from math import sqrt
from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
[docs]
def wc_heavy_ball_momentum(mu, L, alpha, beta, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the convex minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f` is :math:`L`-smooth and :math:`\\mu`-strongly 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, \\mu, \\alpha, \\beta)` such that the guarantee
.. math:: f(x_n) - f_\\star \\leqslant \\tau(n, L, \\mu, \\alpha, \\beta) (f(x_0) - f_\\star)
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`, :math:`L` and :math:`\\mu`,
:math:`\\tau(n, L, \\mu, \\alpha, \\beta)` is computed as the worst-case value of
:math:`f(x_n)-f_\\star` when :math:`f(x_0) - f_\\star \\leqslant 1`.
**Algorithm**:
.. math:: x_{t+1} = x_t - \\alpha \\nabla f(x_t) + \\beta (x_t-x_{t-1})
with
.. math:: \\alpha \\in (0, \\frac{1}{L}]
and
.. math:: \\beta = \\sqrt{(1 - \\alpha \\mu)(1 - L \\alpha)}
**Theoretical guarantee**:
The **upper** guarantee obtained in [2, Theorem 4] is
.. math:: f(x_n) - f_\\star \\leqslant (1 - \\alpha \\mu)^n (f(x_0) - f_\\star).
**References**: This methods was first introduce in [1, Section 2],
and convergence upper bound was proven in [2, Theorem 4].
`[1] B.T. Polyak (1964).
Some methods of speeding up the convergence of iteration method.
URSS Computational Mathematics and Mathematical Physics.
<https://www.sciencedirect.com/science/article/pii/0041555364901375>`_
`[2] 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>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong convexity parameter.
alpha (float): parameter of the scheme.
beta (float): parameter of the scheme such that :math:`0<\\beta<1` and :math:`0<\\alpha<2(1+\\beta)`.
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:
>>> mu = 0.1
>>> L = 1.
>>> alpha = 1 / (2 * L) # alpha \in [0, 1 / L]
>>> beta = sqrt((1 - alpha * mu) * (1 - L * alpha))
>>> pepit_tau, theoretical_tau = wc_heavy_ball_momentum(mu=mu, L=L, alpha=alpha, beta=beta, n=2, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 5x5
(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 12 scalar constraint(s) ...
Function 1 : 12 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: 0.7534930184723507
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 8.447704542447025e-09
All the primal scalar constraints are verified up to an error of 2.3525640133886805e-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
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.1216875770062036e-07
(PEPit) Final upper bound (dual): 0.7534930169804492 and lower bound (primal example): 0.7534930184723507
(PEPit) Duality gap: absolute: -1.4919014912351258e-09 and relative: -1.979980510316926e-09
*** Example file: worst-case performance of the Heavy-Ball method ***
PEPit guarantee: f(x_n)-f_* <= 0.753493 (f(x_0) - f(x_*))
Theoretical guarantee: f(x_n)-f_* <= 0.9025 (f(x_0) - f(x_*))
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth strongly convex function
func = problem.declare_function(SmoothStronglyConvexFunction, mu=mu, L=L)
# Start by defining its unique optimal point xs = x_* and corresponding function value fs = f_*
xs = func.stationary_point()
fs = func(xs)
# Then define the starting point x0 of the algorithm as well as corresponding function value f0
x0 = problem.set_initial_point()
f0 = func(x0)
# Set the initial constraint that is the distance between f(x0) and f(x^*)
problem.set_initial_condition((f0 - fs) <= 1)
# Run one step of the heavy ball method
x_new = x0
x_old = x0
for _ in range(n):
x_next = x_new - alpha * func.gradient(x_new) + beta * (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(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 = (1 - alpha * mu) ** n
# 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} (f(x_0) - f(x_*))'.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_n)-f_* <= {:.6} (f(x_0) - f(x_*))'.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__":
mu = 0.1
L = 1.
alpha = 1 / (2 * L) # alpha \in [0, 1 / L]
beta = sqrt((1 - alpha * mu) * (1 - L * alpha))
pepit_tau, theoretical_tau = wc_heavy_ball_momentum(mu=mu, L=L, alpha=alpha, beta=beta, n=2,
wrapper="cvxpy", solver=None,
verbose=1)