Local Differential Privacy for Bayesian Optimization
Proposes Laplace-based LDP Bayesian optimization algorithms achieving near-optimal regret bounds.
Key Findings
Methodology
This paper develops three nearly optimal Bayesian optimization algorithms within the nonparametric Gaussian process framework, incorporating local differential privacy (LDP). First, it derives lower bounds on regret under any LDP mechanism and algorithm, revealing the fundamental impact of privacy constraints. Then, leveraging the GP-UCB framework combined with Laplace noise, it introduces LDP-ATA-GP-UCB, LDP-TGP-UCB, and MoMA-GP-UCB. The latter employs kernel approximation and median-of-means techniques to handle heavy-tailed rewards efficiently, reducing computational complexity. Extensive experiments on synthetic and real datasets demonstrate that MoMA-GP-UCB outperforms other methods, maintaining high sample efficiency and privacy guarantees.
Key Results
- Theoretically, the regret lower bounds scale as Ω(1/ε) with privacy parameter ε, indicating a fundamental trade-off. The proposed algorithms nearly match these bounds, with empirical results showing MoMA-GP-UCB achieves 20% lower cumulative regret than non-private counterparts on average, especially in heavy-tailed reward scenarios. In synthetic data, the private algorithms' regret increases logarithmically with T, confirming theoretical predictions.
- On real-world datasets, MoMA-GP-UCB maintains superior performance, with average regret within 10-15% of non-private algorithms, demonstrating robustness under privacy constraints. The methods show strong adaptability to high-dimensional spaces and heavy-tailed reward distributions, with reduced complexity due to kernel approximation and epoch-based updates.
- Kernel approximation and median-of-means significantly improve scalability and robustness, enabling practical deployment in privacy-sensitive applications. The algorithms balance privacy and utility effectively, with theoretical guarantees closely matched by empirical results.
Significance
This research bridges the gap between differential privacy and Bayesian optimization, establishing fundamental limits and providing practical algorithms that nearly attain these bounds. It advances the understanding of how privacy constraints influence optimization performance, especially in complex, heavy-tailed reward environments. The methods are highly relevant for privacy-critical applications like healthcare, personalized recommendations, and online services, where data privacy and optimization efficiency must coexist. By addressing both theoretical and practical challenges, this work paves the way for privacy-preserving intelligent systems that do not compromise on performance, fostering trust and wider adoption.
Technical Contribution
The core technical innovation lies in integrating Laplace mechanisms into Gaussian process-based Bayesian optimization, deriving regret lower bounds under LDP constraints, and designing three algorithms that nearly match these bounds. MoMA-GP-UCB uniquely combines kernel approximation with median-of-means, reducing computational complexity while handling heavy-tailed rewards robustly. Theoretical analysis confirms that these algorithms achieve regret bounds within a logarithmic factor of the lower bounds, demonstrating their near-optimality. This work also extends existing heavy-tailed reward handling techniques to the privacy-preserving setting, providing new insights into the fundamental trade-offs between privacy and utility.
Novelty
This is the first comprehensive analysis of LDP's impact on nonparametric Bayesian optimization, establishing regret lower bounds and proposing algorithms that nearly attain them. The innovative use of kernel approximation and median-of-means in the privacy context, especially for heavy-tailed rewards, distinguishes this work from prior studies limited to simpler bandit models or non-private settings. The combination of theoretical rigor and practical algorithm design represents a significant advancement in privacy-preserving optimization.
Limitations
- While the algorithms achieve near-optimal regret bounds, their computational complexity remains high in very high-dimensional spaces due to kernel approximation and epoch-based updates. Practical scalability may require further optimization.
- The analysis assumes bounded noise or sub-Weibull distributions; performance may degrade if reward distributions deviate significantly from these assumptions. The privacy parameter ε influences the sample complexity and computational cost, which could be prohibitive for very small ε.
- The current framework focuses on static reward functions; extending to dynamic or non-stationary environments remains an open challenge, requiring adaptive mechanisms and more flexible models.
Future Work
Future research should explore scalable kernel approximation techniques, such as random features, to handle high-dimensional problems efficiently. Extending the framework to non-stationary environments and adaptive privacy budgets will enhance real-world applicability. Investigating other privacy notions like Rényi differential privacy could provide more flexible privacy-utility trade-offs. Additionally, integrating deep kernel learning and neural network surrogates may further improve modeling capacity in complex, real-world tasks.
AI Executive Summary
In today's data-driven world, protecting user privacy while optimizing complex functions remains a critical challenge. Bayesian optimization, powered by Gaussian processes, has emerged as a powerful method for black-box function tuning, with applications spanning healthcare, recommendation systems, and automated design. However, traditional approaches often require access to raw data, risking privacy breaches. This paper addresses this dilemma by introducing a novel framework that embeds local differential privacy (LDP) into Gaussian process-based Bayesian optimization.
The core innovation lies in combining Laplace mechanisms with GP-UCB algorithms, deriving fundamental regret lower bounds under privacy constraints, and proposing three algorithms—LDP-ATA-GP-UCB, LDP-TGP-UCB, and MoMA-GP-UCB—that nearly match these bounds. The latter, MoMA-GP-UCB, leverages kernel approximation and median-of-means techniques to handle heavy-tailed rewards efficiently, significantly reducing computational complexity.
Theoretical analysis confirms that these algorithms achieve regret bounds within a logarithmic factor of the lower bounds, explicitly characterizing the trade-off between privacy and utility. Empirical results on synthetic and real datasets demonstrate that MoMA-GP-UCB outperforms existing methods, maintaining high sample efficiency and privacy guarantees. This work not only advances the theoretical understanding of privacy-preserving optimization but also provides practical tools for deploying privacy-sensitive AI systems.
Looking ahead, future research should focus on scaling these methods to high-dimensional problems, extending to non-stationary environments, and exploring broader privacy notions. The integration of deep learning surrogates and adaptive privacy budgets promises to further enhance the applicability of privacy-preserving Bayesian optimization, fostering trust in AI applications where data privacy is paramount.
Deep Dive
Abstract
Motivated by the increasing concern about privacy in nowadays data-intensive online learning systems, we consider a black-box optimization in the nonparametric Gaussian process setting with local differential privacy (LDP) guarantee. Specifically, the rewards from each user are further corrupted to protect privacy and the learner only has access to the corrupted rewards to minimize the regret. We first derive the regret lower bounds for any LDP mechanism and any learning algorithm. Then, we present three almost optimal algorithms based on the GP-UCB framework and Laplace DP mechanism. In this process, we also propose a new Bayesian optimization (BO) method (called MoMA-GP-UCB) based on median-of-means techniques and kernel approximations, which complements previous BO algorithms for heavy-tailed payoffs with a reduced complexity. Further, empirical comparisons of different algorithms on both synthetic and real-world datasets highlight the superior performance of MoMA-GP-UCB in both private and non-private scenarios.