The gradient method: Smooth and strongly convex

Problem setup

Consider the unconstrained minimization problem

\[\minimize_{x \in \calH} f(x),\]

where \(f : \calH \to \reals\) is \(\mu\)-strongly convex and \(L\)-smooth, with \(0 < \mu < L\).

For an initial point \(x^0 \in \calH\) and step size \(0 < \gamma < 2/L\), the gradient update is

\[(\forall k \in \naturals)\quad x^{k+1} = x^k - \gamma \nabla f(x^k).\]

In this example, we search for the smallest contraction factor \(\rho\in[0,1)\) provable using AutoLyap such that

\[\|x^k - x^\star\|^2 = O(\rho^k) \quad \textup{ as } \quad k\to\infty,\]

where

\[x^\star \in \Argmin_{x \in \calH} f(x).\]

Model the problem in AutoLyap and search for the smallest rho

• \(f\) is modeled by SmoothStronglyConvex.

• The optimization problem is represented by InclusionProblem.

• The update rule is represented by GradientMethod.

• Distance-to-solution parameters are obtained with IterationIndependent.LinearConvergence.get_parameters_distance_to_solution.

• The contraction factor is searched with IterationIndependent.LinearConvergence.bisection_search_rho.

Run the iteration-independent analysis

This example uses the MOSEK Fusion backend (backend="mosek_fusion"). Install the optional MOSEK dependency first:

pip install "autolyap[mosek]"

from autolyap import IterationIndependent, SolverOptions
from autolyap.algorithms import GradientMethod
from autolyap.problemclass import InclusionProblem, SmoothStronglyConvex

mu = 1.0
L = 4.0
gamma = 0.2

problem = InclusionProblem([SmoothStronglyConvex(mu, L)])
algorithm = GradientMethod(gamma=gamma)
solver_options = SolverOptions(backend="mosek_fusion")
# License-free option:
# solver_options = SolverOptions(backend="cvxpy", cvxpy_solver="CLARABEL")

P, p, T, t = IterationIndependent.LinearConvergence.get_parameters_distance_to_solution(
    algorithm
)

result = IterationIndependent.LinearConvergence.bisection_search_rho(
    problem,
    algorithm,
    P,
    T,
    p=p,
    t=t,
    S_equals_T=True,
    s_equals_t=True,
    remove_C3=True,
    solver_options=solver_options,
)

if result["status"] != "feasible":
    raise RuntimeError("No feasible Lyapunov certificate in the requested rho interval.")

rho_autolyap = result["rho"]
rho_theory = max(gamma * L - 1.0, 1.0 - gamma * mu) ** 2

print(f"rho (AutoLyap): {rho_autolyap:.8f}")
print(f"rho (theory):   {rho_theory:.8f}")

What to inspect in result:

• result["status"]: feasible, infeasible, or not_solved.

• result["solve_status"]: raw backend status.

• result["rho"]: certified contraction factor when feasible.

• result["certificate"]: Lyapunov certificate matrices/scalars.

The computed value rho (AutoLyap) matches (up to solver numerical tolerances) the theoretical rate expression for gradient methods [Pol63]:

\[\|x^k - x^\star\|^2 = O(\rho^k), \qquad \rho = \max\{|1-\gamma L|,\;|1-\gamma\mu|\}^2.\]

Equivalently,

\[\|x^k - x^\star\| = O\!\left(\max\{|1-\gamma L|,\;|1-\gamma\mu|\}^k\right).\]

Sweeping over 100 values of \(\gamma\) on \(0 < \gamma \le 2/L\) gives the plot below, with the theoretical rate in black and AutoLyap certificates as blue dots.

References

[Pol63]

B. T. Polyak. Gradient methods for the minimisation of functionals. USSR Computational Mathematics and Mathematical Physics, 3(4):864–878, 1963. doi:10.1016/0041-5553(63)90382-3.