On quantitative bounds in the mean martingale central limit theorem

Röllin, Adrian

doi:10.1016/j.spl.2018.03.004

Cited by 12 publications

(18 citation statements)

References 11 publications

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“…We emphasize that all the constants in the bound are explicit; yet the bound cannot be expressed in terms of the Wasserstein distance since the second derivative of the test function h appears on the right hand side. This is the main difference between our result and that of [Röl18]; the bounds in [Röl18] are for the univariate case and in Wasserstein metric whereas Theorem 1 establishes the bounds in the multivariate setting but the resulting bound cannot be expressed in terms of Wasserstein distance due to a subtlety of the Stein's method in high dimensions.…”

Section: Convergence Rates Of a Multivariate Martingale Cltmentioning

confidence: 66%

“…In the standard literature, convergence rates are established for one-dimensional random variables using Lindeberg's telescoping sum argument [Bol82,Mou13,Fan19]. In order to obtain convergence rates with explicit constants, we adapt an approach from [Röl18] to the multivariate setting, which is based on a combination of Stein's method [Ste86] and Lindeberg's telescoping sum argument [Bol82]. We state the following nonasymptotic multivariate martingale CLT.…”

Section: Convergence Rates Of a Multivariate Martingale Cltmentioning

confidence: 99%

“…Relaxing the assumption (2.1) has been the focus of many papers; see for example [Bol82,Mou13,Röl18]. Since this assumption is conveniently satisfied for the stochastic algorithms that we consider in the current paper, we omit a rigorous relaxation of this condition.…”

Section: Relaxing the Assumptionsmentioning

confidence: 99%

See 2 more Smart Citations

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Anastasiou,

Balasubramanian,

Erdogdu

2019

Preprint

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We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein's method and Lindeberg's argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense. * Authors are listed in alphabetical order. This paper provides a solution for an open problem formulated at the AIM workshop on Stein's method and Applications in High-dimensional Statistics, 2018 (problem (4) in page 2 of https://aimath.org/pastworkshops/steinhdrep.pdf by KB). We thank Jay Bartroff, Larry Goldstein, Stanislav Minsker, and Gesine Reinert for organizing a stimulating workshop, and the participants for several discussions. MAE is partially funded by CIFAR AI Chairs program at the Vector Institute.

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Section: Convergence Rates Of a Multivariate Martingale Cltmentioning

confidence: 66%

Section: Convergence Rates Of a Multivariate Martingale Cltmentioning

confidence: 99%

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Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Anastasiou,

Balasubramanian,

Erdogdu

2019

Preprint

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“…The proof is by Lindeberg's swapping argument and Stein's method. The approach was used by Röllin (2017) for a martingale CLT. See also Song (2017).…”

Section: Proofs Of Theorems 32 and 31mentioning

confidence: 99%

“…Our main tool for proving the rate of convergence for the CLT is a combination of Lindeberg's swapping argument and Stein's method. This approach was used by Röllin (2017) for proving a martingale CLT. The sublinear expectation (1.1) is defined through a class of probability measures, and in general, cannot be represented in a single probability space.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems with rate of convergence under sublinear expectations

Fang¹,

Péng²,

Shao³

et al. 2019

Bernoulli

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Under the sublinear expectation [·] := sup θ∈Θ E θ [·] for a given set of linear expectations {E θ : θ ∈ Θ}, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the G-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.

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On the rate of convergence in the central limit theorem for arrays of random vectors

Dung

Son

2020

Statistics & Probability Letters

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On quantitative bounds in the mean martingale central limit theorem

Abstract: We provide explicit bounds on the Wasserstein distance between discrete time martingales and the standard normal distribution. The proofs are based on a combination of Lindeberg's and Stein's method.

Cited by 12 publications

References 11 publications

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Limit theorems with rate of convergence under sublinear expectations

On the rate of convergence in the central limit theorem for arrays of random vectors

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