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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 2.54961e+00 1.147e+08 1.329e+05 0.000e+00 1.148e+08
2 5.05255e+00 1.117e+08 1.382e+05 0.000e+00 1.119e+08
3 7.55087e+00 1.089e+08 1.215e+05 0.000e+00 1.090e+08
4 1.00694e+01 1.061e+08 1.115e+05 0.000e+00 1.062e+08
5 1.26065e+01 1.034e+08 1.110e+05 0.000e+00 1.035e+08
6 1.51456e+01 1.007e+08 1.144e+05 0.000e+00 1.008e+08
7 1.76685e+01 9.813e+07 1.189e+05 0.000e+00 9.825e+07
8 2.01617e+01 9.562e+07 1.242e+05 0.000e+00 9.574e+07
9 2.26168e+01 9.317e+07 1.306e+05 0.000e+00 9.330e+07
10 2.50296e+01 9.078e+07 1.376e+05 0.000e+00 9.092e+07
31 6.77163e+01 5.295e+07 2.883e+05 0.000e+00 5.324e+07
61 1.11288e+02 2.517e+07 4.542e+05 0.000e+00 2.563e+07
91 1.40757e+02 1.266e+07 5.673e+05 0.000e+00 1.323e+07
121 1.60379e+02 7.016e+06 6.433e+05 0.000e+00 7.659e+06
151 1.73596e+02 4.465e+06 6.942e+05 0.000e+00 5.160e+06
181 1.82419e+02 3.309e+06 7.284e+05 0.000e+00 4.037e+06
211 1.88331e+02 2.782e+06 7.513e+05 0.000e+00 3.533e+06
241 1.92292e+02 2.540e+06 7.666e+05 0.000e+00 3.307e+06
271 1.94945e+02 2.428e+06 7.769e+05 0.000e+00 3.205e+06
292 1.96263e+02 2.388e+06 7.820e+05 0.000e+00 3.170e+06
293 1.96317e+02 2.386e+06 7.822e+05 0.000e+00 3.169e+06
294 1.96371e+02 2.385e+06 7.824e+05 0.000e+00 3.167e+06
295 1.96423e+02 2.384e+06 7.826e+05 0.000e+00 3.166e+06
296 1.96475e+02 2.382e+06 7.828e+05 0.000e+00 3.165e+06
297 1.96526e+02 2.381e+06 7.830e+05 0.000e+00 3.164e+06
298 1.96577e+02 2.380e+06 7.832e+05 0.000e+00 3.163e+06
299 1.96627e+02 2.378e+06 7.834e+05 0.000e+00 3.162e+06
300 1.96676e+02 2.377e+06 7.836e+05 0.000e+00 3.161e+06
Total time (s) = 5.19
---------------------------------------------------------
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 2.54961e+00 1.147e+08 1.329e+05 0.000e+00 1.148e+08
3 7.48999e+00 1.089e+08 1.622e+05 0.000e+00 1.090e+08
4 1.65571e+01 9.877e+07 2.028e+05 0.000e+00 9.898e+07
5 3.19051e+01 8.306e+07 2.861e+05 0.000e+00 8.335e+07
6 5.52202e+01 6.205e+07 4.082e+05 0.000e+00 6.246e+07
7 8.62168e+01 3.911e+07 5.574e+05 0.000e+00 3.966e+07
8 1.10593e+02 2.498e+07 6.663e+05 0.000e+00 2.565e+07
9 1.29773e+02 1.629e+07 7.415e+05 0.000e+00 1.703e+07
10 1.44869e+02 1.093e+07 7.926e+05 0.000e+00 1.172e+07
13 1.71060e+02 4.725e+06 8.581e+05 0.000e+00 5.583e+06
17 1.77932e+02 3.754e+06 8.468e+05 0.000e+00 4.601e+06
21 1.82074e+02 3.304e+06 8.338e+05 0.000e+00 4.137e+06
25 1.86241e+02 2.929e+06 8.436e+05 0.000e+00 3.772e+06
29 1.89192e+02 2.694e+06 8.347e+05 0.000e+00 3.528e+06
33 1.91291e+02 2.554e+06 8.174e+05 0.000e+00 3.371e+06
37 1.93240e+02 2.468e+06 8.066e+05 0.000e+00 3.275e+06
38 1.93720e+02 2.452e+06 8.050e+05 0.000e+00 3.257e+06
39 1.94188e+02 2.438e+06 8.037e+05 0.000e+00 3.242e+06
40 1.94640e+02 2.426e+06 8.027e+05 0.000e+00 3.228e+06
41 1.95070e+02 2.415e+06 8.018e+05 0.000e+00 3.216e+06
42 1.95472e+02 2.405e+06 8.012e+05 0.000e+00 3.206e+06
43 1.95843e+02 2.396e+06 8.007e+05 0.000e+00 3.197e+06
44 1.96181e+02 2.389e+06 8.003e+05 0.000e+00 3.189e+06
45 1.96484e+02 2.382e+06 8.000e+05 0.000e+00 3.182e+06
46 1.96752e+02 2.376e+06 7.998e+05 0.000e+00 3.176e+06
Total time (s) = 0.88
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 6.597 seconds)