AbstractWe consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as Subsampled Newton and Newton Sketch, which can efficiently construct stochastic Hessian estimates for many tasks, e.g., training machine learning models. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We propose to address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all the past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme exhibits local Q-superlinear convergence with a non-asymptotic rate of $$(\varUpsilon \sqrt{\log (t)/t}\,)^{t}$$ ( Υ log ( t ) / t ) t , where $$\varUpsilon $$ Υ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the method, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still exhibits a superlinear convergence rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.
AbstractIn this paper, we present an “order” characterization of completely bounded bimodule maps for bimodules over unital operator algebras. We use this result to prove a bimodule generalization of Wittstock's generalized Hahn-Banach theorem. Our proofs simplify and unify some of Wittstock's arguments.
AbstractA harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.
Ariska Septieny Fauzy, Sakinah Ubudiyah Siregar, Eva Julyanti
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U ovom radu donosimo pregled matematičkih ideja i koncepata, metodički poredanih u smislenu cjelinu, potrebnih za obradu nastavnih jedinica potencije, eksponencijalna i logaritamska funkcija u osnovnoj i srednjoj školi. Također, donosimo i različite zanimljivosti i svojstva ovih funkcija s povijesnog i matematičkog aspekta, ali i iz kuta nematematičara koji koriste ove funkcije u različitim primjenama.
Este artigo tem como objetivo apresentar o desenvolvimento e a aplicação de um jogo digital educacional que aborda conteúdos específicos da Matemática dos anos finais do ensino fundamental I, com foco no 5º ano. Explosion Math, inspirado no jogo clássico Bomberman, foi desenvolvido utilizando o motor gráfico Unity 3D e pretende auxiliar o ensino de matemática de forma diferenciada e dinâmica. O jogo propõe aos alunos uma viagem com o astronauta Albert, um garoto apaixonado por Matemática, que, ao descobrir a existência de planetas matemáticos, embarca em uma aventura espacial. A missão de Albert é explorar os planetas Numérico, Geométrico e Grammy (grandezas e medidas), a fim de obter o conhecimento presente em cada um deles.