from PEPit import PEP
from PEPit.functions import ConvexQGFunction
[docs]
def wc_heavy_ball_momentum_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 **Heavy-ball (HB)** method, aka **Polyak momentum** method.
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 **Heavy-ball (HB)** method,
and where :math:`x_\\star` is the 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**:
This method is described in [1]
.. math:: x_{t+1} = x_t - \\alpha_t \\nabla f(x_t) + \\beta_t (x_t-x_{t-1})
with
.. math:: \\alpha_t = \\frac{1}{L} \\frac{1}{t+2}
and
.. math:: \\beta_t = \\frac{t}{t+2}
**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}\\frac{1}{n+1} \\|x_0 - x_\\star\\|^2.
**References**:
This methods was first introduce in [1, section 3],
and convergence **tight** bound was proven in [2, Theorem 2.3] (lower) and [2, Theorem 2.4] (upper).
`[1] E. Ghadimi, H. R. Feyzmahdavian, M. Johansson (2015).
Global convergence of the Heavy-ball method for convex optimization.
European Control Conference (ECC).
<https://arxiv.org/pdf/1412.7457.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_heavy_ball_momentum_qg_convex(L=1, n=5, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 8x8
(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 48 scalar constraint(s) ...
Function 1 : 48 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: 0.08333333312648067
(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 1.0360445487633818e-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 9.384283773139436e-11
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 2.3121924686696765e-09
(PEPit) Final upper bound (dual): 0.08333333333812117 and lower bound (primal example): 0.08333333312648067
(PEPit) Duality gap: absolute: 2.1164049679445185e-10 and relative: 2.539685967837512e-09
*** Example file: worst-case performance of the Heavy-Ball method ***
PEPit guarantee: f(x_n)-f_* <= 0.0833333 ||x_0 - x_*||^2
Theoretical guarantee: f(x_n)-f_* <= 0.0833333 ||x_0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth strongly 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 as well as corresponding function value f0
x0 = problem.set_initial_point()
# Set the initial constraint that is the distance between f(x0) and f(x^*)
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
# Run one step of the heavy ball method
x_new = x0
x_old = x0
for t in range(n):
x_next = x_new - 1 / (L * (t + 2)) * func.gradient(x_new) + t / (t + 2) * (x_new - x_old)
x_old = x_new
x_new = x_next
# Set the performance metric to the final distance to optimum
problem.set_performance_metric(func.value(x_new) - 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 the Heavy-Ball 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_heavy_ball_momentum_qg_convex(L=1, n=5, wrapper="cvxpy", solver=None, verbose=1)