Solving L1-regularized problems with l-bfgs-b

lbfgs_l1logistic.py

"""l-bfgs-b L1-Logistic Regression solver"""

# Author: Vlad Niculae <[email protected]>
# Suggested by Mathieu Blondel

from __future__ import division, print_function

import numpy as np
from scipy.optimize import fmin_l_bfgs_b

from sklearn.base import BaseEstimator, ClassifierMixin
from sklearn.utils.extmath import safe_sparse_dot, log_logistic
from sklearn.utils.fixes import expit


def _l1_logistic_loss_grad(w_extended, X, y, alpha):
    _, n_features = X.shape
    w = w_extended[:n_features] - w_extended[n_features:]

    yz = y * safe_sparse_dot(X, w)

    # Logistic loss is the negative of the log of the logistic function.
    out = -np.sum(log_logistic(yz))
    # out += .5 * alpha * np.dot(w, w)  # L2
    out += alpha * w_extended.sum()  # L1, w_extended is non-negative

    z = expit(yz)
    z0 = (z - 1) * y

    grad = safe_sparse_dot(X.T, z0)
    grad = np.concatenate([grad, -grad])
    # grad += alpha * w  # L2
    grad += alpha  # L1

    return out, grad

class LbfgsL1Logistic(BaseEstimator, ClassifierMixin):

    def __init__(self, tol=1e-3, alpha=1.0):
        """Logistic Regression Lasso solved by L-BFGS-B

        Solves the same objective as sklearn.linear_model.LogisticRegression

        Parameters
        ----------
        alpha: float, default: 1.0
            The amount of regularization to use.

        tol: float, default: 1e-3
            Convergence tolerance for L-BFGS-B.
        """
        self.tol = tol
        self.alpha = alpha

    def fit(self, X, y):
        n_samples, n_features = X.shape

        coef0 = np.zeros(2 * n_features)
        w, f, d = fmin_l_bfgs_b(_l1_logistic_loss_grad, x0=coef0, fprime=None,
                                pgtol=self.tol,
                                bounds=[(0, None)] * n_features * 2,
                                args=(X, y, self.alpha))
        self.coef_ = w[:n_features] - w[n_features:]

        return self

    def predict(self, X):
        return np.sign(safe_sparse_dot(X, self.coef_))


if __name__ == '__main__':
    from scipy.spatial.distance import jaccard
    from sklearn.linear_model import LogisticRegression
    from time import time

    # Generate data with known sparsity pattern
    n_samples, n_features, n_relevant = 100, 80, 20
    X = np.random.randn(n_samples, n_features)
    X /= np.linalg.norm(X, axis=1)[:, np.newaxis]
    true_coef = np.zeros(n_features)
    nonzero_idx = np.random.randint(n_features, size=n_relevant)
    true_coef[nonzero_idx] = np.random.randn(n_relevant)
    y = np.dot(X, true_coef) + np.random.randn(n_samples) * 0.01

    # classification, note: y must be {-1, +1}
    y = np.sign(y)

    C = 1.0
    # Run this solver
    t0 = time()
    lasso_1 = LbfgsL1Logistic(alpha=1. / C, tol=1e-8).fit(X, y)
    t0 = time() - t0
    print("l-bfgs-b:  time = {:.4f}s acc = {:.8f}  ||w - w_true|| = {:.6f}  "
          "Jacc. sparsity = {:.2f}".format(t0, lasso_1.score(X, y),
            np.linalg.norm(lasso_1.coef_ - true_coef),
            jaccard(true_coef > 0, lasso_1.coef_ > 0)))

    # run liblinear
    t0 = time()
    lasso_2 = LogisticRegression(penalty='l1', C=C, tol=1e-8,
                                 fit_intercept=False).fit( X, y)
    t0 = time() - t0
    print("liblinear: time = {:.4f}s acc = {:.8f}  ||w - w_true|| = {:.6f}  "
          "Jacc. sparsity = {:.2f}".format(t0, lasso_2.score(X, y),
            np.linalg.norm(lasso_2.coef_ - true_coef),
            jaccard(true_coef > 0, lasso_2.coef_ > 0)))

lbfgs_lasso.py

"""l-bfgs-b Lasso solver"""

# Author: Vlad Niculae <[email protected]>
# Suggested by Mathieu Blondel

from __future__ import division, print_function

import numpy as np
from scipy.optimize import fmin_l_bfgs_b

from sklearn.base import BaseEstimator, RegressorMixin
from sklearn.utils.extmath import safe_sparse_dot


class LbfgsLasso(BaseEstimator, RegressorMixin):

    def __init__(self, tol=1e-3, alpha=1.0):
        """Lasso solved by L-BFGS-B

        Solves the same objective as sklearn.linear_model.Lasso

        Parameters
        ----------
        alpha: float, default: 1.0
            The amount of regularization to use.

        tol: float, default: 1e-3
            Convergence tolerance for L-BFGS-B.
        """
        self.tol = tol
        self.alpha = alpha

    def fit(self, X, y):
        n_samples, n_features = X.shape

        def f(w_extended, *args):
            # w_extended is [w_pos -w_neg]
            # assert (w_extended >= 0).all()
            # abs(w).sum() = w_extended.sum()
            w = w_extended[:n_features] - w_extended[n_features:]
            return (np.sum((safe_sparse_dot(X, w) - y) ** 2) +
                    (2 * self.alpha * n_samples) * w_extended.sum())

        def fprime(w_extended, *args):
            # assert (w_extended >= 0).all()
            # dloss_extended = [X' -X']' * ([X -X]' * [w_pos -w_neg] - y)
            #                = [X' -X']' * (X * w - y)
            #                = [dloss -dloss]
            w = w_extended[:n_features] - w_extended[n_features:]
            dloss = 2 * safe_sparse_dot(X.T, safe_sparse_dot(X, w) - y)
            grad_extended = np.concatenate([dloss, -dloss])
            grad_extended += 2 * self.alpha * n_samples # to match skl lasso
            return grad_extended

        coef0 = np.zeros(2 * n_features)
        w, f, d = fmin_l_bfgs_b(f, x0=coef0, fprime=fprime, pgtol=self.tol,
                                bounds=[(0, None)] * n_features * 2)
        self.coef_ = w[:n_features] - w[n_features:]

        return self

    def predict(self, X):
        return safe_sparse_dot(X, self.coef_)


if __name__ == '__main__':
    from scipy.spatial.distance import jaccard
    from sklearn.linear_model import Lasso
    from time import time

    # Generate data with known sparsity pattern
    X = np.random.randn(100, 80)
    X /= np.linalg.norm(X, axis=1)[:, np.newaxis]
    true_coef = np.zeros(80)
    nonzero_idx = np.random.randint(80, size=20)
    true_coef[nonzero_idx] = np.random.randn(20)
    y = np.dot(X, true_coef) + np.random.randn(100) * 0.01

    # Run this solver
    t0 = time()
    lasso_1 = LbfgsLasso(alpha=0.001, tol=1e-8).fit(X, y)
    t0 = time() - t0
    print("LBFGSB: time = {:.4f}s R2 = {:.8f}  ||w - w_true|| = {:.6f}  "
          "Jacc. sparsity = {:.2f}".format(t0, lasso_1.score(X, y),
            np.linalg.norm(lasso_1.coef_ - true_coef),
            jaccard(true_coef > 0, lasso_1.coef_ > 0)))

    # Run coordinate descent Lasso from scikit-learn
    t0 = time()
    lasso_2 = Lasso(alpha=0.001, fit_intercept=False, tol=1e-8).fit(X, y)
    t0 = time() - t0
    print("CD:     time = {:.4f}s R2 = {:.8f}  ||w - w_true|| = {:.6f}  "
          "Jacc. sparsity = {:.2f}".format(t0, lasso_2.score(X, y),
            np.linalg.norm(lasso_2.coef_ - true_coef),
            jaccard(true_coef > 0, lasso_2.coef_ > 0)))

Hey @MLnick I'm also wondering that. Do you know the answer already?

I believe the loss function in LbfgsL1Logistic is incorrect. I didn't look at the other class. I can't speak for the gradient aspect of the calculation.

Loss for logistic regression includes terms for both the positive and negative class:

h = safe_sparse_dot(X,w)
(y * np.log(h + eps)) + ((1-y) * np.log(1-h + eps))
... or something similar.

Also, the user may or may not wish to include a bias term (perhaps this could be achieved by mean-centering y prior to fitting, or including bias in the calculation of h per this comment)

Oops sorry I realize I missed some very old questions on here!

@MLnick @jpvcaetano: I think you're correct and you could include an L2 penalty leading to elastic net, but don't, there are much better implementations, this is a proof of concept.

@kyle-pena-nlp I think you missed that in this code y is in {-1, 1} rather than {0, 1} (see line 86). So we have that:

P(Y=positive class} = sigmoid(h)
P(Y=negative class} = sigmoid(-h)

so log P(Y=y) = log_sigmoid(yh).

I agree that a bias is useful in practice, but also you wouldn't want to use this in practice, right? (if you do -- why?)

vene, you are correct. I missed that detail.
All the best!