from math import sqrt
from PEPit import PEP
from PEPit.operators import LipschitzOperator
[docs]
def wc_krasnoselskii_mann_constant_step_sizes(n, gamma, 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`.
This code computes a worst-case guarantee for the **Krasnolselskii-Mann** (KM) method with constant step-size.
That is, it computes the smallest possible :math:`\\tau(n)` such that the guarantee
.. math:: \\frac{1}{4}\\|x_n - Ax_n\\|^2 \\leqslant \\tau(n) \\|x_0 - x_\\star\\|^2
is valid, where :math:`x_n` is the output of the KM method, and :math:`x_\\star` is some fixed point of :math:`A`
(i.e., :math:`x_\\star=Ax_\\star`).
**Algorithm**: The constant step-size KM method is described by
.. math:: x_{t+1} = \\left(1 - \\gamma\\right) x_{t} + \\gamma Ax_{t}.
**Theoretical guarantee**: A theoretical **upper** bound is provided by [1, Theorem 4.9]
.. math:: \\tau(n) = \\left\{
\\begin{eqnarray}
\\frac{1}{n+1}\\left(\\frac{n}{n+1}\\right)^n \\frac{1}{4 \\gamma (1 - \\gamma)}\quad & \\text{if } \\frac{1}{2}\\leqslant \\gamma \\leqslant \\frac{1}{2}\\left(1+\\sqrt{\\frac{n}{n+1}}\\right) \\\\
(\\gamma - 1)^{2n} \quad & \\text{if } \\frac{1}{2}\\left(1+\\sqrt{\\frac{n}{n+1}}\\right) < \\gamma \\leqslant 1.
\\end{eqnarray}
\\right.
**Reference**:
`[1] F. Lieder (2018).
Projection Based Methods for Conic Linear Programming
Optimal First Order Complexities and Norm Constrained Quasi Newton Methods.
PhD thesis, HHU Düsseldorf.
<https://docserv.uni-duesseldorf.de/servlets/DerivateServlet/Derivate-49971/Dissertation.pdf>`_
Args:
n (int): number of iterations.
gamma (float): step-size between 1/2 and 1
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 + solver details.
Returns:
pepit_tau (float): worst-case value
theoretical_tau (float): theoretical value
Example:
>>> pepit_tau, theoretical_tau = wc_krasnoselskii_mann_constant_step_sizes(n=3, gamma=3 / 4, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 6x6
(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 10 scalar constraint(s) ...
Function 1 : 10 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: 0.1406249823498115
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 2.6923151650319117e-09
All the primal scalar constraints are verified up to an error of 1.7378567473969042e-09
(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 4.412159703651858e-08
(PEPit) Final upper bound (dual): 0.14062498615478927 and lower bound (primal example): 0.1406249823498115
(PEPit) Duality gap: absolute: 3.804977777299712e-09 and relative: 2.7057623145755453e-08
*** Example file: worst-case performance of Kranoselskii-Mann iterations ***
PEPit guarantee: 1/4||xN - AxN||^2 <= 0.140625 ||x0 - x_*||^2
Theoretical guarantee: 1/4||xN - AxN||^2 <= 0.140625 ||x0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a non expansive operator
A = problem.declare_function(LipschitzOperator, L=1.)
# Start by defining its unique optimal point xs = x_*
xs, _, _ = A.fixed_point()
# 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)
x = x0
for i in range(n):
x = (1 - gamma) * x + gamma * A.gradient(x)
# Set the performance metric to distance between xN and AxN
problem.set_performance_metric((1 / 2 * (x - A.gradient(x))) ** 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)
if 1 / 2 <= gamma <= 1 / 2 * (1 + sqrt(n / (n + 1))):
theoretical_tau = 1 / (n + 1) * (n / (n + 1)) ** n / (4 * gamma * (1 - gamma))
elif 1 / 2 * (1 + sqrt(n / (n + 1))) < gamma <= 1:
theoretical_tau = (2 * gamma - 1) ** (2 * n)
else:
raise ValueError("{} is not a valid value for the step-size \'gamma\'."
" \'gamma\' must be a number between 1/2 and 1".format(gamma))
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of Kranoselskii-Mann iterations ***')
print('\tPEPit guarantee:\t 1/4||xN - AxN||^2 <= {:.6} ||x0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t 1/4||xN - AxN||^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_krasnoselskii_mann_constant_step_sizes(n=3, gamma=3 / 4,
wrapper="cvxpy", solver=None,
verbose=1)