from math import sqrt
import numpy as np
from PEPit import PEP
from PEPit.point import Point
from PEPit.operators import NonexpansiveOperator
[docs]
def wc_inconsistent_halpern_iteration(n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the fixed point problem
.. math:: \\mathrm{Find}\\, x:\\, x = Ax,
where :math:`A` is a non-expansive operator,
that is a :math:`L`-Lipschitz operator with :math:`L=1`.
When the solution of above problem, or fixed point, does not exist,
behavior of the fixed-point iteration with A can be characterized with
infimal displacement vector :math:`v`.
This code computes a worst-case guarantee for the **Halpern Iteration**,
when `A` is not necessarily consistent, i.e., does not necessarily have fixed point.
That is, it computes the smallest possible :math:`\\tau(n)` such that the guarantee
.. math:: \\|x_n - Ax_n - v\\|^2 \\leqslant \\tau(n) \\|x_0 - x_\\star\\|^2
is valid, where :math:`x_n` is the output of the **Halpern iteration**
and :math:`x_\\star` is the point where :math:`v` is attained, i.e.,
.. math:: v = x_\\star - Ax_\\star
In short, for a given value of :math:`n`,
:math:`\\tau(n)` is computed as the worst-case value of
:math:`\\|x_n - Ax_n - v\\|^2` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: The Halpern iteration can be written as
.. math:: x_{t+1} = \\frac{1}{t + 2} x_0 + \\left(1 - \\frac{1}{t + 2}\\right) Ax_t.
**Theoretical guarantee**: A worst-case guarantee for Halpern iteration can be found in [1, Theorem 8]:
.. math:: \\|x_n - Ax_n - v\\|^2 \\leqslant \\left(\\frac{\\sqrt{Hn + 12} + 1}{n + 1}\\right)^2 \\|x_0 - x_\\star\\|^2.
**References**: The detailed approach is available in [1].
`[1] J. Park, E. Ryu (2023).
Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations.
International Conference on Machine Learning.
<https://arxiv.org/pdf/2303.15876.pdf>`_
Args:
n (int): number of iterations.
wrapper (str): the name of the wrapper to be used.
solver (str): the name of the solver the wrapper should use.
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_inconsistent_halpern_iteration(n=25, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 29x29
(PEPit) Setting up the problem: performance measure is the 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 378 scalar constraint(s) ...
Function 1 : 378 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.02678884717170149
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite
All the primal scalar constraints are verified up to an error of 4.2928149923682213e-10
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual scalar values associated with inequality constraints are nonnegative
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 3.9511359559460285e-06
(PEPit) Final upper bound (dual): 0.02678881575052497 and lower bound (primal example): 0.02678884717170149
(PEPit) Duality gap: absolute: -3.142117652177312e-08 and relative: -1.1729200708183147e-06
*** Example file: worst-case performance of (possibly inconsistent) Halpern Iterations ***
PEPit guarantee: ||xN - AxN - v||^2 <= 0.0267888 ||x0 - x_*||^2
Theoretical guarantee: ||xN - AxN - v||^2 <= 0.0366417 ||x0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a non expansive operator
A = problem.declare_function(NonexpansiveOperator)
# Start by defining point xs where infimal displacement vector v is attained
xs = Point()
Txs = A.gradient(xs)
A.v = xs - Txs
# Then define the starting point x0 of the algorithm
x0 = problem.set_initial_point()
# Set the initial constraint that is the difference between x0 and xs
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
# Run n steps of Halpern Iterations
x = x0
for i in range(n):
x = 1 / (i + 2) * x0 + (1 - 1 / (i + 2)) * A.gradient(x)
# Set the performance metric to distance between xN - AxN and v
problem.set_performance_metric((x - A.gradient(x) - A.v) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(wrapper=wrapper, solver=solver, verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
Hn = np.sum(1 / np.arange(1, n+1))
theoretical_tau = ((sqrt(Hn + 12) + 1) / (n + 1)) ** 2
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of (possibly inconsistent) Halpern Iterations ***')
print('\tPEPit guarantee:\t ||xN - AxN - v||^2 <= {:.6} ||x0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||xN - AxN - v||^2 <= {:.6} ||x0 - x_*||^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_inconsistent_halpern_iteration(n=25, wrapper="cvxpy", solver=None, verbose=1)