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Testing Credibility of Public and Private Surveys through the Lens of Regression

Debabrota Basu, Sourav Chakraborty, Debarshi Chanda, Buddha Dev Das, Arijit Ghosh, Arnab Ray

TL;DR

An algorithm to test the credibility of a sample survey in terms of linear regression, which serves as a mechanism to learn linear regression models from data corrupted with noise coming from any subexponential distribution and achieves the optimal estimation error bound for linear regression, which might be of broader interest.

Abstract

Testing whether a sample survey is a credible representation of the population is an important question to ensure the validity of any downstream research. While this problem, in general, does not have an efficient solution, one might take a task-based approach and aim to understand whether a certain data analysis tool, like linear regression, would yield similar answers both on the population and the sample survey. In this paper, we design an algorithm to test the credibility of a sample survey in terms of linear regression. In other words, we design an algorithm that can certify if a sample survey is good enough to guarantee the correctness of data analysis done using linear regression tools. Nowadays, one is naturally concerned about data privacy in surveys. Thus, we further test the credibility of surveys published in a differentially private manner. Specifically, we focus on Local Differential Privacy (LDP), which is a standard technique to ensure privacy in surveys where the survey participants might not trust the aggregator. We extend our algorithm to work even when the data analysis has been done using surveys with LDP. In the process, we also propose an algorithm that learns with high probability the guarantees a linear regression model on a survey published with LDP. Our algorithm also serves as a mechanism to learn linear regression models from data corrupted with noise coming from any subexponential distribution. We prove that it achieves the optimal estimation error bound for $\ell_1$ linear regression, which might be of broader interest. We prove the theoretical correctness of our algorithms while trying to reduce the sample complexity for both public and private surveys. We also numerically demonstrate the performance of our algorithms on real and synthetic datasets.

Testing Credibility of Public and Private Surveys through the Lens of Regression

TL;DR

An algorithm to test the credibility of a sample survey in terms of linear regression, which serves as a mechanism to learn linear regression models from data corrupted with noise coming from any subexponential distribution and achieves the optimal estimation error bound for linear regression, which might be of broader interest.

Abstract

Testing whether a sample survey is a credible representation of the population is an important question to ensure the validity of any downstream research. While this problem, in general, does not have an efficient solution, one might take a task-based approach and aim to understand whether a certain data analysis tool, like linear regression, would yield similar answers both on the population and the sample survey. In this paper, we design an algorithm to test the credibility of a sample survey in terms of linear regression. In other words, we design an algorithm that can certify if a sample survey is good enough to guarantee the correctness of data analysis done using linear regression tools. Nowadays, one is naturally concerned about data privacy in surveys. Thus, we further test the credibility of surveys published in a differentially private manner. Specifically, we focus on Local Differential Privacy (LDP), which is a standard technique to ensure privacy in surveys where the survey participants might not trust the aggregator. We extend our algorithm to work even when the data analysis has been done using surveys with LDP. In the process, we also propose an algorithm that learns with high probability the guarantees a linear regression model on a survey published with LDP. Our algorithm also serves as a mechanism to learn linear regression models from data corrupted with noise coming from any subexponential distribution. We prove that it achieves the optimal estimation error bound for linear regression, which might be of broader interest. We prove the theoretical correctness of our algorithms while trying to reduce the sample complexity for both public and private surveys. We also numerically demonstrate the performance of our algorithms on real and synthetic datasets.
Paper Structure (33 sections, 28 theorems, 75 equations, 4 figures, 2 tables, 5 algorithms)

This paper contains 33 sections, 28 theorems, 75 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Testing Credibility of Surveys.
  4. Learning Linear Regression on Private Surveys.
  5. Our Contributions
  6. Notations and Preliminaries
  7. Linear Regression
  8. Local Differential Privacy
  9. $\texttt{SurVerify}$: A Framework for Testing
  10. High-level Overview of the Correctness of $\texttt{SurVerify}$.
  11. Publishing Surveys & Regression with LDP
  12. Publishing Survey with LDP
  13. Regression on Noisy Covariates
  14. Experimental Analysis
  15. Experimental Setup.
  16. ...and 18 more sections

Key Result

Lemma 6

For any $\alpha >0$ and $\beta\in (0,1]$, Gaussian mechanism $\mathcal{M}$ ensuring $(\alpha,\beta)$-DP for any ${f} : {\mathcal{X}} \rightarrow {\mathbb{R}^d}$ with $\ell_2$-sensitivity $\Delta_2(f)$ is $\mathcal{M}(\mathbf{x}) \triangleq f(\mathbf{x}) + Z$, where $Z \sim \mathcal{N}(0,\sigma^2\mat

Figures (4)

  • Figure 1: Rejection rate of $\texttt{SurVerify}$ on Synthetic_1 vs. model distance (mean over 30 runs) for $\delta$ = 0.1 and different tolerance parameters $\epsilon$.
  • Figure 2: Estimation error of LASSO-SEN (mean $\pm$ std. over 30 runs) for $(\alpha,\beta)$-LDP version of ACS_ Income with $\beta = 0.1$ and different $\alpha$.
  • Figure 3: Estimation error of LASSO-SEN (mean $\pm$ std. over 30 runs) for $\alpha$-LDP version of ACS_ Income for different $\alpha$.
  • Figure 4: Estimation error of LASSO-SEN (mean $\pm$ std. over 30 runs) for subgaussian and subexponential noise for different dimensions of covariates.

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: $(\alpha,\beta)$-Local Differential Privacy
  • Definition 5: $\ell_p$-Sensitivity
  • Lemma 6: Gaussian Mechanism
  • Lemma 7: Laplacian Mechanism
  • Theorem 8: Correctness of $\texttt{SurVerify}$
  • Remark 1: Extensions to LDP Surveys
  • Remark 2: One-Sidedness of $\texttt{SurVerify}$
  • ...and 47 more