Efficient MCMC Sampling for Private Learning with Optimal Utility under Pure and Gaussian Differential Privacy

Core Concepts

The authors propose the Approximate Sample Perturbation (ASAP) algorithm, which leverages an MCMC sampler to maintain pure differential privacy (DP) and pure Gaussian DP guarantees, while achieving the optimal utility rates for strongly convex and smooth losses in nearly linear time.

Abstract

The paper addresses the challenge of efficiently implementing posterior sampling mechanisms for private learning, which provide strong privacy guarantees (pure DP and pure Gaussian DP) but suffer from computational intractability when using exact sampling. The key contributions are: The Approximate Sample Perturbation (ASAP) algorithm, which perturbs an MCMC sample with noise proportional to its Wasserstein-infinity distance from a reference distribution that satisfies pure DP or pure Gaussian DP. This allows ASAP to maintain the strong privacy guarantees. A novel lemma that converts Total Variation (TV) distance bounds to Wasserstein-infinity distance bounds, enabling the use of TV-convergent MCMC samplers while preserving pure DP. The Metropolis-adjusted Langevin algorithm (MALA) with constraint, designed to converge in Wasserstein-infinity distance, which is then used within the ASAP framework. An end-to-end algorithm that combines localization, MALA with constraint, and ASAP to achieve the optimal utility rates for strongly convex and smooth losses in nearly linear time, while satisfying pure DP or pure Gaussian DP. The authors show that their approach is the first to obtain the optimal rates for DP-ERM under pure DP or pure Gaussian DP in nearly linear time.

Stats

The individual loss functions ℓi are G-Lipschitz, β-smooth, and α-strongly convex. The domain Θ is a convex set such that B(θ*, R1) ⊂ Θ, where R1 ≥ 8√(d/(γnα)).

Quotes

"We propose the Approximate SAample Perturbation (abbr. ASAP) algorithm which perturbs an MCMC sample with noise proportional to its Wasserstein-infinity (W∞) distance from a reference distribution that satisfies pure DP or pure Gaussian DP (i.e., δ = 0)." "We show that by combining our new techniques with a careful localization step, we obtain the first nearly linear-time algorithm that achieves the optimal rates in the DP-ERM problem with strongly convex and smooth losses."

Key Insights Distilled From

Tractable MCMC for Private Learning with Pure and Gaussian Differential Privacy

by Yingyu Lin,Y... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2310.14661.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to achieve optimal rates for DP-ERM with general convex and Lipschitz losses, beyond the strongly convex and smooth setting

To extend the techniques developed in this paper to achieve optimal rates for DP-ERM with general convex and Lipschitz losses, beyond the strongly convex and smooth setting, several adjustments and enhancements can be made: Adapting the Localization Technique: The localization technique used in the paper can be modified to handle general convex and Lipschitz losses. This may involve incorporating additional constraints or regularization terms to account for the specific characteristics of the loss functions. Exploring Different Perturbation Strategies: Instead of relying solely on the ASAP framework, exploring alternative perturbation strategies tailored to the characteristics of general convex and Lipschitz losses could lead to improved performance and optimality in DP-ERM. Incorporating Advanced Optimization Methods: Leveraging advanced optimization methods that are specifically designed for convex and Lipschitz functions can enhance the efficiency and convergence properties of the algorithm, ultimately leading to optimal rates in DP-ERM. Fine-tuning Parameters: Fine-tuning the parameters used in the algorithms, such as step sizes, regularization terms, and initialization values, can significantly impact the performance and optimality of the algorithm for general convex and Lipschitz losses. By incorporating these adjustments and enhancements, the techniques developed in this paper can be extended to achieve optimal rates for DP-ERM with general convex and Lipschitz losses.

Can the ASAP framework be adapted to other private learning problems beyond DP-ERM, such as private Bayesian inference or private reinforcement learning

The ASAP framework can indeed be adapted to other private learning problems beyond DP-ERM, such as private Bayesian inference or private reinforcement learning. Here's how the ASAP framework can be applied to these scenarios: Private Bayesian Inference: In private Bayesian inference, the ASAP framework can be used to perturb the MCMC samples to ensure privacy guarantees while maintaining the utility of the inference. By adding noise proportional to the Wasserstein-infinity distance, the ASAP algorithm can provide pure differential privacy guarantees in the Bayesian setting. Private Reinforcement Learning: In private reinforcement learning, the ASAP framework can be utilized to perturb the policy updates or value function estimates to ensure privacy while learning from interactions with the environment. By incorporating noise based on the distance from a reference distribution, ASAP can enable private reinforcement learning algorithms to achieve optimal utility under differential privacy constraints. By adapting the ASAP framework to these contexts, researchers and practitioners can develop efficient and privacy-preserving algorithms for a wide range of private learning problems.

What are the potential applications and real-world implications of having an efficient private learning algorithm that achieves optimal utility under pure differential privacy guarantees

The development of an efficient private learning algorithm that achieves optimal utility under pure differential privacy guarantees has significant applications and real-world implications, including: Privacy-Preserving Machine Learning: The algorithm can be applied in various machine learning tasks where privacy is a concern, such as healthcare, finance, and social media, ensuring that sensitive data is protected while maintaining the quality of the learning models. Compliance with Data Regulations: Organizations handling sensitive data can use this algorithm to comply with data privacy regulations like GDPR and HIPAA, ensuring that user information is kept confidential during the learning process. Secure Collaborative Learning: The algorithm enables secure collaborative learning scenarios where multiple parties can contribute data without compromising individual privacy, fostering collaboration in research and development. Personalized Services: By preserving privacy while learning from user data, companies can offer personalized services and recommendations without infringing on user privacy, enhancing user experience and satisfaction. Overall, having an efficient private learning algorithm under pure differential privacy guarantees opens up opportunities for secure and ethical data-driven applications across various industries and domains.