from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
from PEPit.primitive_steps import inexact_gradient_step
[docs]def wc_inexact_gradient(L, mu, epsilon, n, verbose=1):
"""
Consider the convex minimization problem
.. math:: f_\\star \\triangleq \min_x f(x),
where :math:`f` is :math:`L`-smooth and :math:`\\mu`-strongly convex.
This code computes a worst-case guarantee for an **inexact gradient method** and looks for a low-dimensional
worst-case example nearly achieving this worst-case guarantee using :math:`10` iterations of the logdet heuristic.
That is, it computes the smallest possible :math:`\\tau(n,L,\\mu,\\varepsilon)` such that the guarantee
.. math:: f(x_n) - f_\\star \\leqslant \\tau(n,L,\\mu,\\varepsilon) (f(x_0) - f_\\star)
is valid, where :math:`x_n` is the output of the gradient descent with an inexact descent direction,
and where :math:`x_\\star` is the minimizer of :math:`f`. Then, it looks for a low-dimensional nearly achieving this
performance.
The inexact descent direction is assumed to satisfy a relative inaccuracy
described by (with :math:`0 \\leqslant \\varepsilon \\leqslant 1`)
.. math:: \|\\nabla f(x_t) - d_t\| \\leqslant \\varepsilon \\|\\nabla f(x_t)\\|,
where :math:`\\nabla f(x_t)` is the true gradient, and :math:`d_t` is the approximate descent direction that is used.
**Algorithm**:
The inexact gradient descent under consideration can be written as
.. math:: x_{t+1} = x_t - \\frac{2}{L_{\\varepsilon} + \\mu_{\\varepsilon}} d_t
where :math:`d_t` is the inexact search direction, :math:`L_{\\varepsilon} = (1 + \\varepsilon)L`
and :math:`\mu_{\\varepsilon} = (1-\\varepsilon) \\mu`.
**Theoretical guarantee**:
A **tight** worst-case guarantee obtained in [1, Theorem 5.3] or [2, Remark 1.6] is
.. math:: f(x_n) - f_\\star \\leqslant \\left(\\frac{L_{\\varepsilon} - \\mu_{\\varepsilon}}{L_{\\varepsilon} + \\mu_{\\varepsilon}}\\right)^{2n}(f(x_0) - f_\\star ),
with :math:`L_{\\varepsilon} = (1 + \\varepsilon)L` and :math:`\mu_{\\varepsilon} = (1-\\varepsilon) \\mu`. This
guarantee is achieved on one-dimensional quadratic functions.
**References**:The detailed analyses can be found in [1, 2]. The logdet heuristic is presented in [3].
`[1] E. De Klerk, F. Glineur, A. Taylor (2020). Worst-case convergence analysis of
inexact gradient and Newton methods through semidefinite programming performance estimation.
SIAM Journal on Optimization, 30(3), 2053-2082.
<https://arxiv.org/pdf/1709.05191.pdf>`_
`[2] O. Gannot (2021). A frequency-domain analysis of inexact gradient methods.
Mathematical Programming (to appear).
<https://arxiv.org/pdf/1912.13494.pdf>`_
`[3] F. Maryam, H. Hindi, S. Boyd (2003). Log-det heuristic for matrix rank minimization with applications to Hankel
and Euclidean distance matrices. American Control Conference (ACC).
<https://web.stanford.edu/~boyd/papers/pdf/rank_min_heur_hankel.pdf>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong convexity parameter.
epsilon (float): level of inaccuracy
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_inexact_gradient(L=1, mu=0.1, epsilon=0.1, n=6, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 15x15
(PEPit) Setting up the problem: performance measure is 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 62 scalar constraint(s) ...
function 1 : 62 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); optimal value: 0.13989778793516514
(PEPit) Postprocessing: 2 eigenvalue(s) > 1.7005395180119392e-05 before dimension reduction
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.1398878008962302
(PEPit) Postprocessing: 2 eigenvalue(s) > 5.283608596989854e-06 after 1 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988778337004493
(PEPit) Postprocessing: 2 eigenvalue(s) > 5.335098252373141e-06 after 2 dimension reduction step(s)
(PEPit) Solver status: optimal (solver: SCS); objective value: 0.1398927512487368
(PEPit) Postprocessing: 2 eigenvalue(s) > 1.2372028101610534e-05 after 3 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988824650439619
(PEPit) Postprocessing: 2 eigenvalue(s) > 2.006867894032787e-05 after 4 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988779568391294
(PEPit) Postprocessing: 2 eigenvalue(s) > 5.416953129163531e-06 after 5 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.1398889451757595
(PEPit) Postprocessing: 2 eigenvalue(s) > 3.983502472713177e-05 after 6 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988780180833413
(PEPit) Postprocessing: 2 eigenvalue(s) > 5.4785759855262395e-06 after 7 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988778218159367
(PEPit) Postprocessing: 2 eigenvalue(s) > 5.360843247635456e-06 after 8 dimension reduction step(s)
(PEPit) Solver status: optimal (solver: SCS); objective value: 0.13988478099895965
(PEPit) Postprocessing: 2 eigenvalue(s) > 9.59529914206238e-06 after 9 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988791535665998
(PEPit) Postprocessing: 2 eigenvalue(s) > 9.339529753603287e-06 after 10 dimension reduction step(s)
(PEPit) Solver status: optimal_inaccurate (solver: SCS); objective value: 0.13988791535665998
(PEPit) Postprocessing: 2 eigenvalue(s) > 9.339529753603287e-06 after dimension reduction
*** Example file: worst-case performance of inexact gradient ***
PEPit example: f(x_n)-f_* == 0.139888 (f(x_0)-f_*)
Theoretical guarantee: f(x_n)-f_* <= 0.139731 (f(x_0)-f_*)
"""
# Instantiate PEP
problem = PEP()
# Declare a strongly convex smooth function
func = problem.declare_function(SmoothStronglyConvexFunction, 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
# as well as corresponding inexact gradient and function value g0 and f0
x0 = problem.set_initial_point()
# Set the initial constraint that is the distance between f0 and f_*
problem.set_initial_condition(func(x0) - fs <= 1)
# Run n steps of the inexact gradient method
Leps = (1 + epsilon) * L
meps = (1 - epsilon) * mu
gamma = 2 / (Leps + meps)
x = x0
for i in range(n):
x, dx, fx = inexact_gradient_step(x, func, gamma=gamma, epsilon=epsilon, notion='relative')
# Set the performance metric to the function value accuracy
problem.set_performance_metric(func(x) - fs)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose, dimension_reduction_heuristic="logdet10")
# Compute theoretical guarantee (for comparison)
theoretical_tau = ((Leps - meps) / (Leps + meps)) ** (2 * n)
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of inexact gradient ***')
print('\tPEPit example:\t\t f(x_n)-f_* == {:.6} (f(x_0)-f_*)'.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_n)-f_* <= {:.6} (f(x_0)-f_*)'.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_inexact_gradient(L=1, mu=0.1, epsilon=0.1, n=6, verbose=1)