The gradient method: Gradient-dominated and smooth

Problem setup

Consider the unconstrained minimization problem

\[\minimize_{x \in \calH} f(x) \quad \iff \quad \text{find } x \in \calH \;\text{ such that }\; 0 \in \partial f(x) = \set{\nabla f(x)},\]

where \(f : \calH \to \reals\) is \(\mu_{\textup{gd}}\)-gradient dominated and \(L\)-smooth with \(0<\mu_{\textup{gd}}\le L\). This setting is not necessarily convex.

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

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

where

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

Model the problem in AutoLyap and search for the smallest rho

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]"

import math

from autolyap import SolverOptions
from autolyap.algorithms import GradientMethod
from autolyap.problemclass import GradientDominated, InclusionProblem, Smooth
from autolyap.iteration_independent import IterationIndependent


def rho_theory_theorem3_specialized(mu_gd: float, L: float, gamma: float) -> float:
    # Theorem 3 in abbaszadehpeivasti2023conditionslinearconvergence,
    # specialized to the L-smooth (possibly nonconvex) setting via mu = -L.
    threshold_1 = 1.0 / L
    threshold_2 = math.sqrt(3.0) / L

    if gamma < threshold_1:
        discriminant = 4.0 * L * L - (
            2.0 * L * (L + mu_gd) * mu_gd * gamma * (2.0 - L * gamma)
        )
        discriminant = max(discriminant, 0.0)
        numerator = mu_gd * (1.0 - L * gamma) + math.sqrt(discriminant)
        denominator = 2.0 * L + mu_gd
        return (numerator / denominator) ** 2

    if gamma <= threshold_2:
        return 1.0 - (mu_gd * gamma * (4.0 - (L * gamma) ** 2)) / (2.0 + mu_gd * gamma)

    numerator = (L * gamma - 1.0) ** 2
    denominator = numerator + mu_gd * gamma * (2.0 - L * gamma)
    return numerator / denominator


mu_gd = 0.5
L = 1.0
gamma = 1.0

problem = InclusionProblem(
    [
        [GradientDominated(mu_gd=mu_gd), Smooth(L=L)],  # f in intersection
    ]
)
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_function_value_suboptimality(
    algorithm,
    tau=0,
)

result = IterationIndependent.LinearConvergence.bisection_search_rho(
    problem,
    algorithm,
    P,
    T,
    p=p,
    t=t,
    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 = rho_theory_theorem3_specialized(mu_gd=mu_gd, L=L, gamma=gamma)

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

In the \(L\)-smooth (possibly nonconvex) setting, we can specialize [AdKZ23, Theorem 3] by setting \(\mu=-L\). This gives the reference rate \(\rho_{\textup{AdKZ}}\):

\[\begin{split}\rho_{\textup{AdKZ}}= \begin{cases} \left( \dfrac{ \mu_{\textup{gd}}(1-L\gamma) +\sqrt{ 4L^2-2L(L+\mu_{\textup{gd}})\mu_{\textup{gd}}\gamma(2-L\gamma) } }{2L+\mu_{\textup{gd}}} \right)^2, & 0 < \gamma < \dfrac{1}{L},\\[1.0em] 1-\dfrac{\mu_{\textup{gd}}\gamma\bigl(4-(L\gamma)^2\bigr)}{2+\mu_{\textup{gd}}\gamma}, & \dfrac{1}{L} \le \gamma \le \dfrac{\sqrt{3}}{L},\\[1.0em] \dfrac{(L\gamma-1)^2}{(L\gamma-1)^2+\mu_{\textup{gd}}\gamma(2-L\gamma)}, & \dfrac{\sqrt{3}}{L}<\gamma<\dfrac{2}{L}. \end{cases}\end{split}\]

We then sweep \(\gamma \in (0,2)\) with \(\mu_{\textup{gd}}=0.5\) and \(L=1\). In the plot below, the black curve is \(\rho_{\textup{AdKZ}}\), while the blue markers are AutoLyap certificates.

Numerically, AutoLyap gives tighter rates for \(\gamma \in (1.74,2)\). One plausible reason is that [AdKZ23, Theorem 3] does not use interpolation conditions at solutions \(x^\star \in \Argmin_{x\in \calH} f(x)\) when passing from (7) to (10), which introduces a source of a priori conservatism.

Gradient-method rho versus gamma for gradient-dominated smooth objectives, comparing AutoLyap and Theorem 3.

References

[AdKZ23] (1,2)

Hadi Abbaszadehpeivasti, Etienne de Klerk, and Moslem Zamani. Conditions for linear convergence of the gradient method for non-convex optimization. Optimization Letters, 17(5):1105–1125, 2023. doi:10.1007/s11590-023-01981-2.