from PEPit import PEP
from PEPit.functions import SmoothConvexFunction
[docs]
def wc_gradient_descent_lyapunov_2(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` is :math:`L`-smooth and convex.
This code verifies a worst-case guarantee for **gradient descent** with fixed step-size :math:`\\gamma`.
That is, it verifies that the Lyapunov (or potential/energy) function
.. math:: V_n \\triangleq (2n + 1) L \\left(f(x_n) - f_\\star\\right) + n(n+2) \\|\\nabla f(x_n)\\|^2 + L^2 \\|x_n - x_\\star\\|^2
is decreasing along all trajectories and all smooth convex function :math:`f` (i.e., in the worst-case):
.. math :: V_{n+1} \\leqslant V_n,
where :math:`x_{n+1}` is obtained from a gradient step from :math:`x_{n}` with fixed step-size :math:`\\gamma=\\frac{1}{L}`.
**Algorithm**: Onte iteration of radient descent is described by
.. math:: x_{n+1} = x_n - \\gamma \\nabla f(x_n),
where :math:`\\gamma` is a step-size.
**Theoretical guarantee**:
The theoretical guarantee can be found in [1, Theorem 3]:
.. math:: V_{n+1} - V_n \\leqslant 0,
when :math:`\\gamma=\\frac{1}{L}`.
**References**: The detailed potential function and SDP approach can be found in [1].
`[1] A. Taylor, F. Bach (2019).
Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions.
Conference on Learning Theory (COLT).
<https://arxiv.org/pdf/1902.00947.pdf>`_
Args:
L (float): the smoothness parameter.
gamma (float): the step-size.
n (int): current iteration number.
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.
Examples:
>>> L = 1
>>> pepit_tau, theoretical_tau = wc_gradient_descent_lyapunov_2(L=L, gamma=1 / L, n=10, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 4x4
(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 (0 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
Function 1 : Adding 6 scalar constraint(s) ...
Function 1 : 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: 4.129020680920803e-09
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 2.3114964936226548e-11
All the primal scalar constraints are verified up to an error of 2.3376856006507296e-11
(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 4.907374995917855e-13
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 7.181842880899949e-09
(PEPit) Final upper bound (dual): 0.0 and lower bound (primal example): 4.129020680920803e-09
(PEPit) Duality gap: absolute: -4.129020680920803e-09 and relative: -1.0
*** Example file: worst-case performance of gradient descent with fixed step size for a given Lyapunov function***
PEPit guarantee: V_(n+1) - V_(n) <= 0.0
Theoretical guarantee: V_(n+1) - V_(n) <= 0.0
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex function
func = problem.declare_function(SmoothConvexFunction, 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 gradient and function value gn and fn
xn = problem.set_initial_point()
gn, fn = func.oracle(xn)
# Run the GD at iteration (n+1)
xnp1 = xn - gamma * gn
gnp1, fnp1 = func.oracle(xnp1)
# Compute the Lyapunov function at iteration n and at iteration n+1
init_lyapunov = (2 * n + 1) * L * (fn - fs) + n * (n + 2) * gn ** 2 + L ** 2 * (xn - xs) ** 2
final_lyapunov = (2 * n + 3) * L * (fnp1 - fs) + (n + 1) * (n + 3) * gnp1 ** 2 + L ** 2 * (xnp1 - xs) ** 2
# Set the performance metric to the difference between the initial and the final Lyapunov
problem.set_performance_metric(final_lyapunov - init_lyapunov)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(wrapper=wrapper, solver=solver, verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
if gamma == 1 / L:
theoretical_tau = 0.
else:
theoretical_tau = None
# Print conclusion if required
if verbose != -1:
print('*** Example file:'
' worst-case performance of gradient descent with fixed step size for a given Lyapunov function***')
print('\tPEPit guarantee:\t'
'V_(n+1) - V_(n) <= {:.6}'.format(pepit_tau))
if gamma == 1 / L:
print('\tTheoretical guarantee:\t'
'V_(n+1) - V_(n) <= {:.6}'.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 = 1
pepit_tau, theoretical_tau = wc_gradient_descent_lyapunov_2(L=L, gamma=1 / L, n=10,
wrapper="cvxpy", solver=None,
verbose=1)