from math import sqrt
from PEPit import PEP
from PEPit.functions import ConvexLipschitzFunction
[docs]
def wc_subgradient_method(M, n, gamma, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f` is convex and :math:`M`-Lipschitz. This problem is a (possibly non-smooth) minimization problem.
This code computes a worst-case guarantee for the **subgradient method**. That is, it computes
the smallest possible :math:`\\tau(n, M, \\gamma)` such that the guarantee
.. math:: \\min_{0 \leqslant t \leqslant n} f(x_t) - f_\\star \\leqslant \\tau(n, M, \\gamma)
is valid, where :math:`x_t` are the iterates of the **subgradient method** after :math:`t\\leqslant n` steps,
where :math:`x_\\star` is a minimizer of :math:`f`, and when :math:`\\|x_0-x_\\star\\|\\leqslant 1`.
In short, for given values of :math:`M`, the step-size :math:`\\gamma` and the number of iterations :math:`n`,
:math:`\\tau(n, M, \\gamma)` is computed as the worst-case value of
:math:`\\min_{0 \leqslant t \leqslant n} f(x_t) - f_\\star` when :math:`\\|x_0-x_\\star\\| \\leqslant 1`.
**Algorithm**:
For :math:`t\\in \\{0, \\dots, n-1 \\}`
.. math::
:nowrap:
\\begin{eqnarray}
g_{t} & \\in & \\partial f(x_t) \\\\
x_{t+1} & = & x_t - \\gamma g_t
\\end{eqnarray}
**Theoretical guarantee**: The **tight** bound is obtained in [1, Section 3.2.3] and [2, Eq (2)]
.. math:: \\min_{0 \\leqslant t \\leqslant n} f(x_t)- f(x_\\star) \\leqslant \\frac{M}{\\sqrt{n+1}}\|x_0-x_\\star\|,
and tightness follows from the lower complexity bound for this class of problems, e.g., [3, Appendix A].
**References**: Classical references on this topic include [1, 2].
`[1] Y. Nesterov (2003).
Introductory lectures on convex optimization: A basic course.
Springer Science & Business Media.
<https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.693.855&rep=rep1&type=pdf>`_
`[2] S. Boyd, L. Xiao, A. Mutapcic (2003).
Subgradient Methods (lecture notes).
<https://web.stanford.edu/class/ee392o/subgrad_method.pdf>`_
`[3] Y. Drori, M. Teboulle (2016).
An optimal variant of Kelley's cutting-plane method.
Mathematical Programming, 160(1), 321-351.
<https://arxiv.org/pdf/1409.2636.pdf>`_
Args:
M (float): the Lipschitz parameter.
n (int): the number of iterations.
gamma (float): step-size.
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:
>>> M = 2
>>> n = 6
>>> gamma = 1 / (M * sqrt(n + 1))
>>> pepit_tau, theoretical_tau = wc_subgradient_method(M=M, n=n, gamma=gamma, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 9x9
(PEPit) Setting up the problem: performance measure is the minimum of 7 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 64 scalar constraint(s) ...
Function 1 : 64 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: 0.7559287513714278
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite
All the primal scalar constraints are verified
(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 1.0475429120359347e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 8.251581484359109e-08
(PEPit) Final upper bound (dual): 0.7559287543574007 and lower bound (primal example): 0.7559287513714278
(PEPit) Duality gap: absolute: 2.9859729133718815e-09 and relative: 3.950071892297578e-09
*** Example file: worst-case performance of subgradient method ***
PEPit guarantee: min_(0 \leq t \leq n) f(x_i) - f_* <= 0.755929 ||x_0 - x_*||
Theoretical guarantee: min_(0 \leq t \leq n) f(x_i) - f_* <= 0.755929 ||x_0 - x_*||
"""
# Instantiate PEP
problem = PEP()
# Declare a convex lipschitz function
func = problem.declare_function(ConvexLipschitzFunction, M=M)
# 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
x0 = problem.set_initial_point()
# Set the initial constraint that is the distance between x0 and xs
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
# Run n steps of the subgradient method
x = x0
gx, fx = func.oracle(x)
for _ in range(n):
problem.set_performance_metric(fx - fs)
x = x - gamma * gx
gx, fx = func.oracle(x)
# 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 = M / sqrt(n + 1)
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of subgradient method ***')
print('\tPEPit guarantee:\t min_(0 \leq t \leq n) f(x_i) - f_* <= {:.6} ||x_0 - x_*||'.format(pepit_tau))
print('\tTheoretical guarantee:\t min_(0 \leq t \leq n) f(x_i) - f_* <= {:.6} ||x_0 - 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__":
M = 2
n = 6
gamma = 1 / (M * sqrt(n + 1))
pepit_tau, theoretical_tau = wc_subgradient_method(M=M, n=n, gamma=gamma, wrapper="cvxpy", solver=None, verbose=1)