the model with L1 regularization will not be correctly trained when learning_rate is not 1.

In fact, the trained result of GLM could be quite different from the one of sklearn when the learning rate of GLM is NOT 1.

import pyglmnet
import sklearn

print(pyglmnet.__version__)  # 1.1
print(sklearn.__version__)   # 0.23.1

from sklearn.datasets import load_boston
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import Lasso

L1_reg = 1e-1

x_train, y_train = load_boston(return_X_y=True)
x_train = StandardScaler().fit_transform(x_train) / 2

# GLM with `learing_rate=0.2`
model_glm_lr02 = pyglmnet.GLM(distr='gaussian', alpha=1.0, reg_lambda=L1_reg, learning_rate=0.2, max_iter=10000)
model_glm_lr02.fit(x_train, y_train)

# GLM with `learing_rate=1`
model_glm_lr1 = pyglmnet.GLM(distr='gaussian', alpha=1.0, reg_lambda=L1_reg, learning_rate=1, max_iter=10000)
model_glm_lr1.fit(x_train, y_train)

# sklearn
model_sklearn = Lasso(alpha=L1_reg)
model_sklearn.fit(x_train, y_train)

print(model_glm_lr02.beta_)
# [ 0.         -0.          0.         -0.          0.          5.42708383
#   0.          0.          0.          0.         -2.68700433  0.36198452
#  -7.0861965 ]

print(model_glm_lr1.beta_)  
# [-0.67815431  0.75967646 -0.04907772  1.2337921  -2.15730743  5.92672067
#   0.         -3.46014723  0.01498839  0.         -3.5466964   1.34164976
#  -7.43306087]

print(model_sklearn.coef_)
# [-0.67808937  0.75891847 -0.048276    1.23386678 -2.15994748  5.92750325
#  -0.         -3.46026391  0.0155437  -0.         -3.54744689  1.341734
#  -7.43185641]

[Mainak Jas]

@AnchorBlues it's been a while that I dug into the math. Could you share a screenshot where it shows that the learning rate matters in the link you shared? Let me also solicit opinions from @pavanramkumar and @titipata

I think what is happening is that scikit-learn uses an automatically determined learning rate for 'gaussian' that comes from the maximum eigen-value of (X^T)X. Can you verify that with learning_rate=0.2, how does our convergence plot look? I suspect the convergence is too slow.

[Yu Umegaki]

@jasmainak

Could you share a screenshot where it shows that the learning rate matters in the link you shared?

The following site may be easier to understand.

http://www.stat.cmu.edu/~ryantibs/convexopt-S15/scribes/08-prox-grad-scribed.pdf

The target section is 8.1.3.( Iterative soft-thresholding algorithm (ISTA)). Here, t is the learning rate and lambda is the L1 regularization parameter.

As the formula (8.11) and (8.12) shows, the optimization of parameters of the Lasso is executed as follows (The same method applies to the Elastic-Net):

1. Update the parameters using the gradient descend method ignoring the L1 regularization term.
2. Feed the soft-thresholding operator with the updated parameters.
   • The threshold of the soft-thresholding operator is the product of the learning rate and the L1 regularization parameter.

Can you verify that with learning_rate=0.2, how does our convergence plot look?

I updated the version of pyglmnet to 1.2.dev0 to use the method plot_convergence (the trained results was not changed) and plotted the convergence plot of the case learning_rate=0.2.

Converged in 443 iterations.

[Mainak Jas]

humm ... indeed you are right. Have you verified that the fix solves the comparison problem with scikit-learn ? Would be great if you can make a pull request! Thank you so much.

[Yu Umegaki]

@jasmainak

Have you verified that the fix solves the comparison problem with scikit-learn ?

Yes, I have verified it.

Would be great if you can make a pull request!

I made a pull request.
Please confirm.

[Pavan Ramkumar]

Thanks @AnchorBlues for the careful verification and fix!

[Mainak Jas]

closed by #384