Since the prox-mapping of stochastic mirror descent is intractable when directly being applied to the optimization problem (1), we propose the -inexact prox-mapping within the stochastic mirror descent framework in Section 3. Instead of solving the prox-mapping exactly, we approximate the solution with error. In this section, we will show as long as the approximation error is tolerate, the stoc...

In the paper, we develop a composite version of Mirror Prox algorithm for solving convexconcave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no n...

Uniform sampling of training data has been commonly used in traditional stochastic optimization algorithms such as Proximal Stochastic Mirror Descent (prox-SMD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although uniform sampling can guarantee that the sampled stochastic quantity is an unbiased estimate of the corresponding true quantity, the resulting estimator may have a rath...

Like other tissues and organs in vertebrates, multipotential stem cells serve as the origin of diverse cell types during genesis of the mammalian central nervous system (CNS). During early development, stem cells self-renew and increase their total cell numbers without overt differentiation. At later stages, the cells withdraw from this self-renewal mode, and are fated to differentiate into neu...

In this paper, we propose a simple variant of the original stochastic variance reduction gradient (SVRG) [1], where hereafter we refer to as the variance reduced stochastic gradient descent (VR-SGD). Different from the choices of the snapshot point and starting point in SVRG and its proximal variant, Prox-SVRG [2], the two vectors of each epoch in VRSGD are set to the average and last iterate o...

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no ...

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of Rand J -trivial regular languages, and prove that n! and ⌊e(n −...

Slice Fueter-regular functions, originally called slice Dirac-regular are generalized holomorphic functions defined over the octonion algebra \({\mathbb {O}}\), recently introduced by M. Jin, G. Ren and I. Sabadini. A function \(f:\Omega _D\subset {\mathbb {O}}\rightarrow {O}}\) is (quaternionic) if, given any quaternionic subalgebra {H}}_{\mathbb {I}}\) of generated a pair {I}}=(I,J)\) orthogo...