import numpy as np
from PEPit import PEP
from PEPit.functions import ConvexFunction
from PEPit.functions import ConvexIndicatorFunction
from PEPit.primitive_steps import bregman_gradient_step
[docs]
def wc_no_lips_2(L, gamma, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the constrainted composite convex minimization problem
.. math:: F_\\star \\triangleq \\min_x \\{F(x) \equiv f_1(x)+f_2(x) \\}
where :math:`f_2` is a closed convex indicator function and :math:`f_1` is possibly non-convex,
:math:`L`-smooth relatively to :math:`h`,
and :math:`h` is closed proper and convex.
This code computes a worst-case guarantee for the **NoLips** method.
That is, it computes the smallest possible :math:`\\tau(n,L,\\gamma)` such that the guarantee
.. math:: \\min_{0 \\leqslant t \\leqslant n-1} D_h(x_t;x_{t+1}) \\leqslant \\tau(n, L, \\gamma) (F(x_0) - F(x_n))
is valid, where :math:`x_n` is the output of the **NoLips** method,
and where :math:`D_h` is the Bregman distance generated by :math:`h`:
.. math:: D_h(x; y) \\triangleq h(x) - h(y) - \\nabla h (y)^T(x - y).
In short, for given values of :math:`n`, :math:`L`, and :math:`\\gamma`, :math:`\\tau(n, L, \\gamma)` is computed
as the worst-case value of :math:`\\min_{0 \\leqslant t \\leqslant n-1}D_h(x_t;x_{t+1})` when
:math:`F(x_0) - F(x_n) \\leqslant 1`.
**Algorithms**: This method (also known as Bregman Gradient, or Mirror descent) can be found in,
e.g., [1, Section 3]. For :math:`t \\in \\{0, \\dots, n-1\\}`,
.. math:: x_{t+1} = \\arg\\min_{u \\in R^d} \\nabla f(x_t)^T(u - x_t) + \\frac{1}{\\gamma} D_h(u; x_t).
**Theoretical guarantees**: An empirically **tight** worst-case guarantee is
.. math:: \\min_{0 \\leqslant t \\leqslant n-1}D_h(x_t;x_{t+1}) \\leqslant \\frac{\\gamma}{n}(F(x_0) - F(x_n)).
**References**: The detailed setup is presented in [1]. The PEP approach for studying such settings
is presented in [2].
`[1] J. Bolte, S. Sabach, M. Teboulle, Y. Vaisbourd (2018).
First order methods beyond convexity and Lipschitz gradient continuity
with applications to quadratic inverse problems.
SIAM Journal on Optimization, 28(3), 2131-2151.
<https://arxiv.org/pdf/1706.06461.pdf>`_
`[2] R. Dragomir, A. Taylor, A. d’Aspremont, J. Bolte (2021).
Optimal complexity and certification of Bregman first-order methods.
Mathematical Programming, 1-43.
<https://arxiv.org/pdf/1911.08510.pdf>`_
DISCLAIMER: This example requires some experience with PEPit and PEPs (see Section 4 in [2]).
Args:
L (float): relative-smoothness parameter.
gamma (float): step-size (equal to :math:`\\frac{1}{2L}` for guarantee).
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.
Example:
>>> L = 1
>>> gamma = 1 / L
>>> pepit_tau, theoretical_tau = wc_no_lips_2(L=L, gamma=gamma, n=3, 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 3 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 12 scalar constraint(s) ...
Function 1 : 12 scalar constraint(s) added
Function 2 : Adding 12 scalar constraint(s) ...
Function 2 : 12 scalar constraint(s) added
Function 3 : Adding 25 scalar constraint(s) ...
Function 3 : 25 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: 0.3333333333330493
(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 4.196643033083092e-14
(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 4.199523786044634e-14
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 5.914928381589679e-13
(PEPit) Final upper bound (dual): 0.3333333333330737 and lower bound (primal example): 0.3333333333330493
(PEPit) Duality gap: absolute: 2.4369395390522186e-14 and relative: 7.310818617162885e-14
*** Example file: worst-case performance of the NoLips_2 in Bregman distance ***
PEPit guarantee: min_t Dh(x_(t-1), x_(t)) <= 0.333333 (F(x_0) - F(x_n))
Theoretical guarantee: min_t Dh(x_(t-1), x_(t)) <= 0.333333 (F(x_0) - F(x_n))
"""
# Instantiate PEP
problem = PEP()
# Declare two convex functions and a convex function
d1 = problem.declare_function(ConvexFunction, reuse_gradient=True)
d2 = problem.declare_function(ConvexFunction, reuse_gradient=True)
func1 = (d2 - d1) / 2
h = (d1 + d2) / L / 2
func2 = problem.declare_function(ConvexIndicatorFunction, D=np.inf)
# Define the function to optimize as the sum of func1 and func2
func = func1 + func2
# Then define the starting point x0 of the algorithm and its function value f0
x0 = problem.set_initial_point()
gh0, h0 = h.oracle(x0)
gf0, f0 = func1.oracle(x0)
_, F0 = func.oracle(x0)
# Compute n steps of the NoLips starting from x0
x1, x2 = x0, x0
gfx = gf0
ghx = gh0
hx1, hx2 = h0, h0
for i in range(n):
x2, _, _ = bregman_gradient_step(gfx, ghx, func2 + h, gamma)
gfx, _ = func1.oracle(x2)
ghx, hx2 = h.oracle(x2)
Dhx = hx1 - hx2 - ghx * (x1 - x2)
# update the iterates
x1, hx1 = x2, hx2
# Set the performance metric to the Bregman distance to the last iterate
problem.set_performance_metric(Dhx)
_, Fx = func.oracle(x2)
# Set the initial constraint that is the Bregman distance between x0 and x^*
problem.set_initial_condition(F0 - Fx <= 1)
# 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 = gamma / n
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the NoLips_2 in Bregman distance ***')
print('\tPEPit guarantee:\t min_t Dh(x_(t-1), x_(t)) <= {:.6} (F(x_0) - F(x_n))'.format(pepit_tau))
print('\tTheoretical guarantee:\t min_t Dh(x_(t-1), x_(t)) <= {:.6} (F(x_0) - F(x_n))'.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__":
L = 1
gamma = 1 / L
pepit_tau, theoretical_tau = wc_no_lips_2(L=L, gamma=gamma, n=3, wrapper="cvxpy", solver=None, verbose=1)