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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.


Papers
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Journal ArticleDOI
TL;DR: The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation.
Abstract: Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Levy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ -scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN( log(N))2 ) operations and O(N log(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation. Computational examples for various Levy price processes are presented.

161 citations

Journal ArticleDOI
TL;DR: The discrete shear gap (DSG) method, a general framework for formulation of locking-free elements, is applied to develop a new class of NURBS finite elements.

160 citations

Journal ArticleDOI
TL;DR: In this article, the authors present the limiting distribution theory for the GMM estimator when the estimation is based on a population moment condition which is subject to non-local (or fixed) misspecification.

160 citations

Journal Article
TL;DR: In this article, a decentralized double stochastic averaging gradient (DSA) algorithm is proposed to solve large scale machine learning problems where elements of the training set are distributed to multiple computational elements.
Abstract: This paper considers optimization problems where nodes of a network have access to summands of a global objective. Each of these local objectives is further assumed to be an average of a finite set of functions. The motivation for this setup is to solve large scale machine learning problems where elements of the training set are distributed to multiple computational elements. The decentralized double stochastic averaging gradient (DSA) algorithm is proposed as a solution alternative that relies on: (i) The use of local stochastic averaging gradients. (ii) Determination of descent steps as differences of consecutive stochastic averaging gradients. Strong convexity of local functions and Lipschitz continuity of local gradients is shown to guarantee linear convergence of the sequence generated by DSA in expectation. Local iterates are further shown to approach the optimal argument for almost all realizations. The expected linear convergence of DSA is in contrast to the sublinear rate characteristic of existing methods for decentralized stochastic optimization. Numerical experiments on a logistic regression problem illustrate reductions in convergence time and number of feature vectors processed until convergence relative to these other alternatives.

160 citations

Proceedings Article
12 Jul 2020
TL;DR: It is shown that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization, which significantly expands the recent asymptotic convergence results.
Abstract: We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a Łojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-t})$ toward softmax optimal policy. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform Łojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.

160 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023693
20221,530
20212,129
20202,036
20191,995