Steepest Descent¶
Copyright (C) 2020 Andreas Kloeckner
MIT License
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.In [1]:
import numpy as np
import numpy.linalg as la
import scipy.optimize as sopt
import matplotlib.pyplot as pt
from mpl_toolkits.mplot3d import axes3d
Here's a function. It's an oblong bowl made of two quadratic functions.
This is pretty much the easiest 2D optimization job out there.
In [2]:
def f(x):
return 0.5*x[0]**2 + 2.5*x[1]**2
def df(x):
return np.array([x[0], 5*x[1]])
Let's take a look at the function. First in 3D:
In [3]:
fig = pt.figure()
ax = fig.gca(projection="3d")
xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
fmesh = f(np.array([xmesh, ymesh]))
ax.plot_surface(xmesh, ymesh, fmesh)
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And then as a "contour plot":
In [4]:
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh)
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Next, initialize steepest descent with a starting guess:
In [319]:
guesses = [np.array([2, 2./5])]
Next, run Steepest Descent:
In [356]:
x = guesses[-1]
s = -df(x)
def f1d(alpha):
return f(x + alpha*s)
alpha_opt = sopt.golden(f1d)
next_guess = x + alpha_opt * s
guesses.append(next_guess)
print(next_guess)
Here's some plotting code to illustrate what just happened:
In [357]:
pt.figure(figsize=(9, 9))
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh, 50)
it_array = np.array(guesses)
pt.plot(it_array.T[0], it_array.T[1], "x-")
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In [358]:
for i, guess in enumerate(guesses):
print(i, la.norm(guess, 2))
Adding in "momentum" / the "heavy ball" method¶
Steepest descent with added "momentum" term:
$$x_{k+1} = x_k - \alpha \nabla f(x_k) \color{red}{+ \beta (x_{k}-x_{k-1})}$$In [285]:
guesses = [np.array([2, 2./5])]
# beta = 0.01
beta = 0.1
# beta = 0.5
# beta = 1
Explore different choices of the "momentum parameter" $\beta$ above.
In [312]:
x = guesses[-1]
s = -df(x)
def f1d(alpha):
return f(x + alpha*s)
alpha_opt = sopt.golden(f1d)
next_guess = x + alpha_opt * s
if len(guesses) >= 2:
next_guess = next_guess + beta * (guesses[-1] - guesses[-2])
guesses.append(next_guess)
print(next_guess)
In [317]:
pt.figure(figsize=(9, 9))
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh, 50)
it_array = np.array(guesses)
pt.plot(it_array.T[0], it_array.T[1], "x-")
Out[317]:
In [315]:
for i, guess in enumerate(guesses):
print(i, la.norm(guess, 2))
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