import numpy as np
from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
[docs]
def wc_randomized_coordinate_descent_smooth_strongly_convex(L, mu, gamma, d, 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 :math:`\\mu`-strongly convex.
This code computes a worst-case guarantee for **randomized block-coordinate descent**
with step-size :math:`\\gamma`.
That is, it computes the smallest possible :math:`\\tau(L, \\mu, \\gamma, d)` such that the guarantee
.. math:: \\mathbb{E}[\\|x_{t+1} - x_\star \\|^2] \\leqslant \\tau(L, \\mu, \\gamma, d) \\|x_t - x_\\star\\|^2
holds for any fixed step-size :math:`\\gamma` and any number of blocks :math:`d`,
and where :math:`x_\\star` denotes a minimizer of :math:`f`. The notation :math:`\\mathbb{E}`
denotes the expectation over the uniform distribution of the index
:math:`i \\sim \\mathcal{U}\\left([|1, n|]\\right)`.
In short, for given values of :math:`\\mu`, :math:`L`, :math:`d`, and :math:`\\gamma`,
:math:`\\tau(L, \\mu, \\gamma, d)` is computed as the worst-case value of
:math:`\\mathbb{E}[\\|x_{t+1} - x_\star \\|^2]` when :math:`\\|x_t - x_\\star\\|^2 \\leqslant 1`.
**Algorithm**:
Randomized block-coordinate descent is described by
.. math::
\\begin{eqnarray}
\\text{Pick random }i & \\sim & \\mathcal{U}\\left([|1, d|]\\right), \\\\
x_{t+1} & = & x_t - \\gamma \\nabla_i f(x_t),
\\end{eqnarray}
where :math:`\\gamma` is a step-size and :math:`\\nabla_i f(x_t)` is the :math:`i^{\\text{th}}` partial gradient.
**Theoretical guarantee**:
When :math:`\\gamma \\leqslant \\frac{1}{L}`, the **tight** theoretical guarantee
can be found in [1, Appendix I, Theorem 17]:
.. math:: \\mathbb{E}[\\|x_{t+1} - x_\star \\|^2] \\leqslant \\rho^2 \\|x_t-x_\\star\\|^2,
where :math:`\\rho^2 = \\max \\left( \\frac{(\\gamma\\mu - 1)^2 + d - 1}{d},\\frac{(\\gamma L - 1)^2 + d - 1}{d} \\right)`.
**References**:
`[1] A. Taylor, F. Bach (2019). Stochastic first-order methods: non-asymptotic and computer-aided
analyses via potential functions. In Conference on Learning Theory (COLT).
<https://arxiv.org/pdf/1902.00947.pdf>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong-convexity parameter.
gamma (float): the step-size.
d (int): the dimension.
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:
>>> L = 1
>>> mu = 0.1
>>> gamma = 2 / (mu + L)
>>> pepit_tau, theoretical_tau = wc_randomized_coordinate_descent_smooth_strongly_convex(L=L, mu=mu, gamma=gamma, d=2, 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 (1 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) Setting up the problem: 1 partition(s) added
Partition 1 with 2 blocks: Adding 1 scalar constraint(s)...
Partition 1 with 2 blocks: 1 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.8347107438584297
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite
All the primal scalar constraints are verified up to an error of 1.4183154650737606e-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
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 4.950747594358786e-10
(PEPit) Final upper bound (dual): 0.8347107438665666 and lower bound (primal example): 0.8347107438584297
(PEPit) Duality gap: absolute: 8.136935569780235e-12 and relative: 9.748209939370677e-12
*** Example file: worst-case performance of randomized coordinate gradient descent ***
PEPit guarantee: E[||x_(t+1) - x_*||^2] <= 0.834711 ||x_t - x_*||^2
Theoretical guarantee: E[||x_(t+1) - x_*||^2] <= 0.834711 ||x_t - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a partition of the ambient space in d blocks of variables
partition = problem.declare_block_partition(d=d)
# Declare a strongly convex smooth function
func = problem.declare_function(SmoothStronglyConvexFunction, mu=mu, L=L)
# Start by defining its unique optimal point xs = x_*
xs = func.stationary_point()
# 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)
# Compute all the possible outcomes of the randomized coordinate descent step
g0 = func.gradient(x0)
x1_list = [x0 - gamma * partition.get_block(g0, i) for i in range(d)]
# Set the performance metric to the expected value of the distance to optimiser from the different possible outcomes
problem.set_performance_metric(np.mean([(x1 - xs) ** 2 for x1 in x1_list]))
# 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 = max(((mu * gamma - 1) ** 2 + d - 1) / d, ((L * gamma - 1) ** 2 + d - 1) / d)
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of randomized coordinate gradient descent ***')
print('\tPEPit guarantee:\t E[||x_(t+1) - x_*||^2] <= {:.6} ||x_t - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t E[||x_(t+1) - x_*||^2] <= {:.6} ||x_t - x_*||^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__":
L = 1
mu = 0.1
gamma = 2 / (mu + L)
pepit_tau, theoretical_tau = wc_randomized_coordinate_descent_smooth_strongly_convex(L=L, mu=mu, gamma=gamma, d=2,
wrapper="cvxpy", solver=None,
verbose=1)