from PEPit import PEP
from PEPit.functions import ConvexFunction
from PEPit.functions import SmoothStronglyConvexQuadraticFunction
from PEPit.primitive_steps import proximal_step
[docs]
def wc_proximal_gradient_quadratics(L, mu, gamma, 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, :math:`\\mu`-strongly convex and quadratic,
and where :math:`f_2` is closed convex and proper.
This code computes a worst-case guarantee for the **proximal gradient** method (PGM).
That is, it computes the smallest possible :math:`\\tau(n, L, \\mu)` such that the guarantee
.. math :: \\|x_n - x_\\star\\|^2 \\leqslant \\tau(n, L, \\mu) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_n` is the output of the **proximal gradient**,
and where :math:`x_\\star` is a minimizer of :math:`F`.
In short, for given values of :math:`n`, :math:`L` and :math:`\\mu`,
:math:`\\tau(n, L, \\mu)` is computed as the worst-case value of
:math:`\\|x_n - x_\\star\\|^2` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: Proximal gradient is described by
.. math::
\\begin{eqnarray}
y_t & = & x_t - \\gamma \\nabla f_1(x_t), \\\\
x_{t+1} & = & \\arg\\min_x \\left\\{f_2(x)+\\frac{1}{2\gamma}\|x-y_t\|^2 \\right\\},
\\end{eqnarray}
for :math:`t \in \\{ 0, \\dots, n-1\\}` and where :math:`\\gamma` is a step-size.
**Theoretical guarantee**: It is well known that a **tight** guarantee for PGM is provided by
.. math :: \\|x_n - x_\\star\\|^2 \\leqslant \\max\\{(1-L\\gamma)^2,(1-\\mu\\gamma)^2\\}^n \\|x_0 - x_\\star\\|^2,
which can be found in, e.g., [1, Theorem 3.1]. It is a folk knowledge and the result can be found in many references
for gradient descent; see, e.g.,[2, Section 1.4: Theorem 3], [3, Section 5.1] and [4, Section 4.4].
**References**:
`[1] A. Taylor, J. Hendrickx, F. Glineur (2018).
Exact worst-case convergence rates of the proximal gradient method for composite convex minimization.
Journal of Optimization Theory and Applications, 178(2), 455-476.
<https://arxiv.org/pdf/1705.04398.pdf>`_
`[2] B. Polyak (1987).
Introduction to Optimization.
Optimization Software New York.
<https://www.researchgate.net/profile/Boris-Polyak-2/publication/342978480_Introduction_to_Optimization/links/5f1033e5299bf1e548ba4636/Introduction-to-Optimization.pdf>`_
`[3] E. Ryu, S. Boyd (2016).
A primer on monotone operator methods.
Applied and Computational Mathematics 15(1), 3-43.
<https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf>`_
`[4] L. Lessard, B. Recht, A. Packard (2016).
Analysis and design of optimization algorithms via integral quadratic constraints.
SIAM Journal on Optimization 26(1), 57–95.
<https://arxiv.org/pdf/1408.3595.pdf>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong convexity parameter.
gamma (float): proximal step-size.
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_proximal_gradient_quadratics(L=1, mu=.1, gamma=1, n=2, 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 2 function(s)
Function 1 : Adding 10 scalar constraint(s) ...
Function 1 : 10 scalar constraint(s) added
Function 1 : Adding 1 lmi constraint(s) ...
Size of PSD matrix 1: 4x4
Function 1 : 1 lmi constraint(s) added
Function 2 : Adding 6 scalar constraint(s) ...
Function 2 : 6 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: 0.6561000114872446
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 1.796118427878115e-09
All required PSD matrices are indeed positive semi-definite up to an error of 4.525137966284814e-09
All the primal scalar constraints are verified up to an error of 1.511348379779065e-09
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual matrices to lmi are 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 0.548909470924843
(PEPit) Final upper bound (dual): 0.6561000103069384 and lower bound (primal example): 0.6561000114872446
(PEPit) Duality gap: absolute: -1.180306186121527e-09 and relative: -1.798972969755044e-09
*** Example file: worst-case performance of the Proximal Gradient Method in function values***
PEPit guarantee: ||x_n - x_*||^2 <= 0.6561 ||x0 - xs||^2
Theoretical guarantee: ||x_n - x_*||^2 <= 0.6561 ||x0 - xs||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a strongly convex smooth function and a closed convex proper function
f1 = problem.declare_function(SmoothStronglyConvexQuadraticFunction, mu=mu, L=L)
f2 = problem.declare_function(ConvexFunction)
func = f1 + f2
# Start by defining its unique optimal point xs = x_*
xs = func.stationary_point()
# 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 the proximal gradient method starting from x0
x = x0
for _ in range(n):
y = x - gamma * f1.gradient(x)
x, _, _ = proximal_step(y, f2, gamma)
# Set the performance metric to the distance between x and xs
problem.set_performance_metric((x - xs) ** 2)
# 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 = max((1 - mu * gamma) ** 2, (1 - L * gamma) ** 2) ** n
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the Proximal Gradient Method in function values***')
print('\tPEPit guarantee:\t ||x_n - x_*||^2 <= {:.6} ||x0 - xs||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t ||x_n - x_*||^2 <= {:.6} ||x0 - xs||^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_proximal_gradient_quadratics(L=1, mu=.1, gamma=1, n=2,
wrapper="cvxpy", solver=None, verbose=1)