from PEPit import PEP
from PEPit.operators import MonotoneOperator
from PEPit.primitive_steps import proximal_step
[docs]def wc_accelerated_proximal_point(alpha, n, verbose=1):
"""
Consider the monotone inclusion problem
.. math:: \\mathrm{Find}\\, x:\\, 0\\in Ax,
where :math:`A` is maximally monotone. We denote :math:`J_A = (I + A)^{-1}` the resolvents of :math:`A`.
This code computes a worst-case guarantee for the **accelerated proximal point** method proposed in [1].
That, it computes the smallest possible :math:`\\tau(n, \\alpha)` such that the guarantee
.. math:: \\|x_n - y_n\\|^2 \\leqslant \\tau(n, \\alpha) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_\\star` is such that :math:`0 \\in Ax_\\star`.
**Algorithm**: Accelerated proximal point is described as follows, for :math:`t \in \\{ 0, \\dots, n-1\\}`
.. math::
\\begin{eqnarray}
x_{t+1} & = & J_{\\alpha A}(y_t), \\\\
y_{t+1} & = & x_{t+1} + \\frac{t}{t+2}(x_{t+1} - x_{t}) - \\frac{t}{t+1}(x_t - y_{t-1}),
\\end{eqnarray}
where :math:`x_0=y_0=y_{-1}`
**Theoretical guarantee**: A tight theoretical worst-case guarantee can be found in [1, Theorem 4.1], for :math:`n \\geqslant 1`,
.. math:: \\|x_n - y_{n-1}\\|^2 \\leqslant \\frac{1}{n^2} \\|x_0 - x_\\star\\|^2.
**Reference**:
`[1] D. Kim (2021). Accelerated proximal point method for maximally monotone operators.
Mathematical Programming, 1-31.
<https://arxiv.org/pdf/1905.05149v4.pdf>`_
Args:
alpha (float): the step-size
n (int): number of iterations.
verbose (int): Level of information details to print.
- -1: No verbose at all.
- 0: This example's output.
- 1: This example's output + PEPit information.
- 2: This example's output + PEPit information + CVXPY details.
Returns:
pepit_tau (float): worst-case value
theoretical_tau (float): theoretical value
Example:
>>> pepit_tau, theoretical_tau = wc_accelerated_proximal_point(alpha=2, n=10, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 12x12
(PEPit) Setting up the problem: performance measure is minimum of 1 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (1 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
function 1 : Adding 110 scalar constraint(s) ...
function 1 : 110 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); optimal value: 0.010000353550061647
*** Example file: worst-case performance of the Accelerated Proximal Point Method***
PEPit guarantee: ||x_n - y_n||^2 <= 0.0100004 ||x_0 - x_s||^2
Theoretical guarantee: ||x_n - y_n||^2 <= 0.01 ||x_0 - x_s||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a monotone operator
A = problem.declare_function(MonotoneOperator)
# Start by defining its unique optimal point xs = x_*
xs = A.stationary_point()
# Then define the starting point x0 of the algorithm and its function value f0
x0 = problem.set_initial_point()
# Set the initial constraint that is the distance between x0 and x^*
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
# Compute n steps of the Proximal Gradient method starting from x0
x = [x0 for _ in range(n + 1)]
y = [x0 for _ in range(n + 1)]
for i in range(0, n - 1):
x[i + 1], _, _ = proximal_step(y[i + 1], A, alpha)
y[i + 2] = x[i + 1] + i / (i + 2) * (x[i + 1] - x[i]) - i / (i + 2) * (x[i] - y[i])
x[n], _, _ = proximal_step(y[n], A, alpha)
# Set the performance metric to the distance between xn and yn
problem.set_performance_metric((x[n] - y[n]) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
theoretical_tau = 1 / n ** 2
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the Accelerated Proximal Point Method***')
print('\tPEPit guarantee:\t ||x_n - y_n||^2 <= {:.6} ||x_0 - x_s||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||x_n - y_n||^2 <= {:.6} ||x_0 - x_s||^2'.format(theoretical_tau))
# Return the worst-case guarantee of the evaluated method ( and the reference theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
pepit_tau, theoretical_tau = wc_accelerated_proximal_point(alpha=2, n=10, verbose=1)