A Perturbation Subsampling for Large Scale Data

Q: How does perturbation number affect the performance of different subsampling estimators in linear regression models?

The simulation studies show that perturbation number 'm' plays a significant role in the performance of different subsampling estimators in linear regression models. For Algorithm 2, which uses uniform subsampling estimator based on sampling with replacement (unisMUL), a larger perturbation number 'm' is required to achieve better results. On the other hand, for Algorithm 3, which uses perturbation subsampling estimators based on Gamma(1/q n , 1) and Beta(1, 1) distributions, a smaller perturbation number 'm' can be used while still obtaining good results. The empirical coverage of the confidence intervals remains stable when the perturbation number increases from 'm' = 50 to 'm' = 200. This indicates that the perturbation number 'm' directly impacts the accuracy and efficiency of the subsampling estimators in linear regression models.

Q: What kinematic features differentiate supersymmetric particle production?

In the SUSY benchmark dataset, 14 out of 18 kinematic features were found to differentiate the process of new supersymmetric particles production from the background process. Kinematic features 1-12 were highly significant in both logistic regression and probit regression analysis. The proposed repeated perturbation subsampling method yielded similar results, with 1-12 kinematic features being significantly associated with the classification of the two processes. The conditional standard errors were stable across different subset perturbations, and the unconditional variances of perturbation estimates were significantly lower compared to the full dataset. This analysis demonstrates the effectiveness of the proposed approach in logistic regression and probit regression models using the SUSY benchmark dataset.

Question

1. What is the alternative approach to full sample analysis?

2. What is the optimization objective function in perturbation subsampling for large-scale data?

3. What is perturbation subsampling method?

4. What is the purpose of Algorithm 2 in perturbation subsampling?

Accepted Answer

The alternative approach to full sample analysis is subsampling, which is based on the analysis of a selected fraction of the whole data. This approach has been studied for various models based on sampling with or without replacement. Optimal subsampling methods require calculating data-dependent nonuniform sampling probabilities for all data at once, which can be challenging due to the large memory requirement and computational burden. The subsample observations might be independent conditioned on the full data, but they are correlated unconditionally. Current results quantify the difference between the subsample estimator and the full data estimator through conditional distribution and conditional variance. The perturbation subsampling method, developed in this paper, optimizes convex objective functions in the analysis of large scale data by generating stochastic weights through two steps: Bernoulli sampling and random perturbation weights. This method is feasible for distributed computing and provides an empirical distribution for statistical inference.

Accepted Answer

The optimization objective function in perturbation subsampling for large-scale data is a continuous, convex function f(b, x, y) that characterizes the relationship between the response variable Y and covariates X. The unknown parameter b0 is the arg min of this function over a d-dimensional space, denoted as b0Rd. The function f(b, x, y) is assumed to exist and be finite. The estimator of b0 is obtained by minimizing the empirical analog of the convex objective function, denoted as bn, which is the arg min of bRd over n i=1 samples (y i, x i). This resulting estimator is an M-estimator, which can be a maximum likelihood estimator (MLE) or a Lp-norm estimator depending on the function f(b, x, y).

Accepted Answer

Perturbation subsampling is a method that reduces sample size from n to r n via Bernoulli sampling and approximates the full data objective function via perturbation with independently generated stochastic weights. It involves two steps: subsampling using Bernoulli random variables and perturbation using nonnegative stochastic weights from a known probability distribution. This method has been used in subsampling algorithms and can be implemented repeatedly to estimate the variance of the perturbation subsampling estimator. It is a novel approach compared to rescaling with fixed and data-dependent weights in subsampling algorithms. The Bernoulli sampling step is based on survey sampling, while the perturbation step uses a known probability distribution to approximate the full data objective function. The method can be applied using various probability distributions for the stochastic weights. Overall, perturbation subsampling provides a way to reduce sample size while still approximating the full data objective function accurately.

Accepted Answer

Algorithm 2 is a perturbation subsampling algorithm used for variance estimation. It generates n i.i.d. random variables, calculates the expected value and variance of V n, and minimizes the perturbed objective function to obtain bn,k. The algorithm estimates the conditional variance of bn and the unconditional variance using formulas (2.5) and (2.6). This procedure allows for the repetitive use of stochastic weighting to estimate the variance of the perturbation subsampling estimator for statistical inference, as described in Jin, Ying, and Wei (2001). Overall, Algorithm 2 provides a method for variance estimation in perturbation subsampling, enhancing the accuracy and reliability of statistical inference.

Accepted Answer

Repeated perturbation subsampling is an algorithm used for large scale data analysis. It involves generating n i.i.d. random variables {U n,i,l} following a Bernoulli distribution with parameter qn. The algorithm then estimates the expected value E(V n,i,l,k) and variance var(V n,i,l,k) for each subset. Point estimation is achieved by minimizing the perturbed objective function, while variance estimation is done using the conditional and unconditional variance formulas. The algorithm can be implemented in parallel computing architectures, allowing for efficient processing of large datasets.

Accepted Answer

The conditions for the consistency and asymptotic normality of the estimator obtained from Algorithm 1 are Assumptions 1-3. Assumption 1 states that the parameter space of b is compact in R d, and b 0 satisfying (2.1) is an interior point of the parameter space and is unique. Assumption 2 requires the first and second gradients of the convex objective function f (b, x, y) with respect to b in a neighborhood of b 0 to exist and be finite. Assumption 3 specifies the requirements for the stochastic weights U n ~ Bern(q n ), E(V n ) = 1/q n, and the existence of a > 0 such that lim sup n- q 2+a n EV 2+a n <. These assumptions guarantee the consistency of the minimizer of the convex objective function and the asymptotic normality of the full data estimator.

Accepted Answer

The simulation studies show that perturbation number 'm' plays a significant role in the performance of different subsampling estimators in linear regression models. For Algorithm 2, which uses uniform subsampling estimator based on sampling with replacement (unisMUL), a larger perturbation number 'm' is required to achieve better results. On the other hand, for Algorithm 3, which uses perturbation subsampling estimators based on Gamma(1/q n , 1) and Beta(1, 1) distributions, a smaller perturbation number 'm' can be used while still obtaining good results. The empirical coverage of the confidence intervals remains stable when the perturbation number increases from 'm' = 50 to 'm' = 200. This indicates that the perturbation number 'm' directly impacts the accuracy and efficiency of the subsampling estimators in linear regression models.

Accepted Answer

In the SUSY benchmark dataset, 14 out of 18 kinematic features were found to differentiate the process of new supersymmetric particles production from the background process. Kinematic features 1-12 were highly significant in both logistic regression and probit regression analysis. The proposed repeated perturbation subsampling method yielded similar results, with 1-12 kinematic features being significantly associated with the classification of the two processes. The conditional standard errors were stable across different subset perturbations, and the unconditional variances of perturbation estimates were significantly lower compared to the full dataset. This analysis demonstrates the effectiveness of the proposed approach in logistic regression and probit regression models using the SUSY benchmark dataset.

A Perturbation Subsampling for Large Scale Data

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Most frequently asked questions

1. What is the alternative approach to full sample analysis?

2. What is the optimization objective function in perturbation subsampling for large-scale data?

3. What is perturbation subsampling method?

4. What is the purpose of Algorithm 2 in perturbation subsampling?

5. What is repeated perturbation subsampling?

6. What are the conditions for the consistency and asymptotic normality of the estimator obtained from Algorithm 1?

7. How does perturbation number affect the performance of different subsampling estimators in linear regression models?

8. What kinematic features differentiate supersymmetric particle production?

References

R: A language and environment for statistical computing.

UCI Machine Learning Repository

Searching for exotic particles in high-energy physics with deep learning

Communication-Efficient Distributed Statistical Inference

A simple resampling method by perturbing the minimand

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