Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: It is shown that some versions of boosting work especially well in high-dimensional logistic additive models, and adding a limited labelling noise to the training data may in certain cases improve the convergence.
Abstract: A regularized boosting method is introduced, for which regularization is obtained through a penalization function. It is shown through oracle inequalities that this method is model adaptive. The rate of convergence of the probability of misclassification is investigated. It is shown that for quite a large class of distributions, the probability of error converges to the Bayes risk at a rate faster than n-(V+2)/(4(V+1)) where V is the VC dimension of the "base" class whose elements are combined by boosting methods to obtain an aggregated classifier. The dimension-independent nature of the rates may partially explain the good behavior of these methods in practical problems. Under Tsybakov's noise condition the rate of convergence is even faster. We investigate the conditions necessary to obtain such rates for different base classes. The special case of boosting using decision stumps is studied in detail. We characterize the class of classifiers realizable by aggregating decision stumps. It is shown that some versions of boosting work especially well in high-dimensional logistic additive models. It appears that adding a limited labelling noise to the training data may in certain cases improve the convergence, as has been also suggested by other authors.
134 citations
TL;DR: In this article, a hybrid of Picard and Mann iterative processes is proposed, which converges faster than all of Picard, Mann, and Ishikawa iterative process in the sense of Berinde (Iterative Approximation of fixed points, 2002) for contractions.
Abstract: We introduce a new iterative process which can be seen as a hybrid of Picard and Mann iterative processes. We show that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde (Iterative Approximation of Fixed Points, 2002) for contractions. We support our analytical proof by a numerical example. We prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. Our weak convergence results are proved when the underlying space satisfies Opial’s condition or has Frechet differentiable norm or its dual satisfies the Kadec-Klee property.
134 citations
TL;DR: This study shows that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA and can reduce the number of Krylov iterations substantially.
Abstract: In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.
134 citations
TL;DR: In this paper, a version of the fundamental mean-square convergence theorem for stochastic differential equations (SDEs) is proved for SDEs in which coefficients are allowed to grow polynomially at infinity and satisfy a one-sided Lipschitz condition.
Abstract: A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDEs) in which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. The proposed special balanced scheme is explicit and its mean-square order of convergence is 1/2. Some numerical tests are presented.
133 citations
TL;DR: Newton's method and a class of basic fixed-point iterations can be used to find its minimal positive solution whenever it has a positive solution of any equation in this class of nonsymmetric algebraic Riccati equations.
Abstract: We consider the iterative solution of a class of nonsymmetric algebraic Riccati equations, which includes a class of algebraic Riccati equations arising in transport theory. For any equation in this class, Newton's method and a class of basic fixed-point iterations can be used to find its minimal positive solution whenever it has a positive solution. The properties of these iterative methods are studied and some practical issues are addressed. An algorithm is then proposed to find the minimal positive solution efficiently. Numerical results are also given.
133 citations