Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem

Bobkov, Sergey G.; Chistyakov, G. P.; Götze, Friedrich

doi:10.1214/12-aop780

Cited by 43 publications

(43 citation statements)

References 23 publications

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“…These assumptions can be found in Bobkov et al . (). We remark that we are focusing on the fixed dimension setting.…”

Section: Statistical Inference Via Langevin Diffusionmentioning

confidence: 97%

“…The proof is based on the entropic CLT (Barron, 1986;Bobkov et al, 2013Bobkov et al, , 2014. The classic CLT based on convergence in distribution is too weak for our purpose: we need to translate the non-asymptotic bounds at each step to the whole stochastic process.…”

Section: Statistical Inference Via Langevin Diffusionmentioning

confidence: 99%

“…We first state the standard assumptions for entropic CLT. These assumptions can be found in Bobkov et al (2013). We remark that we are focusing on the fixed dimension setting.…”

Section: Statistical Inference Via Langevin Diffusionmentioning

confidence: 99%

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Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients

Liang

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Modern statistical inference tasks often require iterative optimization methods to compute the solution. Convergence analysis from an optimization viewpoint informs us only how well the solution is approximated numerically but overlooks the sampling nature of the data. In contrast, recognizing the randomness in the data, statisticians are keen to provide uncertainty quantification, or confidence, for the solution obtained by using iterative optimization methods. The paper makes progress along this direction by introducing moment‐adjusted stochastic gradient descent: a new stochastic optimization method for statistical inference. We establish non‐asymptotic theory that characterizes the statistical distribution for certain iterative methods with optimization guarantees. On the statistical front, the theory allows for model misspecification, with very mild conditions on the data. For optimization, the theory is flexible for both convex and non‐convex cases. Remarkably, the moment adjusting idea motivated from ‘error standardization’ in statistics achieves a similar effect to acceleration in first‐order optimization methods that are used to fit generalized linear models. We also demonstrate this acceleration effect in the non‐convex setting through numerical experiments.

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“…These assumptions can be found in Bobkov et al . (). We remark that we are focusing on the fixed dimension setting.…”

Section: Statistical Inference Via Langevin Diffusionmentioning

confidence: 97%

Section: Statistical Inference Via Langevin Diffusionmentioning

confidence: 99%

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Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients

Liang

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…This representation should be compared to the one obtained in Proposition 2.4 (iii) of [25]. At the core of its proof stand the Bismut-type representation of ∂ σ x (P (α,λ) τ (f )) as well as the intertwining relation (9). We assume that the random variable X has a density f X such that f = f X /γ α,λ is smooth enough for the different analytical arguments to hold.…”

Section: A New Hsi-type Inequalitymentioning

confidence: 98%

A stroll along the gamma

Arras

Swan

2017

Stochastic Processes and their Applications

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We provide the first in-depth study of the "smart path" interpolation between an arbitrary probability measure and the gamma-(α, λ) distribution. We propose new explicit representation formulae for the ensuing process as well as a new notion of relative Fisher information with a gamma target distribution. We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's Γ-calculus. Exploiting a specific representation of the "smart path", we obtain a new proof of the logarithmic Sobolev inequality for the gamma law with α ≥ 1/2 as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with α ≥ 1/2.

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“…More recent work of Bobkov, Chistyakov and Götze [19,20] used properties of characteristic functions to remove the assumption of finite Poincaré constant, and even extended this theory to include convergence to other stable laws [18,21]. This is a very useful result in many cases, and gives us well-known facts such as that entropy is maximised under a variance constraint by the Gaussian density.…”

Section: [Normal Approximation]mentioning

confidence: 99%

Entropy and Thinning of Discrete Random Variables

Johnson

2017

Convexity and Concentration

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We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev) inequalities, monotonicity of entropy and concavity of entropy in the Shepp-Olkin regime.

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Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem

Abstract: An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.

Cited by 43 publications

References 23 publications

Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients

Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients

A stroll along the gamma

Entropy and Thinning of Discrete Random Variables

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