SGD: Penalties

Plot the contours of the three penalties.

All of the above are supported by sklearn.linear_model.stochastic_gradient.

Python source code: plot_sgd_penalties.py

from __future__ import division
print(__doc__)

import numpy as np
import pylab as pl


def l1(xs):
    return np.array([np.sqrt((1 - np.sqrt(x ** 2.0)) ** 2.0) for x in xs])


def l2(xs):
    return np.array([np.sqrt(1.0 - x ** 2.0) for x in xs])


def el(xs, z):
    return np.array([(2 - 2 * x - 2 * z + 4 * x * z -
                      (4 * z ** 2
                       - 8 * x * z ** 2
                       + 8 * x ** 2 * z ** 2
                       - 16 * x ** 2 * z ** 3
                       + 8 * x * z ** 3 + 4 * x ** 2 * z ** 4) ** (1. / 2)
                      - 2 * x * z ** 2) / (2 - 4 * z) for x in xs])


def cross(ext):
    pl.plot([-ext, ext], [0, 0], "k-")
    pl.plot([0, 0], [-ext, ext], "k-")

xs = np.linspace(0, 1, 100)

alpha = 0.501  # 0.5 division throuh zero

cross(1.2)

pl.plot(xs, l1(xs), "r-", label="L1")
pl.plot(xs, -1.0 * l1(xs), "r-")
pl.plot(-1 * xs, l1(xs), "r-")
pl.plot(-1 * xs, -1.0 * l1(xs), "r-")

pl.plot(xs, l2(xs), "b-", label="L2")
pl.plot(xs, -1.0 * l2(xs), "b-")
pl.plot(-1 * xs, l2(xs), "b-")
pl.plot(-1 * xs, -1.0 * l2(xs), "b-")

pl.plot(xs, el(xs, alpha), "y-", label="Elastic Net")
pl.plot(xs, -1.0 * el(xs, alpha), "y-")
pl.plot(-1 * xs, el(xs, alpha), "y-")
pl.plot(-1 * xs, -1.0 * el(xs, alpha), "y-")

pl.xlabel(r"$w_0$")
pl.ylabel(r"$w_1$")
pl.legend()

pl.axis("equal")
pl.show()