Convergence Proofs

2001 ◽  
pp. 267-299
Author(s):  
Harold J. Kushner ◽  
Paul Dupuis
Keyword(s):  
Convergence Proofs

scholarly journals CertRL: formalizing convergence proofs for value and policy iteration in Coq

2021 ◽  
Author(s):  
Koundinya Vajjha ◽  
Avraham Shinnar ◽  
Barry Trager ◽  
Vasily Pestun ◽  
Nathan Fulton
Keyword(s):  
Policy Iteration ◽  
Convergence Proofs

scholarly journals Limits and Convergence properties of the Sequentially Markovian Coalescent

2020 ◽  
Author(s):  
Thibaut Sellinger ◽  
Diala Abu Awad ◽  
Aurélien Tellier
Keyword(s):  
Sequence Data ◽  
Demographic History ◽  
Simulated Data ◽  
Simultaneous Estimation ◽  
Data Sets ◽  
Performance Limits ◽  
Biological Variables ◽  
Convergence Proofs ◽  
New Interpretation ◽  
Population Demographic

AbstractMany methods based on the Sequentially Markovian Coalescent (SMC) have been and are being developed. These methods make use of genome sequence data to uncover population demographic history. More recently, new methods have extended the original theoretical framework, allowing the simultaneous estimation of the demographic history and other biological variables. These methods can be applied to many different species, under different model assumptions, in hopes of unlocking the population/species evolutionary history. Although convergence proofs in particular cases have been given using simulated data, a clear outline of the performance limits of these methods is lacking. We here explore the limits of this methodology, as well as present a tool that can be used to help users quantify what information can be confidently retrieved from given datasets. In addition, we study the consequences for inference accuracy violating the hypotheses and the assumptions of SMC approaches, such as the presence of transposable elements, variable recombination and mutation rates along the sequence and SNP call errors. We also provide a new interpretation of the SMC through the use of the estimated transition matrix and offer recommendations for the most efficient use of these methods under budget constraints, notably through the building of data sets that would be better adapted for the biological question at hand.

Prediction Games and Arcing Algorithms

1999 ◽  
Vol 11 (7) ◽  
pp. 1493-1517 ◽  
Author(s):  
Leo Breiman
Keyword(s):  
Optimal Strategy ◽  
Minimax Theorem ◽  
Maximum Error ◽  
The Other ◽  
Generalization Error ◽  
Empirical Comparison ◽  
Training Set ◽  
Finite Set ◽  
Convergence Proofs ◽  
Convex Linear Combination

The theory behind the success of adaptive reweighting and combining algorithms (arcing) such as Adaboost (Freund & Schapire, 1996a, 1997) and others in reducing generalization error has not been well understood. By formulating prediction as a game where one player makes a selection from instances in the training set and the other a convex linear combination of predictors from a finite set, existing arcing algorithms are shown to be algorithms for finding good game strategies. The minimax theorem is an essential ingredient of the convergence proofs. An arcing algorithm is described that converges to the optimal strategy. A bound on the generalization error for the combined predictors in terms of their maximum error is proven that is sharper than bounds to date. Schapire, Freund, Bartlett, and Lee (1997) offered an explanation of why Adaboost works in terms of its ability to produce generally high margins. The empirical comparison of Adaboost to the optimal arcing algorithm shows that their explanation is not complete.

scholarly journals On the computational content of convergence proofs via Banach limits

2012 ◽  
Vol 370 (1971) ◽  
pp. 3449-3463 ◽  
Author(s):  
U. Kohlenbach ◽  
L. Leuştean
Keyword(s):  
Proof Theory ◽  
Quantitative Information ◽  
Prima Facie ◽  
New Developments ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
Banach Limits ◽  
Convergence Proofs ◽  
The Axiom Of Choice ◽  
Highly Uniform

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X . Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

scholarly journals Equilong invariants and convergence proofs

1917 ◽  
Vol 23 (8) ◽  
pp. 341-348
Author(s):  
Edward Kasner
Keyword(s):  
Convergence Proofs

scholarly journals Michael J. D. Powell. 29 July 1936—19 April 2015

2018 ◽  
Vol 64 ◽  
pp. 341-366
Author(s):  
M. D. Buhmann ◽  
R. Fletcher ◽  
A. Iserles ◽  
P. Toint
Keyword(s):  
Optimization Theory ◽  
Subject Area ◽  
Basis Functions ◽  
Spline Functions ◽  
Energy Research ◽  
Design Of Algorithms ◽  
Science And Engineering ◽  
Computational Mathematics ◽  
Convergence Proofs ◽  
Mathematical Foundations

Michael James David Powell was a British numerical analyst who was among the pioneers of computational mathematics. During a long and distinguished career, first at the Atomic Energy Research Establishment (AERE) Harwell and subsequently as the John Humphrey Plummer Professor of Applied Numerical Analysis in Cambridge, he contributed decisively towards establishing optimization theory as an effective tool of scientific enquiry, replete with highly effective methods and mathematical sophistication. He also made crucial contributions to approximation theory, in particular to the theory of spline functions and of radial basis functions. In a subject that roughly divides into practical designers of algorithms and theoreticians who seek to underpin algorithms with solid mathematical foundations, Mike Powell refused to follow this dichotomy. His achievements span the entire range from difficult and intricate convergence proofs to the design of algorithms and production of software. He was among the leaders of a subject area that is at the nexus of mathematical enquiry and applications throughout science and engineering.

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