from math import sqrt
from PEPit import PEP
from PEPit.functions import ConvexFunction
from PEPit.functions import SmoothConvexFunction
from PEPit.primitive_steps import inexact_proximal_step
[docs]
def wc_accelerated_inexact_forward_backward(L, zeta, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the composite convex minimization problem,
.. math:: F_\\star \\triangleq \\min_x \\left\\{F(x) \\equiv f(x) + g(x) \\right\\},
where :math:`f` is :math:`L`-smooth convex, and :math:`g` is closed, proper, and convex.
We further assume that one can readily evaluate the gradient of :math:`f` and that one has access to an inexact
version of the proximal operator of :math:`g` (whose level of accuracy is controlled by some
parameter :math:`\\zeta\\in (0,1)`).
This code computes a worst-case guarantee for an **accelerated inexact forward backward** (AIFB) method (a.k.a.,
inexact accelerated proximal gradient method). That is, it computes the smallest possible
:math:`\\tau(n, L, \\zeta)` such that the guarantee
.. math :: F(x_n) - F(x_\\star) \\leqslant \\tau(n, L, \\zeta) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_n` is the output of the IAFB, and where :math:`x_\\star` is a minimizer of :math:`F`.
In short, for given values of :math:`n`, :math:`L` and :math:`\\zeta`, :math:`\\tau(n, L, \\zeta)` is computed as
the worst-case value of :math:`F(x_n) - F(x_\\star)` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: Let :math:`t\\in\\{0,1,\\ldots,n\\}`. The method is presented in, e.g., [1, Algorithm 3.1].
For simplicity, we instantiate [1, Algorithm 3.1] using simple values for its parameters and for the problem
setting (in the notation of [1]: :math:`A_0\\triangleq 0`, :math:`\\mu=0`, :math:`\\xi_t \\triangleq0`,
:math:`\\sigma_t\\triangleq 0`, :math:`\\lambda_t \\triangleq\\gamma\\triangleq\\tfrac{1}{L}`,
:math:`\\zeta_t\\triangleq\\zeta`, :math:`\\eta \\triangleq (1-\\zeta^2) \\gamma`), and without backtracking,
arriving to:
.. math::
:nowrap:
\\begin{eqnarray}
A_{t+1} && = A_t + \\frac{\\eta+\\sqrt{\\eta^2+4\\eta A_t}}{2},\\\\
y_{t} && = x_t + \\frac{A_{t+1}-A_t}{A_{t+1}} (z_t-x_t),\\\\
(x_{t+1},v_{t+1}) && \\approx_{\\varepsilon_t} \\left(\\mathrm{prox}_{\\gamma g}\\left(y_t-\\gamma \\nabla f(y_t)\\right),\,
\\mathrm{prox}_{ g^*/\\gamma}\\left(\\frac{y_t-\\gamma \\nabla f(y_t)}{\\gamma}\\right)\\right),\\\\
&& \\text{with } \\varepsilon_t = \\frac{\\zeta^2\\gamma^2}{2}\|v_{t+1}+\\nabla f(y_t) \|^2,\\\\
z_{t+1} && = z_t-(A_{t+1}-A_t)\\left(v_{t+1}+\\nabla f(y_t)\\right),\\\\
\\end{eqnarray}
where :math:`\\{\\varepsilon_t\\}_{t\\geqslant 0}` is some sequence of accuracy parameters (whose values are fixed
within the algorithm as it runs), and :math:`\\{A_t\\}_{t\\geqslant 0}` is some scalar sequence of parameters
for the method (typical of accelerated methods).
The line with ":math:`\\approx_{\\varepsilon}`" can be described as the pair :math:`(x_{t+1},v_{t+1})` satisfying
an accuracy requirement provided by [1, Definition 2.3]. More precisely (but without providing any intuition),
it requires the existence of some :math:`w_{t+1}` such that :math:`v_{t+1} \\in \\partial g(w_{t+1})`
and for which the accuracy requirement
.. math:: \\gamma^2 || x_{t+1} - y_t + \\gamma v_{t+1} ||^2 + \\gamma (g(x_{t+1}) - g(w_{t+1}) - v_{t+1}(x_{t+1} - w_{t+1})) \\leqslant \\varepsilon_t,
is valid.
**Theoretical guarantee**: A theoretical upper bound is obtained in [1, Corollary 3.5]:
.. math:: F(x_n)-F_\\star\\leqslant \\frac{2L \\|x_0-x_\\star\\|^2}{(1-\\zeta^2)n^2}.
**References**: The method and theoretical result can be found in [1, Section 3].
`[1] M. Barre, A. Taylor, F. Bach (2021).
A note on approximate accelerated forward-backward methods with
absolute and relative errors, and possibly strongly convex objectives.
arXiv:2106.15536v2.
<https://arxiv.org/pdf/2106.15536v2.pdf>`_
Args:
L (float): smoothness parameter.
zeta (float): relative approximation parameter in (0,1).
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_accelerated_inexact_forward_backward(L=1.3, zeta=.45, n=11, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 59x59
(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 156 scalar constraint(s) ...
Function 1 : 156 scalar constraint(s) added
Function 2 : Adding 506 scalar constraint(s) ...
Function 2 : 506 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 1 function(s)
Function 1 : Adding 22 scalar constraint(s) ...
Function 1 : 22 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.018734101450651804
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 6.169436183689734e-09
All the primal scalar constraints are verified up to an error of 2.3055138501440475e-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 up to an error of 3.9318808809398555e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.038962372016442e-06
(PEPit) Final upper bound (dual): 0.018734107018754872 and lower bound (primal example): 0.018734101450651804
(PEPit) Duality gap: absolute: 5.5681030688981e-09 and relative: 2.9721751446501176e-07
*** Example file: worst-case performance of an inexact accelerated forward backward method ***
PEPit guarantee: F(x_n)-F_* <= 0.0187341 ||x_0 - x_*||^2
Theoretical guarantee: F(x_n)-F_* <= 0.0269437 ||x_0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex and a convex function
f = problem.declare_function(SmoothConvexFunction, L=L)
h = problem.declare_function(ConvexFunction)
F = f + h
# Start by defining its unique optimal point xs = x_* and its function value Fs = F(x_*)
xs = F.stationary_point()
Fs = F(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)
# Some algorithmic parameters (for convenience)
gamma = 1 / L
eta = (1 - zeta ** 2) * gamma
A = [0]
# Compute n steps of the IAFB method starting from x0
x = x0
z = x0
for i in range(n):
A.append(A[i] + (eta + sqrt(eta ** 2 + 4 * eta * A[i])) / 2)
y = x + (1 - A[i] / A[i + 1]) * (z - x)
gy = f.gradient(y)
x, sx, hx, _, vx, _, epsVar = inexact_proximal_step(y - gamma * gy, h, gamma, opt='PD_gapI')
h.add_constraint(epsVar <= (zeta * gamma) ** 2 / 2 * (vx + gy) ** 2) # this is the accuracy requirement
z = z - (A[i + 1] - A[i]) * (vx + gy)
# Set the performance metric to the function value accuracy
problem.set_performance_metric((f(x) + hx) - 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 = 2 * L / (1 - zeta ** 2) / n ** 2
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of an inexact accelerated forward backward method ***')
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 upper theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
pepit_tau, theoretical_tau = wc_accelerated_inexact_forward_backward(L=1.3, zeta=.45, n=11,
wrapper="cvxpy", solver=None,
verbose=1)