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Adaptive and Stratified Subsampling Techniques for High Dimensional Non-Standard Data Environments

Prateek Mittal, Jai Dalmotra, Joohi Chauhan

TL;DR

This work proposes robust subsampling techniques, specifically Adaptive Importance Sampling (AIS) and Stratified Subsampling, designed to enhance the reliability and efficiency of parameter estimation.

Abstract

This paper addresses the challenge of estimating high-dimensional parameters in non-standard data environments, where traditional methods often falter due to issues such as heavy-tailed distributions, data contamination, and dependent observations. We propose robust subsampling techniques, specifically Adaptive Importance Sampling (AIS) and Stratified Subsampling, designed to enhance the reliability and efficiency of parameter estimation. Under some clearly outlined conditions, we establish consistency and asymptotic normality for the proposed estimators, providing non-asymptotic error bounds that quantify their performance. Our theoretical foundations are complemented by controlled experiments demonstrating the superiority of our methods over conventional approaches. By bridging the gap between theory and practice, this work offers significant contributions to robust statistical estimation, paving the way for advancements in various applied domains.

Adaptive and Stratified Subsampling Techniques for High Dimensional Non-Standard Data Environments

TL;DR

This work proposes robust subsampling techniques, specifically Adaptive Importance Sampling (AIS) and Stratified Subsampling, designed to enhance the reliability and efficiency of parameter estimation.

Abstract

This paper addresses the challenge of estimating high-dimensional parameters in non-standard data environments, where traditional methods often falter due to issues such as heavy-tailed distributions, data contamination, and dependent observations. We propose robust subsampling techniques, specifically Adaptive Importance Sampling (AIS) and Stratified Subsampling, designed to enhance the reliability and efficiency of parameter estimation. Under some clearly outlined conditions, we establish consistency and asymptotic normality for the proposed estimators, providing non-asymptotic error bounds that quantify their performance. Our theoretical foundations are complemented by controlled experiments demonstrating the superiority of our methods over conventional approaches. By bridging the gap between theory and practice, this work offers significant contributions to robust statistical estimation, paving the way for advancements in various applied domains.

Paper Structure

This paper contains 18 sections, 7 theorems, 60 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Theoretical Analysis
  4. Proposed Algorithms
  5. Adaptive Importance Subsampling (AIS) for High Dimensional Data
  6. Algorithm Overview
  7. Computational Complexity
  8. Theoretical Justification
  9. Stratified Subsampling for Non-Standard Environments
  10. Algorithm Overview
  11. Computational Complexity
  12. Theoretical Justification and Robustness
  13. Statistical Analysis
  14. Numerical studies
  15. Experimental Setup
  16. ...and 3 more sections

Key Result

Theorem 1

Convergence Rates for Subsampled Estimators in High Dimensions: Let $X_1, \ldots, X_n \in \mathbb{R}^p$ be i.i.d. samples from a distribution $P$ on $\mathbb{R}^p$, where $p$ may be large relative to $n$. Let $\theta^* \in \mathbb{R}^p$ be the parameter of interest, and let $\hat{\theta}_n$ be the f Then, for any $\delta \in (0,1)$, with probability at least $1 - \delta$, where $C$ and $C'$ are c

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 3.1
  • proof
  • Theorem 4
  • proof
  • ...and 4 more