Preserving Privacy in Multi-Armed Bandits through Concentrated Differential Privacy

Core Concepts

Designing multi-armed bandit algorithms that preserve privacy through zero Concentrated Differential Privacy (zCDP) while maintaining near-optimal regret.

Abstract

The paper investigates the problem of preserving privacy in multi-armed bandit (MAB) problems through the framework of Differential Privacy (DP). It focuses on the relaxation of pure DP, known as zero Concentrated Differential Privacy (zCDP), and its implications on the regret of MAB algorithms.
The key contributions are:

Formalizing and comparing different adaptations of DP to the bandit setting, including Table DP and View DP, and highlighting the differences between them, especially for relaxations of pure DP.

Proposing three private MAB algorithms, AdaC-UCB, AdaC-GOPE, and AdaC-OFUL, for finite-armed bandits, linear bandits, and linear contextual bandits, respectively. These algorithms share a common blueprint of adding Gaussian noise and running in adaptive episodes to ensure zCDP.

Analyzing the regret of the proposed algorithms and showing that the price of zCDP is asymptotically negligible compared to the non-private regret. Specifically, the additional regret due to zCDP is ˜O(ρ^(-1/2) log(T)), where ρ is the zCDP parameter and T is the horizon.

Proving the first minimax lower bounds on the regret of bandits with zCDP, which quantify the hardness of preserving privacy in these settings. The lower bounds show that the proposed algorithms are optimal, up to poly-logarithmic factors.

Experimentally validating the theoretical insights on the performance of the proposed private algorithms in different bandit settings.

Stats

The paper does not contain any explicit numerical data or statistics. The key results are presented in the form of regret upper and lower bounds.

Quotes

"Bandits serve as the theoretical foundation of sequential learning and an algorithmic foundation of modern recommender systems. However, recommender systems often rely on user-sensitive data, making privacy a critical concern."
"The goal of the policy is to reveal the sequence of actions while protecting the privacy of the users and achieving minimal regret."
"Our analysis shows that in all of these settings, the prices of imposing zCDP are (asymptotically) negligible in comparison with the regrets incurred oblivious to privacy."

Key Insights Distilled From

Concentrated Differential Privacy for Bandits

by Achraf Azize... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2309.00557.pdf

Deeper Inquiries

How can the proposed private bandit algorithms be extended to settings where the contexts are also considered private information

To extend the proposed private bandit algorithms to settings where contexts are also considered private information, we need to adapt the privacy definitions and algorithm design to account for the additional privacy constraints. In the contextual bandit setting where contexts are private, both the contexts and the rewards need to be protected to ensure comprehensive privacy.
One approach to address this extension is to modify the privacy definitions to include the privacy of both contexts and rewards. This would involve formulating a new privacy definition that considers the joint privacy of contexts and rewards, ensuring that the algorithm's output does not reveal sensitive information about either the contexts or the rewards. This new definition could be a combination of the existing definitions for context privacy and reward privacy, tailored to the specific requirements of the contextual bandit setting with private contexts.
In terms of algorithm design, the extension to private contexts would require adjustments in how the algorithm handles and processes context information. The algorithm would need to incorporate mechanisms to protect the privacy of contexts while still making effective decisions based on the available information. This could involve introducing additional noise or encryption techniques to mask the context information while maintaining the utility of the algorithm.
Overall, extending the private bandit algorithms to settings with private contexts would involve redefining the privacy constraints and adapting the algorithm design to ensure the privacy of both contexts and rewards throughout the decision-making process.

Can the techniques used to prove the minimax lower bounds for zCDP be applied to other relaxations of pure Differential Privacy, such as (ε, δ)-DP

The techniques used to prove the minimax lower bounds for zero Concentrated Differential Privacy (zCDP) can potentially be applied to other relaxations of pure Differential Privacy, such as (ε, δ)-DP. The key lies in understanding the fundamental principles of privacy-preserving algorithms and the trade-offs between privacy and utility in different settings.
When applying these techniques to other relaxations of Differential Privacy, researchers would need to consider the specific characteristics and requirements of the privacy definition being used. For (ε, δ)-DP, which allows for a controlled amount of privacy loss, the minimax lower bounds would need to account for this additional parameter and its impact on the algorithm's performance.
The general approach would involve adapting the proof techniques to accommodate the specific parameters and constraints of the chosen privacy definition. This may involve modifying the analysis to incorporate the epsilon and delta values, adjusting the privacy guarantees accordingly, and demonstrating the trade-offs between privacy, utility, and the complexity of the algorithm.
By leveraging the foundational concepts and methodologies used in proving minimax lower bounds for zCDP, researchers can apply similar principles to explore the privacy-utility trade-offs in other variants of Differential Privacy, providing valuable insights into the optimal design and performance of privacy-preserving algorithms.

What are the potential applications of the private bandit algorithms beyond recommender systems, and how can the privacy-utility trade-offs be further optimized for those applications

The private bandit algorithms proposed in the context of recommender systems have the potential for various applications beyond this domain. Some potential applications include healthcare decision-making, financial portfolio management, and personalized marketing strategies. In healthcare, private bandit algorithms could be used to recommend treatment options while protecting patient privacy. In finance, these algorithms could assist in making investment decisions while preserving the confidentiality of financial data. In marketing, private bandit algorithms could optimize personalized recommendations without compromising customer privacy.
To further optimize the privacy-utility trade-offs for these applications, researchers can explore advanced privacy-enhancing techniques such as homomorphic encryption, secure multi-party computation, and differential privacy mechanisms tailored to specific use cases. By integrating these techniques into the algorithm design, it is possible to enhance privacy protections while maintaining the utility and effectiveness of the recommendations.
Additionally, conducting thorough empirical evaluations and real-world case studies in diverse application domains can provide valuable insights into the performance and scalability of private bandit algorithms. By iteratively refining the algorithms based on empirical results and feedback from domain experts, researchers can fine-tune the privacy-utility trade-offs to meet the requirements of specific applications and ensure the successful deployment of private bandit algorithms in practical settings.