from PEPit import PEP
from PEPit.operators import MonotoneOperator
from PEPit.primitive_steps import proximal_step
[docs]def wc_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 **proximal point** method, and looks for a low-dimensional
worst-case example nearly achieving this worst-case guarantee using the trace heuristic.
That is, it computes the smallest possible :math:`\\tau(n, \\alpha)` such that the guarantee
.. math:: \\|x_n - x_{n-1}\\|^2 \\leqslant \\tau(n, \\alpha) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_\\star` is such that :math:`0 \\in Ax_\\star`. Then, it looks for a low-dimensional nearly achieving this
performance.
**Algorithm**: The proximal point algorithm for monotone inclusions is described as follows, for :math:`t \in \\{ 0, \\dots, n-1\\}`,
.. math:: x_{t+1} = J_{\\alpha A}(x_t),
where :math:`\\alpha` is a step-size.
**Theoretical guarantee**: A tight theoretical guarantee can be found in [1, section 4].
.. math:: \\|x_n - x_{n-1}\\|^2 \\leqslant \\frac{\\left(1 - \\frac{1}{n}\\right)^{n - 1}}{n} \\|x_0 - x_\\star\\|^2.
**Reference**:
`[1] G. Gu, J. Yang (2020). Tight sublinear convergence rate of the proximal point algorithm for maximal
monotone inclusion problem. SIAM Journal on Optimization, 30(3), 1905-1921.
<https://epubs.siam.org/doi/pdf/10.1137/19M1299049>`_
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_proximal_point(alpha=2.2, n=11, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 13x13
(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 132 scalar constraint(s) ...
function 1 : 132 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); optimal value: 0.03504735907840766
(PEPit) Postprocessing: 2 eigenvalue(s) > 1.885183851963194e-06 before dimension reduction
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); objective value: 0.03503739338571882
(PEPit) Postprocessing: 2 eigenvalue(s) > 1.9044504527414672e-06 after dimension reduction
*** Example file: worst-case performance of the Proximal Point Method***
PEPit example: ||x(n) - x(n-1)||^2 == 0.0350374 ||x0 - xs||^2
Theoretical guarantee: ||x(n) - x(n-1)||^2 <= 0.0350494 ||x0 - xs||^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 gradient value g0
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):
previous_x = x
x, _, _ = proximal_step(previous_x, A, alpha)
# Set the performance metric to the distance between x(n) and x(n-1)
problem.set_performance_metric((x - previous_x) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose, dimension_reduction_heuristic="trace")
# Compute theoretical guarantee (for comparison)
theoretical_tau = (1 - 1 / n) ** (n - 1) / n
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the Proximal Point Method***')
print('\tPEPit example:\t\t ||x(n) - x(n-1)||^2 == {:.6} ||x0 - xs||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||x(n) - x(n-1)||^2 <= {:.6} ||x0 - xs||^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_proximal_point(alpha=2.2, n=11, verbose=1)