Efficient and Differentially Private High-dimensional Model Selection via Best Subset Selection

Core Concepts

A computationally efficient and differentially private algorithm for high-dimensional sparse model selection using the best subset selection approach.

Abstract

The paper considers the problem of model selection in a high-dimensional sparse linear regression model under privacy constraints. The authors propose a differentially private best subset selection (BSS) method with strong utility properties by adopting the well-known exponential mechanism for selecting the best model.
Key highlights:

The authors establish that the proposed exponential mechanism enjoys polynomial mixing time to its stationary distribution and provides approximate differential privacy for the final estimates.
They design an efficient Metropolis-Hastings algorithm that also enjoys desirable utility similar to the exponential mechanism under the differential privacy framework.
The authors provide theoretical guarantees for the utility of the proposed methods, showing that accurate model recovery is possible under certain signal strength conditions.
Illustrative experiments demonstrate the strong utility of the proposed algorithms compared to non-private best subset selection.

Stats

There exist positive constants r and xmax such that supy∈Y |y| ≤ r and supx∈X ∥x∥∞ ≤ xmax.
The true parameter vector β satisfies ∥β∥1 ≤ bmax.
The design matrix X satisfies the Sparse Riesz Condition with positive constants κ- and κ+.
The true sparsity level s follows s ≤ n/(log p).

Quotes

"We propose a differentially private best subset selection method with strong utility properties by adopting the well-known exponential mechanism for selecting the best model."
"We propose an efficient Metropolis-Hastings algorithm and establish that it enjoys polynomial mixing time to its stationary distribution."
"Furthermore, we also establish approximate differential privacy for the final estimates of the Metropolis-Hastings random walk using its mixing property."

Key Insights Distilled From

On the Computational Complexity of Private High-dimensional Model Selection

by Saptarshi Ro... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2310.07852.pdf

Deeper Inquiries

How can the proposed methods be extended to handle more general loss functions beyond the squared error loss

The proposed methods can be extended to handle more general loss functions beyond the squared error loss by incorporating the loss function into the score function used in the exponential mechanism. The score function can be tailored to reflect the specific loss function being used, allowing for the optimization of model selection under different loss functions. By adapting the score function to the loss function, the differential privacy framework can be applied to a wider range of statistical models and optimization problems, enabling more flexible and versatile applications in various domains.

Can the signal strength requirement for model consistency be further improved under the differential privacy constraint

The signal strength requirement for model consistency under the differential privacy constraint can potentially be further improved by refining the algorithm's utility properties and optimizing the privacy parameters. By enhancing the efficiency of the Metropolis-Hastings algorithm and fine-tuning the privacy budget allocation, it may be possible to reduce the signal strength requirement for model consistency. Additionally, exploring advanced privacy-preserving techniques and incorporating them into the model selection framework could lead to more stringent privacy guarantees without compromising the utility of the model selection process.

What are the potential applications of the differentially private model selection framework in domains such as healthcare, finance, or social sciences

The differentially private model selection framework has numerous potential applications in various domains such as healthcare, finance, and social sciences. In healthcare, the framework can be utilized for privacy-preserving analysis of sensitive patient data, enabling researchers to perform statistical analyses while protecting patient privacy. In finance, the framework can be applied to secure financial data for accurate model selection in risk assessment and investment strategies. In social sciences, the framework can support privacy-preserving data analysis for research studies involving sensitive information, ensuring confidentiality while extracting valuable insights from the data. Overall, the framework offers a robust solution for conducting statistical analyses in a privacy-preserving manner across diverse fields.