from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
from PEPit.functions import ConvexFunction
from PEPit.primitive_steps import proximal_step
[docs]def wc_accelerated_proximal_gradient(mu, L, n, verbose=1):
"""
Consider the composite convex minimization problem
.. math:: F_\\star \\triangleq \\min_x \\{F(x) \equiv f(x) + h(x)\\},
where :math:`f` is :math:`L`-smooth and :math:`\\mu`-strongly convex,
and where :math:`h` is closed convex and proper.
This code computes a worst-case guarantee for the **accelerated proximal gradient** method,
also known as **fast proximal gradient (FPGM)** method.
That is, it computes the smallest possible :math:`\\tau(n, L, \\mu)` such that the guarantee
.. math :: F(x_n) - F(x_\\star) \\leqslant \\tau(n, L, \\mu) \\|x_0 - x_\\star\\|^2,
is valid, where :math:`x_n` is the output of the **accelerated proximal gradient** method,
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:`F(x_n) - F(x_\\star)` when :math:`\\|x_0 - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**: Accelerated proximal gradient is described as follows, for :math:`t \in \\{ 0, \\dots, n-1\\}`,
.. math::
:nowrap:
\\begin{eqnarray}
x_{t+1} & = & \\arg\\min_x \\left\\{h(x)+\\frac{L}{2}\|x-\\left(y_{t} - \\frac{1}{L} \\nabla f(y_t)\\right)\\|^2 \\right\\}, \\\\
y_{t+1} & = & x_{t+1} + \\frac{i}{i+3} (x_{t+1} - x_{t}),
\\end{eqnarray}
where :math:`y_{0} = x_0`.
**Theoretical guarantee**: A **tight** (empirical) worst-case guarantee for FPGM is obtained in [1, method FPGM1 in Sec. 4.2.1, Table 1 in sec 4.2.2], for :math:`\\mu=0`:
.. math:: F(x_n) - F_\\star \\leqslant \\frac{2 L}{n^2+5n+2} \\|x_0 - x_\\star\\|^2,
which is attained on simple one-dimensional constrained linear optimization problems.
**References**:
`[1] A. Taylor, J. Hendrickx, F. Glineur (2017). Exact worst-case performance of first-order methods for composite
convex optimization. SIAM Journal on Optimization, 27(3):1283–1313.
<https://arxiv.org/pdf/1512.07516.pdf>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong convexity parameter.
n (int): number of iterations.
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:
>>> pepit_tau, theoretical_tau = wc_accelerated_proximal_gradient(L=1, mu=0, n=4, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 6x6
(PEPit) Setting up the problem: performance measure is minimum of 1 element(s)
(PEPit) Setting up the problem: initial conditions (1 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 2 function(s)
function 1 : 6 constraint(s) added
function 2 : 2 constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
*** Example file: worst-case performance of the Fast Proximal Gradient Method in function values***
PEPit guarantee: f(x_n)-f_* <= 0.0526302 ||x0 - xs||^2
Theoretical guarantee: f(x_n)-f_* <= 0.0526316 ||x0 - xs||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a strongly convex smooth function and a convex function
f = problem.declare_function(SmoothStronglyConvexFunction, mu=mu, 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
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)
# Compute n steps of the accelerated proximal gradient method starting from x0
x_new = x0
y = x0
for i in range(n):
x_old = x_new
x_new, _, hx_new = proximal_step(y - 1 / L * f.gradient(y), h, 1 / L)
y = x_new + i / (i + 3) * (x_new - x_old)
# Set the performance metric to the function value accuracy
problem.set_performance_metric((f(x_new) + hx_new) - Fs)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
if mu == 0:
theoretical_tau = 2 * L / (n ** 2 + 5 * n + 2) # tight, see [2], Table 1 (column 1, line 1)
else:
theoretical_tau = 2 * L / (n ** 2 + 5 * n + 2) # not tight (bound for smooth convex functions)
print('Warning: momentum is tuned for non-strongly convex functions.')
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the Accelerated Proximal Gradient Method in function values***')
print('\tPEPit guarantee:\t f(x_n)-f_* <= {:.6} ||x0 - xs||^2'.format(pepit_tau))
print('\tTheoretical guarantee :\t f(x_n)-f_* <= {:.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_accelerated_proximal_gradient(L=1, mu=0, n=4, verbose=1)