Central limit theorems for high dimensional dependent data

Chang, Jinyuan; Chen, Xiaohui; Wu, Mingcong

doi:10.48550/arxiv.2104.12929

“…, θp ) T where θ is the proposed moment-based estimator given in (3.5) based on the sanitized data Z. As shown in Proposition 4, the leading term of θ − θ cannot be formulated as a partial sum of independent (or weakly dependent) random vectors, which is different from the standard framework of Gaussian approximation (Chernozhukov, Chetverikov and Kato, 2013;Chang, Chen and Wu, 2021).…”

Section: Bootstrap Algorithm and Simultaneous Inferencementioning

confidence: 99%

Edge differentially private estimation in the β-model via jittering and method of moments

Chang,

Hu,

Kolaczyk

et al. 2024

Ann. Statist.

0

View full text Add to dashboard Cite

A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here we conduct an in-depth study of this trade-off for parameter estimation in the β-model (Chatterjee, Diaconis and Sly, 2011) for edge differentially private network data released via jittering (Karwa, Krivitsky and Slavković, 2017). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via method of moments. This choice facilitates our exploration of a substantially broader range of privacy levels -corresponding to stricter privacy -than has been to date. Over this new range we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult to impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the β-model (i.e., equal to the number of vertices), which would appear to be the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.

show abstract

“…, θp ) T where θℓ is the proposed moment-based estimator given in (3.5) based on the sanitized data Z. As shown in Proposition 4, the leading term of θ − θ cannot be formulated as a partial sum of independent (or weakly dependent) random vectors, which is different from the standard framework of Gaussian approximation (Chernozhukov, Chetverikov and Kato, 2013;Chang, Chen and Wu, 2021).…”

Section: Bootstrap Algorithm and Simultaneous Inferencementioning

confidence: 99%

Edge differentially private estimation in the $β$-model via jittering and method of moments

Chang¹,

Qiao²,

Kolaczyk³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here we conduct an in-depth study of this trade-off for parameter estimation in the β-model (Chatterjee, Diaconis and Sly, 2011) for edge differentially private network data released via jittering (Karwa, Krivitsky and Slavković, 2017). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via method of moments. This choice facilitates our exploration of a substantially broader range of privacy levels -corresponding to stricter privacy -than has been to date. Over this new range we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult to impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the β-model (i.e., equal to the number of vertices), which would appear to be the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.

show abstract

“…In the case of sums of independent random vectors, high-dimensional Gaussian comparison has been developed in the milestone work of Chernozhukov et al (2013) (see also Bentkus, 2005). Since then, similar results have been developed for U -statistics (Chen, 2018) and stochastic processes with weak dependence such as mixing or spatial process (Kurisu et al, 2021;Chang et al, 2021). However, these extensions do not cover the cross-validation case, where each term in the summation is dependent of each other with a similar magnitude of dependence, violating the sparsity of dependence (U -statistics) and fast decaying dependence (mixing and spatial processes).…”

Section: Introductionmentioning

confidence: 99%

On High-Dimensional Gaussian Comparisons For Cross-Validation

Kissel¹,

Lei²

2022

Preprint

0

View full text Add to dashboard Cite

We derive high-dimensional Gaussian comparison results for the standard V -fold cross-validated risk estimates. Our result combines a recent stability-based argument for the low-dimensional central limit theorem of cross-validation with the high-dimensional Gaussian comparison framework for sums of independent random variables. These results give new insights into the joint sampling distribution of cross-validated risks in the context of model comparison and tuning parameter selection, where the number of candidate models and tuning parameters can be larger than the fitting sample size. As a consequence, our results provide theoretical support for a recent methodological development that constructs model confidence sets using cross-validation.

show abstract

Central limit theorems for high dimensional dependent data

Cited by 4 publications

References 61 publications

Edge differentially private estimation in the β-model via jittering and method of moments

Edge differentially private estimation in the β-model via jittering and method of moments

Edge differentially private estimation in the $β$-model via jittering and method of moments

On High-Dimensional Gaussian Comparisons For Cross-Validation

Contact Info

Product

Resources

About