from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction
[docs]
def wc_polyak_steps_in_function_value(L, mu, gamma, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the minimization problem
.. math:: f_\\star \\triangleq \\min_x f(x),
where :math:`f` is :math:`L`-smooth and :math:`\\mu`-strongly convex, and :math:`x_\\star=\\arg\\min_x f(x)`.
This code computes a worst-case guarantee for a variant of a **gradient method** relying on **Polyak step-sizes**.
That is, it computes the smallest possible :math:`\\tau(L, \\mu, \\gamma)` such that the guarantee
.. math:: f(x_{t+1}) - f_\\star \\leqslant \\tau(L, \\mu, \\gamma) (f(x_t) - f_\\star)
is valid, where :math:`x_t` is the output of the gradient method with PS and :math:`\\gamma` is the effective value
of the step-size of the gradient method.
In short, for given values of :math:`L`, :math:`\\mu`, and :math:`\\gamma`, :math:`\\tau(L, \\mu, \\gamma)` is
computed as the worst-case value of :math:`f(x_{t+1})-f_\\star` when :math:`f(x_t)-f_\\star \\leqslant 1`.
**Algorithm**:
Gradient descent is described by
.. math:: x_{t+1} = x_t - \\gamma \\nabla f(x_t),
where :math:`\\gamma` is a step-size. The Polyak step-size rule under consideration here corresponds to choosing
of :math:`\\gamma` satisfying:
.. math:: \\|\\nabla f(x_t)\\|^2 = 2 L (2 - L \\gamma) (f(x_t) - f_\\star).
**Theoretical guarantee**:
The gradient method with the variant of Polyak step-sizes under consideration enjoys the
**tight** theoretical guarantee [1, Proposition 2]:
.. math:: f(x_{t+1})-f_\\star \\leqslant \\tau(L,\\mu,\\gamma) (f(x_{t})-f_\\star),
where :math:`\\gamma` is the effective step-size used at iteration :math:`t` and
.. math::
:nowrap:
\\begin{eqnarray}
\\tau(L,\\mu,\\gamma) & = & \\left\\{\\begin{array}{ll} (\\gamma L - 1) (L \\gamma (3 - \\gamma (L + \\mu)) - 1) & \\text{if } \\gamma\in[\\tfrac{1}{L},\\tfrac{2L-\mu}{L^2}],\\\\
0 & \\text{otherwise.} \\end{array}\\right.
\\end{eqnarray}
**References**:
`[1] M. Barré, A. Taylor, A. d’Aspremont (2020).
Complexity guarantees for Polyak steps with momentum.
In Conference on Learning Theory (COLT).
<https://arxiv.org/pdf/2002.00915.pdf>`_
Args:
L (float): the smoothness parameter.
mu (float): the strong convexity parameter.
gamma (float): the step-size.
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 / (L + mu)
>>> pepit_tau, theoretical_tau = wc_polyak_steps_in_function_value(L=L, mu=mu, gamma=gamma, 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 (2 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
Function 1 : Adding 6 scalar constraint(s) ...
Function 1 : 6 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: 0.6694214253294206
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 2.474995615842516e-09
All the primal scalar constraints are verified up to an error of 1.1975611058367974e-09
(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 6.514273074953545e-08
(PEPit) Final upper bound (dual): 0.6694214228930617 and lower bound (primal example): 0.6694214253294206
(PEPit) Duality gap: absolute: -2.4363588924103396e-09 and relative: -3.6394994247628294e-09
*** Example file: worst-case performance of Polyak steps ***
PEPit guarantee: f(x_1) - f_* <= 0.669421 (f(x_0) - f_*)
Theoretical guarantee: f(x_1) - f_* <= 0.669421 (f(x_0) - f_*)
"""
# Instantiate PEP
problem = PEP()
# Declare a smooth convex function
func = problem.declare_function(SmoothStronglyConvexFunction, L=L, mu=mu)
# 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 gradient and function value gn and fn
x0 = problem.set_initial_point()
g0, f0 = func.oracle(x0)
# Set the initial condition to the distance betwenn x0 and xs
problem.set_initial_condition(f0 - fs <= 1)
# Set the initial condition to the Polyak step-size
problem.add_constraint(g0 ** 2 == 2 * L * (2 - L * gamma) * (f0 - fs))
# Run the Polayk steps at iteration 1
x1 = x0 - gamma * g0
g1, f1 = func.oracle(x1)
# Set the performance metric to the distance in function values between x_1 and x_* = xs
problem.set_performance_metric(f1 - 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)
if 1 / L <= gamma <= (2 * L - mu) / L ** 2:
theoretical_tau = (gamma * L - 1) * (L * gamma * (3 - gamma * (L + mu)) - 1)
else:
theoretical_tau = 0.
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of Polyak steps ***')
print('\tPEPit guarantee:\t f(x_1) - f_* <= {:.6} (f(x_0) - f_*) '.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_1) - 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__":
L = 1
mu = 0.1
gamma = 2 / (L + mu)
pepit_tau, theoretical_tau = wc_polyak_steps_in_function_value(L=L, mu=mu, gamma=gamma,
wrapper="cvxpy", solver=None,
verbose=1)