import numpy as np
from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexQuadraticFunction
[docs]
def wc_gradient_descent_quadratics(mu, L, gamma, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the convex minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f=\\frac{1}{2} x^T Q x` is :math:`L`-smooth and :math:`\\mu`-strongly convex (i.e. :math:`\\mu I \\preceq Q \\preceq LI`).
This code computes a worst-case guarantee for **gradient descent** with fixed step-size :math:`\\gamma`.
That is, it computes the smallest possible :math:`\\tau(n, \\mu, L, \\gamma)` such that the guarantee
.. math:: f(x_n) - f_\\star \\leqslant \\tau(n, \\mu, L, \\gamma) \\|x_0 - x_\\star\\|^2
is valid, where :math:`x_n` is the output of gradient descent with fixed step-size :math:`\\gamma`, and
where :math:`x_\\star` is a minimizer of :math:`f`.
In short, for given values of :math:`n`, :math:`\\mu`, :math:`L`, and :math:`\\gamma`, :math:`\\tau(n, L, \\gamma)` is computed as the worst-case
value of :math:`f(x_n)-f_\\star` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**:
Gradient descent is described by
.. math:: x_{t+1} = x_t - \\gamma \\nabla f(x_t),
where :math:`\\gamma` is a step-size.
**Theoretical guarantee**:
When :math:`\\gamma \\leqslant \\frac{2}{L}` and :math:`0 \\leqslant \\mu \\leqslant L`,
the **tight** theoretical conjecture can be found in [1, Equation (4.17)]:
.. math:: f(x_n)-f_\\star \\leqslant \\frac{L}{2} \\max\\left\\{\\alpha(1-\\alpha L\\gamma)^{2n}, (1-L\\gamma)^{2n} \\right\\} \\|x_0-x_\\star\\|^2,
where :math:`\\alpha = \\mathrm{proj}_{[\\frac{\\mu}{L},1]} \\left(\\frac{1}{L\\gamma (2n+1)}\\right)`.
**References**:
`[1] N. Bousselmi, J. Hendrickx, F. Glineur (2023).
Interpolation Conditions for Linear Operators and applications to Performance Estimation Problems.
arXiv preprint
<https://arxiv.org/pdf/2302.08781.pdf>`_
Args:
mu (float): the strong convexity parameter.
L (float): the smoothness parameter.
gamma (float): 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 + CVXPY details.
Returns:
pepit_tau (float): worst-case value
theoretical_tau (float): theoretical value
Example:
>>> L = 3.
>>> mu = 0.3
>>> pepit_tau, theoretical_tau = wc_gradient_descent_quadratics(mu=mu, L=L, gamma=1 / L, n=4, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 7x7
(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 21 scalar constraint(s) ...
Function 1 : 21 scalar constraint(s) added
Function 1 : Adding 1 lmi constraint(s) ...
Size of PSD matrix 1: 6x6
Function 1 : 1 lmi 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: 0.0649573836866313
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 6.909122375079227e-10
All required PSD matrices are indeed positive semi-definite up to an error of 8.456976505538092e-09
All the primal scalar constraints are verified up to an error of 8.495850689627105e-10
(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 2.5432361579460934
(PEPit) Final upper bound (dual): 0.06495738812047232 and lower bound (primal example): 0.0649573836866313
(PEPit) Duality gap: absolute: 4.4338410165600806e-09 and relative: 6.825769088776581e-08
*** Example file: worst-case performance of gradient descent on quadratics with fixed step-sizes ***
PEPit guarantee: f(x_n)-f_* <= 0.0649574 ||x_0 - x_*||^2
Theoretical guarantee: f(x_n)-f_* <= 0.0649574 ||x_0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a strongly convex smooth function
func = problem.declare_function(SmoothStronglyConvexQuadraticFunction, 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
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 GD method
x = x0
for _ in range(n):
x = x - gamma * func.gradient(x)
# Set the performance metric to the function values accuracy
problem.set_performance_metric(func(x) - fs)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
t = 1 / (L * gamma * (2 * n + 1))
if t < mu / L:
alpha = mu / L
elif t > 1:
alpha = 1
else:
alpha = t
theoretical_tau = 0.5 * L * np.max((alpha * (1 - alpha * L * gamma) ** (2 * n), (1 - L * gamma) ** (2 * n)))
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of gradient descent on quadratics with fixed step-sizes ***')
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__":
L = 3
mu = 0.3
pepit_tau, theoretical_tau = wc_gradient_descent_quadratics(mu=mu, L=L, gamma=1 / L, n=4,
wrapper="cvxpy", solver=None,
verbose=1)