Iteration Complexity Analysis of Block Coordinate Descent Methods

In this paper, we provide a unified iteration complexity analysis for a family of general block coordinate descent (BCD) methods, covering popular methods such as the block coordinate gradient descent (BCGD) and the block coordinate proximal gradient (BCPG), under various different coordinate update rules. We unify these algorithms under the so-called Block Successive Upper-bound Minimization (BSUM) framework, and show that for a broad class of multi-block nonsmooth convex problems, all algorithms covered by the BSUM framework achieve a global sublinear iteration complexity of $O(1/r)$, where r is the iteration index. Moreover, for the case of block coordinate minimization (BCM) where each block is minimized exactly, we establish the sublinear convergence rate of $O(1/r)$ without per block strong convexity assumption. Further, we show that when there are only two blocks of variables, a special BSUM algorithm with Gauss-Seidel rule can be accelerated to achieve an improved rate of $O(1/r^2)$

Solving Multiple-Block Separable Convex Minimization Problems Using Two-Block Alternating Direction Method of Multipliers

In this paper, we consider solving multiple-block separable convex minimization problems using alternating direction method of multipliers (ADMM). Motivated by the fact that the existing convergence theory for ADMM is mostly limited to the two-block case, we analyze in this paper, both theoretically and numerically, a new strategy that first transforms a multi-block problem into an equivalent two-block problem (either in the primal domain or in the dual domain) and then solves it using the standard two-block ADMM. In particular, we derive convergence results for this two-block ADMM approach to solve multi-block separable convex minimization problems, including an improved O(1/\epsilon) iteration complexity result. Moreover, we compare the numerical efficiency of this approach with the standard multi-block ADMM on several separable convex minimization problems which include basis pursuit, robust principal component analysis and latent variable Gaussian graphical model selection. The numerical results show that the multiple-block ADMM, although lacks theoretical convergence guarantees, typically outperforms two-block ADMMs

Renormalization group improved pQCD prediction for Υ(1S)\Upsilon(1S) leptonic decay

The complete next-to-next-to-next-to-leading order short-distance and bound-state QCD corrections to $\Upsilon(1S)$ leptonic decay rate $\Gamma(\Upsilon(1S)\to \ell^+\ell^-)$ has been finished by Beneke {\it et al.} \cite{Beneke:2014qea}. Based on those improvements, we present a renormalization group (RG) improved pQCD prediction for $\Gamma(\Upsilon(1S)\to \ell^+\ell^-)$ by applying the principle of maximum conformality (PMC). The PMC is based on RG-invariance and is designed to solve the pQCD renormalization scheme and scale ambiguities. After applying the PMC, all known-type of $\beta$-terms at all orders, which are controlled by the RG-equation, are resummed to determine optimal renormalization scale for its strong running coupling at each order. We then achieve a more convergent pQCD series, a scheme- independent and more accurate pQCD prediction for $\Upsilon(1S)$ leptonic decay, i.e. $\Gamma_{\Upsilon(1S) \to e^+ e^-}|_{\rm PMC} = 1.270^{+0.137}_{-0.187}$ keV, where the uncertainty is the squared average of the mentioned pQCD errors. This RG-improved pQCD prediction agrees with the experimental measurement within errors.Comment: 11 pages, 4 figures. Numerical results and discussions improved, references updated, to be published in JHE

When Does Paternalistic Control Positively Relate to Job Satisfaction and Citizenship Behavior in Taiwan?:The Role of Follower Expectation

Although prior research predicts mainly that followers expect leaders to exert less paternalistic control (such as emphasis on discipline, didactic instruction, and belittling followers), we argue that such an expectation may not be stable overtime or across settings. Based on the connectionist perspectives of implicit leadership theories, we propose a follower expectation model of paternalistic control, in which followers compare their perceived with expected levels of paternalistic control. Two inconsistent conditions—insufficient and excessive control—are identified, and the consistency between perceived and expected paternalistic control is predicted to relate to favorable follower outcomes. We examine this model by conducting two daily experience sampling studies in Taiwan. Our findings indicate that insufficient control is as unfavorable as excessive control in lowering followers’ job satisfaction and citizenship behavior, and this pattern is particularly salient in terms of emphasis on discipline and the belittling of followers. A supplemental, qualitative analysis additionally demonstrated the conditions under which the expectation–perception consistency regarding belittling followers relates to favorable follower responses. (PsycInfo Database Record (c) 2023 APA, all rights reserved