Asymptotic Properties of $\mathcal {S}$-$\mathcal {AB}$ Method With Diminishing Step-Size

The popular
$\mathcal {AB}$
/push–pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of
$\mathcal {AB}$
/push–pull method (
$\mathcal {S}$
-
$\mathcal {AB}$
) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This article is devoted to the asymptotic properties of
$\mathcal {S}$
-
$\mathcal {AB}$
with diminishing step-size. Specifically, under the condition that each local objective is smooth and the global objective is strongly convex, we first present the boundedness of the iterates of
$\mathcal {S}$
-
$\mathcal {AB}$
and then show that the iterates converge to the global minimizer with the rate
$\mathcal {O}(1/k)$
in the mean square sense. Furthermore, the asymptotic normality of Polyak–Ruppert averaged
$\mathcal {S}$
-
$\mathcal {AB}$
is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.