from PEPit import PEP
from PEPit.functions import ConvexIndicatorFunction
from PEPit.functions import SmoothConvexFunction
from PEPit.primitive_steps import linear_optimization_step
[docs]
def wc_frank_wolfe(L, D, 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` is :math:`L`-smooth and convex
and where :math:`f_2` is a convex indicator function on :math:`\\mathcal{D}` of diameter at most :math:`D`.
This code computes a worst-case guarantee for the **conditional gradient** method, aka **Frank-Wolfe** method.
That is, it computes the smallest possible :math:`\\tau(n, L)` such that the guarantee
.. math :: F(x_n) - F(x_\\star) \\leqslant \\tau(n, L) D^2,
is valid, where x_n is the output of the **conditional gradient** method,
and where :math:`x_\\star` is a minimizer of :math:`F`.
In short, for given values of :math:`n` and :math:`L`, :math:`\\tau(n, L)` is computed as the worst-case value of
:math:`F(x_n) - F(x_\\star)` when :math:`D \\leqslant 1`.
**Algorithm**:
This method was first presented in [1]. A more recent version can be found in, e.g., [2, Algorithm 1].
For :math:`t \\in \\{0, \\dots, n-1\\}`,
.. math::
\\begin{eqnarray}
y_t & = & \\arg\\min_{s \\in \\mathcal{D}} \\langle s \\mid \\nabla f_1(x_t) \\rangle, \\\\
x_{t+1} & = & \\frac{t}{t + 2} x_t + \\frac{2}{t + 2} y_t.
\\end{eqnarray}
**Theoretical guarantee**:
An **upper** guarantee obtained in [2, Theorem 1] is
.. math :: F(x_n) - F(x_\\star) \\leqslant \\frac{2L D^2}{n+2}.
**References**:
[1] M .Frank, P. Wolfe (1956).
An algorithm for quadratic programming.
Naval research logistics quarterly, 3(1-2), 95-110.
`[2] M. Jaggi (2013).
Revisiting Frank-Wolfe: Projection-free sparse convex optimization.
In 30th International Conference on Machine Learning (ICML).
<http://proceedings.mlr.press/v28/jaggi13.pdf>`_
Args:
L (float): the smoothness parameter.
D (float): diameter of :math:`f_2`.
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:
>>> pepit_tau, theoretical_tau = wc_frank_wolfe(L=1, D=1, n=10, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 26x26
(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 (0 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 2 function(s)
Function 1 : Adding 132 scalar constraint(s) ...
Function 1 : 132 scalar constraint(s) added
Function 2 : Adding 325 scalar constraint(s) ...
Function 2 : 325 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: 0.07828953904645822
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 4.140365475126263e-09
All the primal scalar constraints are verified up to an error of 7.758491793463662e-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 up to an error of 3.474580080029191e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.2084335345375351e-07
(PEPit) Final upper bound (dual): 0.07828954284798424 and lower bound (primal example): 0.07828953904645822
(PEPit) Duality gap: absolute: 3.801526024527213e-09 and relative: 4.855726666459652e-08
*** Example file: worst-case performance of the Conditional Gradient (Frank-Wolfe) in function value ***
PEPit guarantee: f(x_n)-f_* <= 0.0782895 ||x0 - xs||^2
Theoretical guarantee: f(x_n)-f_* <= 0.166667 ||x0 - xs||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex function and a convex indicator of rayon D
func1 = problem.declare_function(function_class=SmoothConvexFunction, L=L)
func2 = problem.declare_function(function_class=ConvexIndicatorFunction, D=D)
# Define the function to optimize as the sum of func1 and func2
func = func1 + func2
# Start by defining its unique optimal point xs = x_* and its function value fs = F(x_*)
xs = func.stationary_point()
fs = func(xs)
# Then define the starting point x0 of the algorithm and its function value f0
x0 = problem.set_initial_point()
# Enforce the feasibility of x0 : there is no initial constraint on x0
_ = func1(x0)
_ = func2(x0)
# Compute n steps of the Conditional Gradient / Frank-Wolfe method starting from x0
x = x0
for i in range(n):
g = func1.gradient(x)
y, _, _ = linear_optimization_step(g, func2)
lam = 2 / (i + 2)
x = (1 - lam) * x + lam * y
# Set the performance metric to the final distance in function values to optimum
problem.set_performance_metric((func(x)) - 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)
# when theta = 1
theoretical_tau = 2 * L * D ** 2 / (n + 2)
# Print conclusion if required
if verbose != -1:
print('*** Example file:'
' worst-case performance of the Conditional Gradient (Frank-Wolfe) in function value ***')
print('\tPEPit guarantee:\t f(x_n)-f_* <= {:.6} ||x0 - xs||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_n)-f_* <= {:.6} ||x0 - xs||^2'.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_frank_wolfe(L=1, D=1, n=10, wrapper="cvxpy", solver=None, verbose=1)