Orders On Free Metabelian Groups

A bi-order on a group $G$ is a total, bi-multiplication invariant order. Such an order is regular if the positive cone associated to the order can be recognised by a regular language. A subset $S$ in an orderable group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we study the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order. As an application, we prove that non-abelian free metabelian groups of finite rank do not admit a regular bi-order while they are computably bi-orderable.Comment: 19 Pages, 1 figure. Comments are welcome

Are All Item Response Functions Monotonically Increasing?

Item response functions of the parametric logistic IRT models follow the logistic form which is monotonically increasing. However, item response functions of some real items are nonmonotonic which might lead to examinees with lower proficiency levels receiving higher scores. This study compared three nonparametric IRF estimation methods--the nonparametric smooth regression method, the item-ability regression method, and the B-spline nonparametric IRF method--to determine whether they could detect the nonmonotonic IRF accurately using simulated data. In addition, these methods were used to identify items with nonmonotonic IRFs on real assessments. Results present that three nonparametric methods can detect the nonmonotonic IRF equally and each real assessment has some items with nonmonotonic IRFs. Investigations on the reasons for and the consequences of the nonmonotonicity were conducted for several items and indicate that the nonmonotonicity can affect the fairness and comparability of the test score. Thus, the nonmonotonicity should be checked before applying the parametric logistic models

Dehn Function of Finitely Presented Metabelian Groups

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.Comment: 41 pages, 4 figures. Fix an issue with the trivial case and improve the theorem. Comments are welcome

Fast Convergence Federated Learning with Aggregated Gradients

Federated Learning (FL) is a novel machine learning framework, which enables multiple distributed devices cooperatively training a shared model scheduled by a central server while protecting private data locally. However, the non-independent-and-identically-distributed (Non-IID) data samples and frequent communication among participants will slow down the convergent rate and increase communication costs. To achieve fast convergence, we ameliorate the local gradient descend approach in conventional local update rule by introducing the aggregated gradients at each local update epoch, and propose an adaptive learning rate algorithm that further takes the deviation of local parameter and global parameter into consideration at each iteration. The above strategy requires all clients' local parameters and gradients at each local iteration, which is challenging as there is no communication during local update epochs. Accordingly, we utilize mean field approach by introducing two mean field terms to estimate the average local parameters and gradients respectively, which does not require clients to exchange their private information with each other at each local update epoch. Numerical results show that our proposed framework is superior to the state-of-art schemes in model accuracy and convergent rate on both IID and Non-IID dataset.Comment: 7 pages, 2 figure