Thompson Sampling Itself is Differentially Private (2407.14879v1)

Abstract: In this work we first show that the classical Thompson sampling algorithm for multi-arm bandits is differentially private as-is, without any modification. We provide per-round privacy guarantees as a function of problem parameters and show composition over $T$ rounds; since the algorithm is unchanged, existing $O(\sqrt{NT\log N})$ regret bounds still hold and there is no loss in performance due to privacy. We then show that simple modifications -- such as pre-pulling all arms a fixed number of times, increasing the sampling variance -- can provide tighter privacy guarantees. We again provide privacy guarantees that now depend on the new parameters introduced in the modification, which allows the analyst to tune the privacy guarantee as desired. We also provide a novel regret analysis for this new algorithm, and show how the new parameters also impact expected regret. Finally, we empirically validate and illustrate our theoretical findings in two parameter regimes and demonstrate that tuning the new parameters substantially improve the privacy-regret tradeoff.

• The paper proves that classical Thompson Sampling, with Gaussian priors, inherently satisfies differential privacy without algorithmic modifications.
• It explores modifications like pre-pulling and increased sampling variance to fine-tune the privacy-regret trade-off while maintaining strong performance.
• Empirical results validate theoretical privacy guarantees and demonstrate the applicability of TS to various online learning problems with minimal changes.

Thompson Sampling Itself is Differentially Private

The paper "Thompson Sampling Itself is Differentially Private" by Tingting Ou, Marco Avella Medina, and Rachel Cummings presents a significant analysis and a set of theoretical results demonstrating that the classical Thompson Sampling (TS) algorithm for multi-arm bandits inherently satisfies differential privacy (DP) without any modifications. This insight is crucial for researchers considering privacy in sequential decision-making processes, particularly in online learning and multi-arm bandit problems.

Key Contributions

The paper offers several key contributions:

1. Privacy Guarantees for Thompson Sampling: The authors show that the TS algorithm, when initialized with Gaussian priors on reward distributions, is differentially private as-is. More specifically, they provide per-round privacy guarantees in terms of Gaussian Differential Privacy (GDP) and extend these to cumulative privacy guarantees for $T$ rounds. This analysis capitalizes on the noise implicitly added in the TS sampling process, which aligns with the Gaussian Mechanism in DP literature.
2. Tuning Privacy-Performance Trade-off:

Additionally, the paper explores modifications to the TS algorithm to provide tighter privacy guarantees. Two modifications are discussed: - Pre-pulling each arm a fixed number of times before starting the TS process. - Increasing the variance in the sampling process.

These modifications allow for a better privacy-regret trade-off, enabling researchers to tune the privacy parameters according to their needs. The authors also provide novel regret bounds for these modified algorithms, showing how these parameters impact expected regret.

1. Empirical Validation: Theoretical findings are empirically validated through experiments with various parameter regimes. By tuning the new parameters (pre-pulling count and sampling variance), substantial improvements in the privacy-regret tradeoff are demonstrated, affirming the theoretical results.

Theoretical Implications

The paper’s findings align with and extend existing work on the privacy properties of randomized algorithms. Specifically:

• Privacy without Performance Loss:

The authors show that since the TS algorithm does not need modification to ensure privacy, the existing regret bounds $\mathcal{O}(\sqrt{NT\log N})$ for TS hold without incurring additional loss due to privacy.

• Applicability to Wide Range of Online Learning Problems:

The DP guarantees for TS extend to various other applications of TS, including contextual bandits, combinatorial semi-bandits, and online optimization problems.

Practical Implications

From a practical standpoint, the results presented in the paper have substantial implications:

• Direct Applicability:

Many systems already employing TS can claim differential privacy guarantees with minimal changes. This is particularly beneficial for applications where privacy is of utmost concern, such as clinical trials, recommendation systems, and personalized online services.

• Enhanced Privacy Controls:

Introducing pre-pulling and variance scaling provides practitioners with tools to adjust the privacy levels of their systems without extensive overhauls. This tunability is valuable for balancing between privacy constraints and system performance requirements.

Future Developments

The results achieved in this paper open several avenues for future research:

• Further Refinement of Privacy Analysis:

Future work may aim to refine the privacy guarantees even further, possibly tightening bounds under specific conditions or leveraging different mathematical techniques.

• Expansion to Other Algorithms:

The methodologies and findings of this work can inspire similar analyses and modifications for other algorithms in online learning and adaptive data collection, broadening the scope of differentially private algorithmic tools available to practitioners.

• Extended Empirical Studies:

Additional empirical studies across diverse application domains can help in understanding the practical performance of these modifications and in validating the theoretical guarantees under varying real-world conditions.

Conclusion

This paper makes a significant contribution to the understanding of inherent privacy guarantees within classical algorithms and provides practical methods for improving the privacy-performance trade-off. By showing that Thompson Sampling is differentially private as-is and suggesting modifications for tighter privacy, the authors provide powerful tools for researchers and practitioners to ensure privacy in sequential decision-making systems without sacrificing performance.