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Adaptive Primal-Dual#
This tutorial compares the traditional Chambolle-Pock Primal-dual algorithm with the Adaptive Primal-Dual Hybrid Gradient of Goldstein and co-authors.
By adaptively changing the step size in the primal and the dual directions, this algorithm shows faster convergence, which is of great importance for some of the problems that the Primal-Dual algorithm can solve - especially those with an expensive proximal operator.
For this example, we consider a simple denoising problem.
import numpy as np
import matplotlib.pyplot as plt
import pylops
from skimage.data import camera
import pyproximal
plt.close('all')
def callback(x, f, g, K, cost, xtrue, err):
cost.append(f(x) + g(K.matvec(x)))
err.append(np.linalg.norm(x - xtrue))
Let’s start by loading a sample image and adding some noise
We can now define a pylops.Gradient operator as well as the
different proximal operators to be passed to our solvers
# Gradient operator
sampling = 1.
Gop = pylops.Gradient(dims=(ny, nx), sampling=sampling, edge=False,
kind='forward', dtype='float64')
L = 8. / sampling ** 2 # maxeig(Gop^H Gop)
# L2 data term
lamda = .04
l2 = pyproximal.L2(b=noise_img.ravel(), sigma=lamda)
# L1 regularization (isotropic TV)
l1iso = pyproximal.L21(ndim=2)
To start, we solve our denoising problem with the original Primal-Dual algorithm
# Primal-dual
tau = 0.95 / np.sqrt(L)
mu = 0.95 / np.sqrt(L)
cost_fixed = []
err_fixed = []
iml12_fixed = \
pyproximal.optimization.primaldual.PrimalDual(l2, l1iso, Gop,
tau=tau, mu=mu, theta=1.,
x0=np.zeros_like(img.ravel()),
gfirst=False, niter=300, show=True,
callback=lambda x: callback(x, l2, l1iso,
Gop, cost_fixed,
img.ravel(),
err_fixed))
iml12_fixed = iml12_fixed.reshape(img.shape)
Primal-dual: min_x f(Ax) + x^T z + g(x)
---------------------------------------------------------
Proximal operator (f): <class 'pyproximal.proximal.L2.L2'>
Proximal operator (g): <class 'pyproximal.proximal.L21.L21'>
Linear operator (A): <class 'pylops.basicoperators.gradient.Gradient'>
Additional vector (z): None
tau = 0.33587572106361 mu = 0.33587572106361
theta = 1.00 niter = 300
Itn x[0] f g z^x J = f + g + z^x
1 2.55225e+00 1.147e+08 1.331e+05 0.000e+00 1.149e+08
2 4.95447e+00 1.118e+08 1.384e+05 0.000e+00 1.119e+08
3 7.32107e+00 1.089e+08 1.216e+05 0.000e+00 1.090e+08
4 9.70895e+00 1.061e+08 1.114e+05 0.000e+00 1.062e+08
5 1.21188e+01 1.034e+08 1.107e+05 0.000e+00 1.035e+08
6 1.45291e+01 1.008e+08 1.141e+05 0.000e+00 1.009e+08
7 1.69219e+01 9.819e+07 1.186e+05 0.000e+00 9.830e+07
8 1.92914e+01 9.567e+07 1.239e+05 0.000e+00 9.580e+07
9 2.16414e+01 9.322e+07 1.303e+05 0.000e+00 9.335e+07
10 2.39798e+01 9.084e+07 1.373e+05 0.000e+00 9.097e+07
31 6.70599e+01 5.298e+07 2.885e+05 0.000e+00 5.327e+07
61 1.10590e+02 2.519e+07 4.549e+05 0.000e+00 2.564e+07
91 1.39931e+02 1.267e+07 5.685e+05 0.000e+00 1.323e+07
121 1.59453e+02 7.019e+06 6.447e+05 0.000e+00 7.663e+06
151 1.72525e+02 4.466e+06 6.957e+05 0.000e+00 5.162e+06
181 1.81239e+02 3.309e+06 7.300e+05 0.000e+00 4.039e+06
211 1.87092e+02 2.782e+06 7.529e+05 0.000e+00 3.535e+06
241 1.91023e+02 2.540e+06 7.683e+05 0.000e+00 3.308e+06
271 1.93648e+02 2.428e+06 7.786e+05 0.000e+00 3.207e+06
292 1.94953e+02 2.387e+06 7.838e+05 0.000e+00 3.171e+06
293 1.95007e+02 2.386e+06 7.840e+05 0.000e+00 3.170e+06
294 1.95060e+02 2.384e+06 7.842e+05 0.000e+00 3.169e+06
295 1.95113e+02 2.383e+06 7.844e+05 0.000e+00 3.167e+06
296 1.95166e+02 2.382e+06 7.846e+05 0.000e+00 3.166e+06
297 1.95217e+02 2.380e+06 7.848e+05 0.000e+00 3.165e+06
298 1.95268e+02 2.379e+06 7.850e+05 0.000e+00 3.164e+06
299 1.95318e+02 2.378e+06 7.852e+05 0.000e+00 3.163e+06
300 1.95368e+02 2.377e+06 7.854e+05 0.000e+00 3.162e+06
Total time (s) = 6.65
---------------------------------------------------------
We do the same with the adaptive algorithm
cost_ada = []
err_ada = []
iml12_ada, steps = \
pyproximal.optimization.primaldual.AdaptivePrimalDual(l2, l1iso, Gop,
tau=tau, mu=mu,
x0=np.zeros_like(img.ravel()),
niter=45, show=True, tol=0.05,
callback=lambda x: callback(x, l2, l1iso,
Gop, cost_ada,
img.ravel(),
err_ada))
iml12_ada = iml12_ada.reshape(img.shape)
Adaptive Primal-dual: min_x f(Ax) + x^T z + g(x)
---------------------------------------------------------
Proximal operator (f): <class 'pyproximal.proximal.L2.L2'>
Proximal operator (g): <class 'pyproximal.proximal.L21.L21'>
Linear operator (A): <class 'pylops.basicoperators.gradient.Gradient'>
Additional vector (z): None
tau0 = 3.358757e-01 mu0 = 3.358757e-01
alpha0 = 5.000000e-01 eta = 9.500000e-01
s = 1.000000e+00 delta = 1.500000e+00
niter = 45 tol = 5.000000e-02
Itn x[0] f g z^x J = f + g + z^x
2 2.55225e+00 1.147e+08 1.331e+05 0.000e+00 1.149e+08
3 7.29382e+00 1.089e+08 1.624e+05 0.000e+00 1.091e+08
4 1.59759e+01 9.883e+07 2.027e+05 0.000e+00 9.903e+07
5 3.06967e+01 8.311e+07 2.854e+05 0.000e+00 8.340e+07
6 5.30254e+01 6.209e+07 4.068e+05 0.000e+00 6.250e+07
7 8.26519e+01 3.913e+07 5.551e+05 0.000e+00 3.969e+07
8 1.05944e+02 2.500e+07 6.635e+05 0.000e+00 2.566e+07
9 1.24300e+02 1.630e+07 7.386e+05 0.000e+00 1.704e+07
10 1.38814e+02 1.094e+07 7.896e+05 0.000e+00 1.173e+07
13 1.64419e+02 4.730e+06 8.553e+05 0.000e+00 5.585e+06
17 1.71883e+02 3.758e+06 8.445e+05 0.000e+00 4.602e+06
21 1.77391e+02 3.307e+06 8.321e+05 0.000e+00 4.139e+06
25 1.82697e+02 2.931e+06 8.428e+05 0.000e+00 3.774e+06
29 1.87108e+02 2.695e+06 8.346e+05 0.000e+00 3.530e+06
33 1.89833e+02 2.555e+06 8.179e+05 0.000e+00 3.372e+06
37 1.91793e+02 2.468e+06 8.076e+05 0.000e+00 3.276e+06
38 1.92212e+02 2.452e+06 8.061e+05 0.000e+00 3.258e+06
39 1.92599e+02 2.438e+06 8.048e+05 0.000e+00 3.243e+06
40 1.92953e+02 2.426e+06 8.039e+05 0.000e+00 3.230e+06
41 1.93278e+02 2.415e+06 8.031e+05 0.000e+00 3.218e+06
42 1.93577e+02 2.405e+06 8.026e+05 0.000e+00 3.208e+06
43 1.93857e+02 2.396e+06 8.022e+05 0.000e+00 3.198e+06
44 1.94120e+02 2.389e+06 8.019e+05 0.000e+00 3.190e+06
45 1.94371e+02 2.382e+06 8.016e+05 0.000e+00 3.183e+06
46 1.94611e+02 2.376e+06 8.014e+05 0.000e+00 3.177e+06
Total time (s) = 1.19
Let’s now compare the final results as well as the convergence curves of the two algorithms. We can see how the adaptive Primal-Dual produces a better estimate of the clean image in a much smaller number of iterations
fig, axs = plt.subplots(1, 4, figsize=(16, 4))
axs[0].imshow(img, cmap='gray', vmin=0, vmax=255)
axs[0].set_title('Original')
axs[0].axis('off')
axs[0].axis('tight')
axs[1].imshow(noise_img, cmap='gray', vmin=0, vmax=255)
axs[1].set_title('Noisy')
axs[1].axis('off')
axs[1].axis('tight')
axs[2].imshow(iml12_fixed, cmap='gray', vmin=0, vmax=255)
axs[2].set_title('PD')
axs[2].axis('off')
axs[2].axis('tight')
axs[3].imshow(iml12_ada, cmap='gray', vmin=0, vmax=255)
axs[3].set_title('Adaptive PD')
axs[3].axis('off')
axs[3].axis('tight')
fig, axs = plt.subplots(2, 1, figsize=(12, 7))
axs[0].plot(cost_fixed, 'k', label='Fixed step')
axs[0].plot(cost_ada, 'r', label='Adaptive step')
axs[0].legend()
axs[0].set_title('Functional')
axs[1].plot(err_fixed, 'k', label='Fixed step')
axs[1].plot(err_ada, 'r', label='Adaptive step')
axs[1].set_title('MSE')
axs[1].legend()
plt.tight_layout()
fig, axs = plt.subplots(3, 1, figsize=(12, 7))
axs[0].plot(steps[0], 'k')
axs[0].set_title(r'$\tau^k$')
axs[1].plot(steps[1], 'k')
axs[1].set_title(r'$\mu^k$')
axs[2].plot(steps[2], 'k')
axs[2].set_title(r'$\alpha^k$')
plt.tight_layout();
Total running time of the script: (0 minutes 8.666 seconds)