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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 2.74009e+00 1.148e+08 1.332e+05 0.000e+00 1.149e+08
2 5.33483e+00 1.118e+08 1.387e+05 0.000e+00 1.120e+08
3 7.79620e+00 1.090e+08 1.220e+05 0.000e+00 1.091e+08
4 1.01855e+01 1.062e+08 1.118e+05 0.000e+00 1.063e+08
5 1.25593e+01 1.035e+08 1.111e+05 0.000e+00 1.036e+08
6 1.49436e+01 1.008e+08 1.146e+05 0.000e+00 1.009e+08
7 1.73372e+01 9.823e+07 1.191e+05 0.000e+00 9.835e+07
8 1.97256e+01 9.571e+07 1.245e+05 0.000e+00 9.584e+07
9 2.20949e+01 9.326e+07 1.309e+05 0.000e+00 9.339e+07
10 2.44390e+01 9.088e+07 1.380e+05 0.000e+00 9.102e+07
31 6.76474e+01 5.301e+07 2.890e+05 0.000e+00 5.330e+07
61 1.11499e+02 2.520e+07 4.555e+05 0.000e+00 2.566e+07
91 1.40329e+02 1.268e+07 5.691e+05 0.000e+00 1.325e+07
121 1.60019e+02 7.031e+06 6.454e+05 0.000e+00 7.677e+06
151 1.73258e+02 4.478e+06 6.965e+05 0.000e+00 5.175e+06
181 1.82082e+02 3.321e+06 7.308e+05 0.000e+00 4.051e+06
211 1.87945e+02 2.793e+06 7.538e+05 0.000e+00 3.547e+06
241 1.91893e+02 2.551e+06 7.692e+05 0.000e+00 3.321e+06
271 1.94521e+02 2.439e+06 7.796e+05 0.000e+00 3.219e+06
292 1.95806e+02 2.399e+06 7.847e+05 0.000e+00 3.183e+06
293 1.95859e+02 2.397e+06 7.849e+05 0.000e+00 3.182e+06
294 1.95912e+02 2.396e+06 7.851e+05 0.000e+00 3.181e+06
295 1.95963e+02 2.394e+06 7.853e+05 0.000e+00 3.180e+06
296 1.96015e+02 2.393e+06 7.855e+05 0.000e+00 3.178e+06
297 1.96065e+02 2.392e+06 7.857e+05 0.000e+00 3.177e+06
298 1.96115e+02 2.390e+06 7.859e+05 0.000e+00 3.176e+06
299 1.96165e+02 2.389e+06 7.861e+05 0.000e+00 3.175e+06
300 1.96214e+02 2.388e+06 7.863e+05 0.000e+00 3.174e+06
Total time (s) = 13.79
---------------------------------------------------------
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 2.74009e+00 1.148e+08 1.332e+05 0.000e+00 1.149e+08
3 7.86167e+00 1.090e+08 1.629e+05 0.000e+00 1.091e+08
4 1.68808e+01 9.887e+07 2.034e+05 0.000e+00 9.908e+07
5 3.18816e+01 8.315e+07 2.865e+05 0.000e+00 8.344e+07
6 5.46224e+01 6.212e+07 4.085e+05 0.000e+00 6.253e+07
7 8.48974e+01 3.915e+07 5.577e+05 0.000e+00 3.971e+07
8 1.08716e+02 2.502e+07 6.667e+05 0.000e+00 2.568e+07
9 1.27445e+02 1.631e+07 7.422e+05 0.000e+00 1.706e+07
10 1.42173e+02 1.095e+07 7.936e+05 0.000e+00 1.175e+07
13 1.67755e+02 4.739e+06 8.598e+05 0.000e+00 5.599e+06
17 1.74749e+02 3.767e+06 8.488e+05 0.000e+00 4.616e+06
21 1.79702e+02 3.316e+06 8.361e+05 0.000e+00 4.152e+06
25 1.85302e+02 2.940e+06 8.460e+05 0.000e+00 3.786e+06
29 1.89319e+02 2.705e+06 8.365e+05 0.000e+00 3.542e+06
33 1.91736e+02 2.566e+06 8.189e+05 0.000e+00 3.384e+06
37 1.93424e+02 2.480e+06 8.083e+05 0.000e+00 3.288e+06
38 1.93811e+02 2.464e+06 8.067e+05 0.000e+00 3.271e+06
39 1.94192e+02 2.450e+06 8.054e+05 0.000e+00 3.255e+06
40 1.94567e+02 2.437e+06 8.045e+05 0.000e+00 3.242e+06
41 1.94933e+02 2.426e+06 8.037e+05 0.000e+00 3.230e+06
42 1.95286e+02 2.417e+06 8.032e+05 0.000e+00 3.220e+06
43 1.95622e+02 2.408e+06 8.028e+05 0.000e+00 3.211e+06
44 1.95937e+02 2.400e+06 8.024e+05 0.000e+00 3.203e+06
45 1.96227e+02 2.394e+06 8.022e+05 0.000e+00 3.196e+06
46 1.96492e+02 2.388e+06 8.019e+05 0.000e+00 3.190e+06
Total time (s) = 2.37
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 17.333 seconds)