from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
from PEPit.operators import CocoerciveOperator
from PEPit.operators import MonotoneOperator
from PEPit.primitive_steps import proximal_step
[docs]def wc_three_operator_splitting(L, mu, beta, alpha, theta, verbose=1):
"""
Consider the monotone inclusion problem
.. math:: \\mathrm{Find}\\, x:\\, 0\\in Ax + Bx + Cx,
where :math:`A` is maximally monotone, :math:`B` is :math:`\\beta`-cocoercive and C is the gradient of some
:math:`L`-smooth :math:`\\mu`-strongly convex function. We denote by :math:`J_{\\alpha A}` and :math:`J_{\\alpha B}`
the resolvent of respectively :math:`A` and :math:`B`, with step-size :math:`\\alpha`.
This code computes a worst-case guarantee for the **three operator splitting** (TOS).
That is, given two initial points :math:`w^{(0)}_t` and :math:`w^{(1)}_t`,
this code computes the smallest possible :math:`\\tau(L, \\mu, \\beta, \\alpha, \\theta)`
(a.k.a. "contraction factor") such that the guarantee
.. math:: \\|w^{(0)}_{t+1} - w^{(1)}_{t+1}\\|^2 \\leqslant \\tau(L, \\mu, \\beta, \\alpha, \\theta) \\|w^{(0)}_{t} - w^{(1)}_{t}\\|^2,
is valid, where :math:`w^{(0)}_{t+1}` and :math:`w^{(1)}_{t+1}` are obtained after one iteration of TOS from
respectively :math:`w^{(0)}_{t}` and :math:`w^{(1)}_{t}`.
In short, for given values of :math:`L`, :math:`\\mu`, :math:`\\beta`, :math:`\\alpha` and :math:`\\theta`,
the contraction factor :math:`\\tau(L, \\mu, \\beta, \\alpha, \\theta)` is computed as the worst-case value of
:math:`\\|w^{(0)}_{t+1} - w^{(1)}_{t+1}\\|^2` when :math:`\\|w^{(0)}_{t} - w^{(1)}_{t}\\|^2 \\leqslant 1`.
**Algorithm**:
One iteration of the algorithm is described in [1]. For :math:`t \in \\{ 0, \\dots, n-1\\}`,
.. math::
:nowrap:
\\begin{eqnarray}
x_{t+1} & = & J_{\\alpha B} (w_t),\\\\
y_{t+1} & = & J_{\\alpha A} (2x_{t+1} - w_t - C x_{t+1}),\\\\
w_{t+1} & = & w_t - \\theta (x_{t+1} - y_{t+1}).
\\end{eqnarray}
**References**: The TOS was proposed in [1], the analysis of such operator splitting methods using PEPs was proposed in [2].
`[1] D. Davis, W. Yin (2017). A three-operator splitting scheme and its optimization applications.
Set-valued and variational analysis, 25(4), 829-858.
<https://arxiv.org/pdf/1504.01032.pdf>`_
`[2] E. Ryu, A. Taylor, C. Bergeling, P. Giselsson (2020). Operator splitting performance estimation:
Tight contraction factors and optimal parameter selection. SIAM Journal on Optimization, 30(3), 2251-2271.
<https://arxiv.org/pdf/1812.00146.pdf>`_
Args:
L (float): smoothness constant of C.
mu (float): strong convexity of C.
beta (float): cocoercivity of B.
alpha (float): step-size (in the resolvants).
theta (float): overrelaxation parameter.
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 (None): no theoretical value.
Example:
>>> pepit_tau, theoretical_tau = wc_three_operator_splitting(L=1, mu=.1, beta=1, alpha=.9, theta=1.3, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 8x8
(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 3 function(s)
function 1 : Adding 2 scalar constraint(s) ...
function 1 : 2 scalar constraint(s) added
function 2 : Adding 2 scalar constraint(s) ...
function 2 : 2 scalar constraint(s) added
function 3 : Adding 2 scalar constraint(s) ...
function 3 : 2 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); optimal value: 0.7796889999218343
*** Example file: worst-case contraction factor of the Three Operator Splitting ***
PEPit guarantee: ||w_(t+1)^0 - w_(t+1)^1||^2 <= 0.779689 ||w_(t)^0 - w_(t)^1||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a monotone operator
A = problem.declare_function(MonotoneOperator)
B = problem.declare_function(CocoerciveOperator, beta=beta)
C = problem.declare_function(SmoothStronglyConvexFunction, L=L, mu=mu)
# Then define the starting points w0 and w1
w0 = problem.set_initial_point()
w1 = problem.set_initial_point()
# Set the initial constraint that is the distance between w0 and w1
problem.set_initial_condition((w0 - w1) ** 2 <= 1)
# Compute one step of the Three Operator Splitting starting from w0
x0, _, _ = proximal_step(w0, B, alpha)
y0, _, _ = proximal_step(2 * x0 - w0 - alpha * C.gradient(x0), A, alpha)
z0 = w0 - theta * (x0 - y0)
# Compute one step of the Three Operator Splitting starting from w1
x1, _, _ = proximal_step(w1, B, alpha)
y1, _, _ = proximal_step(2 * x1 - w1 - alpha * C.gradient(x1), A, alpha)
z1 = w1 - theta * (x1 - y1)
# Set the performance metric to the distance between z0 and z1
problem.set_performance_metric((z0 - z1) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
theoretical_tau = None
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case contraction factor of the Three Operator Splitting ***')
print('\tPEPit guarantee:\t ||w_(t+1)^0 - w_(t+1)^1||^2 <= {:.6} ||w_(t)^0 - w_(t)^1||^2'.format(pepit_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_three_operator_splitting(L=1, mu=.1, beta=1, alpha=.9, theta=1.3, verbose=1)