from PEPit import PEP
from PEPit.functions.convex_qg_function import ConvexQGFunction
from PEPit.primitive_steps import exact_linesearch_step
[docs]
def wc_conjugate_gradient_qg_convex(L, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the convex minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f` is quadratically upper bounded (:math:`\\text{QG}^+` [2]), i.e.
:math:`\\forall x, f(x) - f_\\star \\leqslant \\frac{L}{2} \\|x-x_\\star\\|^2`, and convex.
This code computes a worst-case guarantee for the **conjugate gradient (CG)** method (with exact span searches).
That is, it computes the smallest possible :math:`\\tau(n, L)` such that the guarantee
.. math:: f(x_n) - f_\\star \\leqslant \\tau(n, L) \\|x_0-x_\\star\\|^2
is valid, where :math:`x_n` is the output of the **conjugate 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_\\star` when :math:`\\|x_0-x_\\star\\|^2 \\leqslant 1`.
**Algorithm**:
.. math:: x_{t+1} = x_t - \\sum_{i=0}^t \\gamma_i \\nabla f(x_i)
with
.. math:: (\\gamma_i)_{i \\leqslant t} = \\arg\\min_{(\\gamma_i)_{i \\leqslant t}} f \\left(x_t - \\sum_{i=0}^t \\gamma_i \\nabla f(x_i) \\right)
**Theoretical guarantee**:
The **tight** guarantee obtained in [2, Theorem 2.3] (lower) and [2, Theorem 2.4] (upper) is
.. math:: f(x_n) - f_\\star \\leqslant \\frac{L}{2 (n + 1)} \\|x_0-x_\\star\\|^2.
**References**:
The detailed approach (based on convex relaxations) is available in [1, Corollary 6],
and the result provided in [2, Theorem 2.4].
`[1] Y. Drori and A. Taylor (2020). Efficient first-order methods for convex minimization: a constructive approach.
Mathematical Programming 184 (1), 183-220.
<https://arxiv.org/pdf/1803.05676.pdf>`_
`[2] B. Goujaud, A. Taylor, A. Dieuleveut (2022).
Optimal first-order methods for convex functions with a quadratic upper bound.
<https://arxiv.org/pdf/2205.15033.pdf>`_
Args:
L (float): the quadratic growth parameter.
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_conjugate_gradient_qg_convex(L=1, n=12, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 27x27
(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 195 scalar constraint(s) ...
Function 1 : 195 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 1 function(s)
Function 1 : Adding 156 scalar constraint(s) ...
Function 1 : 156 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.038461537777152104
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 9.102635033370141e-10
All the primal scalar constraints are verified up to an error of 6.985771551504261e-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 6.487940056145134e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 2.3524476901692115e-07
(PEPit) Final upper bound (dual): 0.038461545907145095 and lower bound (primal example): 0.038461537777152104
(PEPit) Duality gap: absolute: 8.129992991323665e-09 and relative: 2.113798215357174e-07
*** Example file: worst-case performance of conjugate gradient method ***
PEPit guarantee: f(x_n)-f_* <= 0.0384615 ||x_0 - x_*||^2
Theoretical guarantee: f(x_n)-f_* <= 0.0384615 ||x_0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex function
func = problem.declare_function(ConvexQGFunction, L=L)
# Start by defining its unique optimal point xs = x_* and corresponding function value fs = f_*
xs = func.stationary_point()
fs = func.value(xs)
# Then define the starting point x0 of the algorithm
x0 = problem.set_initial_point()
# Set the initial constraint that is the distance between x0 and x_*
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
# Run n steps of the Conjugate Gradient method
x_new = x0
g0, f0 = func.oracle(x0)
span = [g0] # list of search directions
for i in range(n):
x_old = x_new
x_new, gx, fx = exact_linesearch_step(x_new, func, span)
span.append(gx)
span.append(x_old - x_new)
# Set the performance metric to the function value accuracy
problem.set_performance_metric(fx - 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 = L / (2 * (n + 1))
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of conjugate gradient method ***')
print('\tPEPit guarantee:\t f(x_n)-f_* <= {:.6} ||x_0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_n)-f_* <= {:.6} ||x_0 - 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_conjugate_gradient_qg_convex(L=1, n=12, wrapper="cvxpy", solver=None, verbose=1)