Note
Go to the end to download the full example code.
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 3.15860e+00 1.148e+08 1.329e+05 0.000e+00 1.149e+08
2 5.95850e+00 1.118e+08 1.384e+05 0.000e+00 1.119e+08
3 8.58035e+00 1.089e+08 1.218e+05 0.000e+00 1.090e+08
4 1.11684e+01 1.061e+08 1.117e+05 0.000e+00 1.063e+08
5 1.37181e+01 1.034e+08 1.110e+05 0.000e+00 1.035e+08
6 1.62291e+01 1.008e+08 1.145e+05 0.000e+00 1.009e+08
7 1.87041e+01 9.820e+07 1.190e+05 0.000e+00 9.832e+07
8 2.11462e+01 9.568e+07 1.244e+05 0.000e+00 9.581e+07
9 2.35565e+01 9.323e+07 1.308e+05 0.000e+00 9.336e+07
10 2.59354e+01 9.085e+07 1.379e+05 0.000e+00 9.099e+07
31 6.91910e+01 5.299e+07 2.889e+05 0.000e+00 5.328e+07
61 1.13307e+02 2.519e+07 4.553e+05 0.000e+00 2.564e+07
91 1.42868e+02 1.267e+07 5.689e+05 0.000e+00 1.324e+07
121 1.62676e+02 7.020e+06 6.451e+05 0.000e+00 7.665e+06
151 1.75948e+02 4.467e+06 6.962e+05 0.000e+00 5.163e+06
181 1.84842e+02 3.310e+06 7.305e+05 0.000e+00 4.040e+06
211 1.90801e+02 2.782e+06 7.535e+05 0.000e+00 3.536e+06
241 1.94795e+02 2.541e+06 7.689e+05 0.000e+00 3.309e+06
271 1.97470e+02 2.429e+06 7.792e+05 0.000e+00 3.208e+06
292 1.98799e+02 2.388e+06 7.844e+05 0.000e+00 3.172e+06
293 1.98853e+02 2.386e+06 7.846e+05 0.000e+00 3.171e+06
294 1.98907e+02 2.385e+06 7.848e+05 0.000e+00 3.170e+06
295 1.98960e+02 2.384e+06 7.850e+05 0.000e+00 3.169e+06
296 1.99012e+02 2.382e+06 7.852e+05 0.000e+00 3.167e+06
297 1.99064e+02 2.381e+06 7.854e+05 0.000e+00 3.166e+06
298 1.99115e+02 2.380e+06 7.856e+05 0.000e+00 3.165e+06
299 1.99165e+02 2.378e+06 7.858e+05 0.000e+00 3.164e+06
300 1.99214e+02 2.377e+06 7.860e+05 0.000e+00 3.163e+06
Total time (s) = 6.89
---------------------------------------------------------
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 3.15860e+00 1.148e+08 1.329e+05 0.000e+00 1.149e+08
3 8.68514e+00 1.089e+08 1.625e+05 0.000e+00 1.091e+08
4 1.82426e+01 9.884e+07 2.033e+05 0.000e+00 9.904e+07
5 3.40762e+01 8.312e+07 2.863e+05 0.000e+00 8.341e+07
6 5.78726e+01 6.209e+07 4.082e+05 0.000e+00 6.250e+07
7 8.92975e+01 3.913e+07 5.573e+05 0.000e+00 3.969e+07
8 1.13902e+02 2.500e+07 6.664e+05 0.000e+00 2.567e+07
9 1.33167e+02 1.630e+07 7.419e+05 0.000e+00 1.704e+07
10 1.48255e+02 1.094e+07 7.932e+05 0.000e+00 1.173e+07
13 1.74190e+02 4.728e+06 8.594e+05 0.000e+00 5.588e+06
17 1.80754e+02 3.756e+06 8.482e+05 0.000e+00 4.604e+06
21 1.84435e+02 3.305e+06 8.354e+05 0.000e+00 4.141e+06
25 1.88410e+02 2.930e+06 8.450e+05 0.000e+00 3.775e+06
29 1.91757e+02 2.695e+06 8.357e+05 0.000e+00 3.530e+06
33 1.94331e+02 2.555e+06 8.183e+05 0.000e+00 3.373e+06
37 1.96311e+02 2.469e+06 8.077e+05 0.000e+00 3.277e+06
38 1.96730e+02 2.453e+06 8.062e+05 0.000e+00 3.260e+06
39 1.97123e+02 2.439e+06 8.049e+05 0.000e+00 3.244e+06
40 1.97491e+02 2.427e+06 8.040e+05 0.000e+00 3.231e+06
41 1.97835e+02 2.416e+06 8.033e+05 0.000e+00 3.219e+06
42 1.98157e+02 2.406e+06 8.028e+05 0.000e+00 3.209e+06
43 1.98459e+02 2.397e+06 8.024e+05 0.000e+00 3.200e+06
44 1.98742e+02 2.390e+06 8.021e+05 0.000e+00 3.192e+06
45 1.99007e+02 2.383e+06 8.018e+05 0.000e+00 3.185e+06
46 1.99254e+02 2.377e+06 8.016e+05 0.000e+00 3.178e+06
Total time (s) = 1.04
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 8.664 seconds)