Asymptotic Properties of <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula> Method With Diminishing Step-Size

The popular <inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula>/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 <inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula>/push–pull method (<inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula>) 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 <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula> and then show that the iterates converge to the global minimizer with the rate <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(1/k)$</tex-math></inline-formula> in the mean square sense. Furthermore, the asymptotic normality of Polyak–Ruppert averaged <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$\mathcal {AB}$</tex-math></inline-formula> is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.