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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.

Interlayered thin-film composite (i-TFC) membranes based on 2D materials have been widely studied due to their high efficiency in mass transfer. However, the randomly stacked 2D nanosheets usually increase the fluid path length to some extent. Herein, in situ-grown quasi-vertically oriented 2D ZIF-L was introduced as an interlayer for preparing high-performance reverse osmosis (RO) membranes. Through the optimization of the crystal growth based on the inert polyethylene substrate, the novel i-TFC RO membrane via interfacial polymerization shows an outstanding water permeance (5.50 L m–2h–1bar–1) and good NaCl rejection (96.3%). The membrane also shows promising potential in domestic water purification and organic solvent separation applications. Compared with the randomly stacked ZIF-L interlayer, the advantages of the vertically oriented one were ascribed to the excellent storage capacity of the amine monomers and the intensified gutter effect. This work will encourage more exploration on the interlayer architectures for high-performance i-TFC membranes.