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 import camera

import pyproximal


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

# Load image
img = camera()
ny, nx = img.shape

# Add noise
sigman = 20
n = np.random.normal(0, sigman, img.shape)
noise_img = img + n

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.,
                                                  gfirst=False, niter=300, show=True,
                                                  callback=lambda x: callback(x, l2, l1iso,
                                                                              Gop, cost_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   2.59605e+00   1.148e+08   1.334e+05   0.000e+00       1.149e+08
     2   5.16785e+00   1.118e+08   1.387e+05   0.000e+00       1.120e+08
     3   7.70666e+00   1.090e+08   1.219e+05   0.000e+00       1.091e+08
     4   1.01947e+01   1.062e+08   1.116e+05   0.000e+00       1.063e+08
     5   1.26277e+01   1.035e+08   1.109e+05   0.000e+00       1.036e+08
     6   1.50206e+01   1.008e+08   1.142e+05   0.000e+00       1.009e+08
     7   1.73989e+01   9.824e+07   1.187e+05   0.000e+00       9.836e+07
     8   1.97851e+01   9.573e+07   1.240e+05   0.000e+00       9.585e+07
     9   2.21893e+01   9.327e+07   1.303e+05   0.000e+00       9.341e+07
    10   2.46073e+01   9.089e+07   1.373e+05   0.000e+00       9.103e+07
    31   6.73083e+01   5.302e+07   2.877e+05   0.000e+00       5.331e+07
    61   1.11401e+02   2.521e+07   4.535e+05   0.000e+00       2.566e+07
    91   1.40663e+02   1.268e+07   5.666e+05   0.000e+00       1.325e+07
   121   1.60374e+02   7.032e+06   6.425e+05   0.000e+00       7.675e+06
   151   1.73405e+02   4.479e+06   6.934e+05   0.000e+00       5.172e+06
   181   1.82280e+02   3.321e+06   7.276e+05   0.000e+00       4.049e+06
   211   1.88176e+02   2.794e+06   7.504e+05   0.000e+00       3.544e+06
   241   1.92130e+02   2.552e+06   7.658e+05   0.000e+00       3.318e+06
   271   1.94768e+02   2.440e+06   7.761e+05   0.000e+00       3.216e+06
   292   1.96083e+02   2.399e+06   7.812e+05   0.000e+00       3.180e+06
   293   1.96136e+02   2.398e+06   7.814e+05   0.000e+00       3.179e+06
   294   1.96190e+02   2.396e+06   7.816e+05   0.000e+00       3.178e+06
   295   1.96242e+02   2.395e+06   7.818e+05   0.000e+00       3.177e+06
   296   1.96294e+02   2.394e+06   7.820e+05   0.000e+00       3.176e+06
   297   1.96345e+02   2.392e+06   7.822e+05   0.000e+00       3.174e+06
   298   1.96395e+02   2.391e+06   7.824e+05   0.000e+00       3.173e+06
   299   1.96444e+02   2.390e+06   7.826e+05   0.000e+00       3.172e+06
   300   1.96493e+02   2.388e+06   7.828e+05   0.000e+00       3.171e+06

Total time (s) = 11.12

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,
                                                          niter=45, show=True, tol=0.05,
                                                          callback=lambda x: callback(x, l2, l1iso,
                                                                                      Gop, cost_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   2.59605e+00   1.148e+08   1.334e+05   0.000e+00       1.149e+08
     3   7.67236e+00   1.090e+08   1.628e+05   0.000e+00       1.092e+08
     4   1.68656e+01   9.888e+07   2.031e+05   0.000e+00       9.909e+07
     5   3.20891e+01   8.316e+07   2.856e+05   0.000e+00       8.345e+07
     6   5.48988e+01   6.213e+07   4.068e+05   0.000e+00       6.253e+07
     7   8.51166e+01   3.916e+07   5.550e+05   0.000e+00       3.971e+07
     8   1.08951e+02   2.502e+07   6.633e+05   0.000e+00       2.568e+07
     9   1.27808e+02   1.632e+07   7.381e+05   0.000e+00       1.705e+07
    10   1.42763e+02   1.095e+07   7.890e+05   0.000e+00       1.174e+07
    13   1.69078e+02   4.741e+06   8.547e+05   0.000e+00       5.596e+06
    17   1.76349e+02   3.769e+06   8.439e+05   0.000e+00       4.613e+06
    21   1.81309e+02   3.318e+06   8.310e+05   0.000e+00       4.149e+06
    25   1.86364e+02   2.942e+06   8.410e+05   0.000e+00       3.783e+06
    29   1.89516e+02   2.706e+06   8.327e+05   0.000e+00       3.539e+06
    33   1.91169e+02   2.566e+06   8.158e+05   0.000e+00       3.382e+06
    37   1.92282e+02   2.480e+06   8.052e+05   0.000e+00       3.286e+06
    38   1.92568e+02   2.464e+06   8.035e+05   0.000e+00       3.268e+06
    39   1.92870e+02   2.450e+06   8.023e+05   0.000e+00       3.253e+06
    40   1.93192e+02   2.438e+06   8.013e+05   0.000e+00       3.239e+06
    41   1.93531e+02   2.427e+06   8.005e+05   0.000e+00       3.227e+06
    42   1.93886e+02   2.417e+06   7.999e+05   0.000e+00       3.217e+06
    43   1.94253e+02   2.408e+06   7.995e+05   0.000e+00       3.208e+06
    44   1.94625e+02   2.401e+06   7.991e+05   0.000e+00       3.200e+06
    45   1.94997e+02   2.394e+06   7.988e+05   0.000e+00       3.193e+06
    46   1.95363e+02   2.388e+06   7.986e+05   0.000e+00       3.186e+06

Total time (s) = 2.02

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[1].imshow(noise_img, cmap='gray', vmin=0, vmax=255)
axs[2].imshow(iml12_fixed, cmap='gray', vmin=0, vmax=255)
axs[3].imshow(iml12_ada, cmap='gray', vmin=0, vmax=255)
axs[3].set_title('Adaptive PD')

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[1].plot(err_fixed, 'k', label='Fixed step')
axs[1].plot(err_ada, 'r', label='Adaptive step')

fig, axs = plt.subplots(3, 1, figsize=(12, 7))
axs[0].plot(steps[0], 'k')
axs[1].plot(steps[1], 'k')
axs[2].plot(steps[2], 'k')
  • Original, Noisy, PD, Adaptive PD
  • Functional, MSE
  • $\tau^k$, $\mu^k$, $\alpha^k$

Total running time of the script: (0 minutes 14.230 seconds)

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