from PEPit import PEP
from PEPit.operators import StronglyMonotoneOperator
from PEPit.primitive_steps import proximal_step
def phi(mu, idx):
if idx == -1:
return 0
return ((1 + 2 * mu) ** (2 * idx + 2) - 1) / ((1 + 2 * mu) ** 2 - 1)
[docs]
def wc_optimal_strongly_monotone_proximal_point(n, mu, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the monotone inclusion problem
.. math:: \\mathrm{Find}\\, x:\\, 0\\in Ax,
where :math:`A` is maximally :math:`\\mu`-strongly monotone.
We denote by :math:`J_{A}` the resolvent of :math:`A`.
For any :math:`x` such that :math:`x = J_{A} y` for some :math:`y`,
define the resolvent residual :math:`\\tilde{A}x = y - J_{A}y \\in Ax`.
This code computes a worst-case guarantee for the **Optimal Strongly-monotone Proximal Point Method** (OS-PPM).
That is, it computes the smallest possible :math:`\\tau(n, \\mu)` such that the guarantee
.. math:: \\|\\tilde{A}x_n\\|^2 \\leqslant \\tau(n, \\mu) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_n` is the output of the **Optimal Strongly-monotone Proximal Point Method**,
and :math:`x_\\star` is a zero of :math:`A`. In short, for a given value of :math:`n, \\mu`,
:math:`\\tau(n, \\mu)` is computed as the worst-case value of
:math:`\\|\\tilde{A}x_n\\|^2` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: The Optimal Strongly-monotone Proximal Point Method can be written as
.. math::
:nowrap:
\\begin{eqnarray}
x_{t+1} & = & J_{A} y_t,\\\\
y_{t+1} & = & x_{t+1} + \\frac{\\varphi_{t} - 1}{\\varphi_{t+1}} (x_{t+1} - x_t) - \\frac{2 \\mu \\varphi_{t}}{\\varphi_{t+1}} (y_t - x_{t+1}) \\\\
& & + \\frac{(1+2\\mu) \\varphi_{t-1}}{\\varphi_{t+1}} (y_{t-1} - x_t).
\\end{eqnarray}
where :math:`\\varphi_k = \sum_{i=0}^k (1+2\\mu)^{2i}` with :math:`\\varphi_{-1}=0`
and :math:`x_0 = y_0 = y_{-1}` is a starting point.
This method is equivalent to the Optimal Contractive Halpern iteration.
**Theoretical guarantee**: A **tight** worst-case guarantee for the Optimal Strongly-monotone Proximal Point Method
can be found in [1, Theorem 3.2, Corollary 4.2]:
.. math:: \\|\\tilde{A}x_n\\|^2 \\leqslant \\left( \\frac{1}{\sum_{k=0}^{N-1} (1+2\\mu)^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.
mu (float): :math:`\\mu \ge 0`. :math:`A` will be maximal :math:`\\mu`-strongly monotone.
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_strongly_monotone_proximal_point(n=10, mu=0.05, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 12x12
(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 55 scalar constraint(s) ...
Function 1 : 55 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: 0.003936989547244047
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 1.0607807556608727e-09
All the primal scalar constraints are verified up to an error of 3.5351675688243754e-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 1.804099035739487e-08
(PEPit) Final upper bound (dual): 0.003936990621254958 and lower bound (primal example): 0.003936989547244047
(PEPit) Duality gap: absolute: 1.0740109105886186e-09 and relative: 2.7280004117370406e-07
*** Example file: worst-case performance of Optimal Strongly-monotone Proximal Point Method ***
PEPit guarantee: ||AxN||^2 <= 0.00393699 ||x0 - x_*||^2
Theoretical guarantee: ||AxN||^2 <= 0.00393698 ||x0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a monotone operator
A = problem.declare_function(StronglyMonotoneOperator, mu=mu)
# Start by defining the zero point xs
xs = A.stationary_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 Strongly-monotone Proximal Point Method
x, y, y_prv = x0, x0, x0
for i in range(n):
x_nxt, _, _ = proximal_step(y, A, 1)
y_nxt = x_nxt + (phi(mu, i) - 1) / phi(mu, i + 1) * (x_nxt - x) - 2 * mu * phi(mu, i) / phi(mu, i + 1) * (
y - x_nxt) + (1 + 2 * mu) * phi(mu, i - 1) / phi(mu, i + 1) * (y_prv - x)
x, y_prv, y = x_nxt, y, y_nxt
# Set the performance metric to length of \tilde{A}xN
problem.set_performance_metric((y_prv - 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 = (2 * mu / ((1 + 2 * mu) ** n - 1)) ** 2
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of Optimal Strongly-monotone Proximal Point Method ***')
print('\tPEPit guarantee:\t ||AxN||^2 <= {:.6} ||x0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||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_strongly_monotone_proximal_point(n=10, mu=0.05,
wrapper="cvxpy", solver=None,
verbose=1)