from PEPit import PEP
from PEPit.functions.convex_function import ConvexFunction
[docs]
def wc_gradient_flow_convex(t, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the convex minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f` is convex.
This code computes a worst-case guarantee for a **gradient** flow.
That is, it verifies the following inequality
.. math:: \\frac{d}{dt}\\mathcal{V}(X_t, t) \\leqslant 0,
is valid, where :math:`\\mathcal{V}(X_t, t) = t(f(X_t) - f(x_\\star)) + \\frac{1}{2} \\|X_t - x_\\star\\|^2`,
:math:`X_t` is the output of the **gradient** flow, and where :math:`x_\\star` is the minimizer of :math:`f`.
In short, for given values of :math:`t`, it verifies :math:`\\frac{d}{dt}\\mathcal{V}(X_t, t)\\leqslant 0`.
**Algorithm**:
For :math:`t \\geqslant 0`,
.. math:: \\frac{d}{dt}X_t = -\\nabla f(X_t),
with some initialization :math:`X_{0}\\triangleq x_0`.
**Theoretical guarantee**:
The following **tight** guarantee can be found in [1, p. 7]:
.. math:: \\frac{d}{dt}\\mathcal{V}(X_t, t) \\leqslant 0.
After integrating between :math:`0` and :math:`T`,
.. math:: f(X_T) - f_\\star \\leqslant \\frac{1}{2T}\\|x_0 - x_\\star\\|^2.
The detailed approach using PEPs is available in [2, Theorem 2.3].
**References**:
`[1] W. Su, S. Boyd, E. J. Candès (2016).
A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights.
In the Journal of Machine Learning Research (JMLR).
<https://jmlr.org/papers/volume17/15-084/15-084.pdf>`_
`[2] C. Moucer, A. Taylor, F. Bach (2022).
A systematic approach to Lyapunov analyses of continuous-time models in convex optimization.
<https://arxiv.org/pdf/2205.12772.pdf>`_
Args:
t (float): time step
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_gradient_flow_convex(t=2.5, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 3x3
(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 2 scalar constraint(s) ...
Function 1 : 2 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: 2.1751469748629293e-09
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 1.3103319446316331e-08
All the primal scalar constraints are verified up to an error of 3.2757248828870714e-08
(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 7.927687298250013e-08
(PEPit) Final upper bound (dual): 0.0 and lower bound (primal example): 2.1751469748629293e-09
(PEPit) Duality gap: absolute: -2.1751469748629293e-09 and relative: -1.0
*** Example file: worst-case performance of the gradient flow ***
PEPit guarantee: d/dt V(X_t) <= 0.0
Theoretical guarantee: d/dt V(X_t) <= 0.0
"""
# Instantiate PEP
problem = PEP()
# Declare a convex function
func = problem.declare_function(ConvexFunction)
# Start by defining its unique optimal point xs = x_* and corresponding function value fs = f_*
xs = func.stationary_point()
fs = func.value(xs)
# Then define the starting point xt (considering the derivative of the Lyapunov function)
xt = problem.set_initial_point()
gt, ft = func.oracle(xt)
# Run the gradient flow (and define the derivative of the starting point)
xt_dot = - gt
# Chose the Lyapunov function and compute its derivative
# lyap = t * (ft - fs) + .5 * (xt - xs)**2
lyap_dot = (ft - fs) + t * gt * xt_dot + (xt - xs) * xt_dot
# Set the performance metric to the function value accuracy
problem.set_performance_metric(lyap_dot)
# 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 = 0.
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the gradient flow ***')
print('\tPEPit guarantee:\t d/dt V(X_t) <= {:.6}'.format(pepit_tau))
print('\tTheoretical guarantee:\t d/dt V(X_t) <= {:.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__":
pepit_tau, theoretical_tau = wc_gradient_flow_convex(t=2.5, wrapper="cvxpy", solver=None, verbose=1)