SGD (Stochastic Gradient Descent) is a popular algorithm for large scale optimization problems due to its low iterative cost. However, SGD can not achieve linear convergence rate as FGD (Full Gradient Descent) because of the inherent gradient variance. To attack the problem, mini-batch SGD was proposed to get a trade-off in terms of convergence rate and iteration cost. In this paper, a general ...

In this paper, we have generalized the definition of vector space by considering the group as a canonical $m$-ary hypergroup, the field as a krasner $(m,n)$-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed $m$-ary hypervector space and introduce the concept of convergence of sequence on $m$-ary hypernormed space...

An explicit representation of the Gamma limit a single-well Modica--Mortola functional is given for one-dimensional space under graph convergence which finer than conventional $L^1$-convergence or in measure. As an application, singular Kobayashi-Warren-Carter energy, popular materials science, given. Some compactness also established. Such formulas, as well compactness, useful to characterize ...

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in R. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the α-order fractional derivatives of the velocity for some α > 0 in the space variables in L, which is independent of the viscosity μ > 0. Then it is shown that this key observation yields the Lequicontinu...

In this paper, the concept of {sl local base with  stratifiedstructure} in $I$-topological vector spaces is introduced. Weprove that every $I$-topological vector space has a balanced localbase with stratified structure. Furthermore, a newcharacterization of $I$-topological vector spaces by means of thelocal base with stratified structure is given.