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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 paper is devoted to the asymptotic properties of $mathcal{S}$-$mathcal{AB}$ with diminishing stepsize. 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 $mathcal{O}left(1/sqrt{k}right)$. 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.
We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain th
We study the Coulomb branch of class $mathcal{S}_k$ $mathcal{N} = 1$ SCFTs by constructing and analyzing their spectral curves.
The experimental results on the ratios of branching fractions $mathcal{R}(D) = {cal B}(bar{B} to D tau^- bar{ u}_{tau})/{cal B}(bar{B} to D ell^- bar{ u}_{ell})$ and $mathcal{R}(D^*) = {cal B}(bar{B} to D^* tau^- bar{ u}_{tau})/{cal B}(bar{B} to D^*
We report a measurement of the ratios of branching fractions $mathcal{R}(D) = {cal B}(bar{B} to D tau^- bar{ u}_{tau})/{cal B}(bar{B} to D ell^- bar{ u}_{ell})$ and $mathcal{R}(D^{ast}) = {cal B}(bar{B} to D^* tau^- bar{ u}_{tau})/{cal B}(bar{B} to D
We propose a generalization of S-folds to 4d $mathcal{N}=2$ theories. This construction is motivated by the classification of rank one 4d $mathcal{N}=2$ super-conformal field theories (SCFTs), which we reproduce from D3-branes probing a configuration