from PEPit import PEP
from PEPit.operators import LipschitzOperator
[docs]
def wc_optimal_contractive_halpern_iteration(n, gamma, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the fixed point problem
.. math:: \\mathrm{Find}\\, x:\\, x = Ax,
where :math:`A` is a :math:`1/\gamma`-contractive operator,
i.e. a :math:`L`-Lipschitz operator with :math:`L=1/\gamma`.
This code computes a worst-case guarantee for the **Optimal Contractive Halpern Iteration**.
That is, it computes the smallest possible :math:`\\tau(n, \\gamma)` such that the guarantee
.. math:: \\|x_n - Ax_n\\|^2 \\leqslant \\tau(n, \\gamma) \\|x_0 - x_\\star\\|^2
is valid, where :math:`x_n` is the output of the **Optimal Contractive Halpern iteration**,
and :math:`x_\\star` is the fixed point of :math:`A`. In short, for a given value of :math:`n, \\gamma`,
:math:`\\tau(n, \\gamma)` is computed as the worst-case value of
:math:`\\|x_n - Ax_n\\|^2` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: The Optimal Contractive Halpern iteration can be written as
.. math:: x_{t+1} = \\left(1 - \\frac{1}{\\varphi_{t+1}} \\right) Ax_t + \\frac{1}{\\varphi_{t+1}} x_0.
where :math:`\\varphi_k = \sum_{i=0}^k \gamma^{2i}` and :math:`x_0` is a starting point.
**Theoretical guarantee**: A **tight** worst-case guarantee for the Optimal Contractive Halpern iteration
can be found in [1, Corollary 3.3, Theorem 4.1]:
.. math:: \\|x_n - Ax_n\\|^2 \\leqslant \\left(1 + \\frac{1}{\\gamma}\\right)^2 \\left( \\frac{1}{\\sum_{k=0}^n \\gamma^k} \\right)^2 \\|x_0 - x_\\star\\|^2.
**References**: The detailed approach and tight bound are available in [1].
`[1] J. Park, E. Ryu (2022).
Exact Optimal Accelerated Complexity for Fixed-Point Iterations.
In 39th International Conference on Machine Learning (ICML).
<https://proceedings.mlr.press/v162/park22c/park22c.pdf>`_
Args:
n (int): number of iterations.
gamma (float): :math:`\\gamma \ge 1`. :math:`A` will be :math:`1/\\gamma`-contractive.
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_optimal_contractive_halpern_iteration(n=10, gamma=1.1, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 13x13
(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 66 scalar constraint(s) ...
Function 1 : 66 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: 0.010613261462073679
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 5.170073408600879e-09
All the primal scalar constraints are verified up to an error of 1.5453453107439064e-08
(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 up to an error of 3.883430104655162e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.396268932559219e-07
(PEPit) Final upper bound (dual): 0.010613268001708536 and lower bound (primal example): 0.010613261462073679
(PEPit) Duality gap: absolute: 6.5396348579438435e-09 and relative: 6.161757986753765e-07
*** Example file: worst-case performance of Optimal Contractive Halpern Iterations ***
PEPit guarantee: ||xN - AxN||^2 <= 0.0106133 ||x0 - x_*||^2
Theoretical guarantee: ||xN - AxN||^2 <= 0.0106132 ||x0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a non expansive operator
A = problem.declare_function(LipschitzOperator, L=1 / gamma)
# 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)
# Run n steps of Optimal Contractive Halpern Iterations
x = x0
for i in range(n):
phi = (gamma ** (2 * i + 4) - 1) / (gamma ** 2 - 1)
x = 1 / phi * x0 + (1 - 1 / phi) * A.gradient(x)
# Set the performance metric to distance between xN and AxN
problem.set_performance_metric((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)
theoretical_tau = (1 + 1 / gamma) ** 2 * ((gamma - 1) / (gamma ** (n + 1) - 1)) ** 2
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of Optimal Contractive Halpern Iterations ***')
print('\tPEPit guarantee:\t ||xN - AxN||^2 <= {:.6} ||x0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||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_optimal_contractive_halpern_iteration(n=10, gamma=1.1,
wrapper="cvxpy", solver=None,
verbose=1)