from PEPit import PEP
from PEPit.functions import ConvexFunction
from PEPit.primitive_steps import bregman_proximal_step
[docs]
def wc_bregman_proximal_point(gamma, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the composite convex minimization problem
.. math:: F_\\star \\triangleq \min_x \\{F(x) \equiv f_1(x)+f_2(x) \\}
where :math:`f_1(x)` and :math:`f_2(x)` are closed convex proper functions.
This code computes a worst-case guarantee for **Bregman Proximal Point** method.
That is, it computes the smallest possible :math:`\\tau(n, \\gamma)` such that the guarantee
.. math:: F(x_n) - F(x_\\star) \\leqslant \\tau(n, \gamma) D_{f_1}(x_\\star; x_0)
is valid, where :math:`x_n` is the output of the **Bregman Proximal Point** (BPP) method,
where :math:`x_\\star` is a minimizer of :math:`F`, and when :math:`D_{f_1}` is the Bregman distance
generated by :math:`f_1`.
**Algorithm**: Bregman proximal point is described in [1, Section 2, equation (9)]. For :math:`t \\in \\{0, \\dots, n-1\\}`,
.. math::
:nowrap:
\\begin{eqnarray}
x_{t+1} & = & \\arg\\min_{u \\in R^n} f_1(u) + \\frac{1}{\\gamma} D_{f_2}(u; x_t), \\\\
D_h(x; y) & = & h(x) - h(y) - \\nabla h (y)^T(x - y).
\\end{eqnarray}
**Theoretical guarantee**: A **tight** empirical guarantee can be guessed from the numerics
.. math:: F(x_n) - F(x_\\star) \\leqslant \\frac{1}{\\gamma n} D_{f_1}(x_\\star, x_0).
**References**:
`[1] Y. Censor, S.A. Zenios (1992).
Proximal minimization algorithm with D-functions.
Journal of Optimization Theory and Applications, 73(3), 451-464.
<https://link.springer.com/content/pdf/10.1007/BF00940051.pdf>`_
Args:
gamma (float): step-size.
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 + solver details.
Returns:
pepit_tau (float): worst-case value.
theoretical_tau (float): theoretical value.
Examples:
>>> pepit_tau, theoretical_tau = wc_bregman_proximal_point(gamma=3, n=5, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 14x14
(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 2 function(s)
Function 1 : Adding 30 scalar constraint(s) ...
Function 1 : 30 scalar constraint(s) added
Function 2 : Adding 42 scalar constraint(s) ...
Function 2 : 42 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: 0.06666666577966435
(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 7.300917023722597e-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 up to an error of 3.627346042650288e-10
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.1600769917201519e-08
(PEPit) Final upper bound (dual): 0.06666666638907502 and lower bound (primal example): 0.06666666577966435
(PEPit) Duality gap: absolute: 6.094106747012162e-10 and relative: 9.1411602421417e-09
*** Example file: worst-case performance of the Bregman Proximal Point in function values ***
PEPit guarantee: F(x_n)-F_* <= 0.0666667 Dh(x_*; x_0)
Theoretical guarantee: F(x_n)-F_* <= 0.0666667 Dh(x_*; x_0)
"""
# Instantiate PEP
problem = PEP()
# Declare three convex functions
func1 = problem.declare_function(ConvexFunction)
func2 = problem.declare_function(ConvexFunction)
# Start by defining its unique optimal point xs = x_* and its function value fs = F(x_*)
xs = func1.stationary_point()
fs = func1(xs)
gf2s, f2s = func2.oracle(xs)
# Then define the starting point x0 of the algorithm and its function value f0
x0 = problem.set_initial_point()
gf20, f20 = func2.oracle(x0)
# Set the initial constraint that is the Bregman distance between x0 and x^*
problem.set_initial_condition(f2s - f20 - gf20 * (xs - x0) <= 1)
# Compute n steps of the Bregman Proximal Point method starting from x0
gf2 = gf20
for i in range(n):
x, gf2, f2x, f1x, f1 = bregman_proximal_step(gf2, func2, func1, gamma)
# Set the performance metric to the final distance in function values to optimum
problem.set_performance_metric(f1 - fs)
# 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 / (gamma * n)
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the Bregman Proximal Point in function values ***')
print('\tPEPit guarantee:\t F(x_n)-F_* <= {:.6} Dh(x_*; x_0)'.format(pepit_tau))
print('\tTheoretical guarantee:\t F(x_n)-F_* <= {:.6} Dh(x_*; x_0)'.format(theoretical_tau))
# Return the worst-case guarantee of the evaluated method (and the upper theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
pepit_tau, theoretical_tau = wc_bregman_proximal_point(gamma=3, n=5, wrapper="cvxpy", solver=None, verbose=1)