Connecting Thompson Sampling and UCB: Towards More Efficient Trade-offs Between Privacy and Regret

TL;DR

This paper introduces DP-TS-UCB, a private bandit algorithm that balances privacy and regret by leveraging Gaussian mechanisms and exploring their connections, with theoretical guarantees.

Contribution

It proposes a novel parametrized private bandit algorithm connecting Thompson Sampling and UCB, enabling adjustable privacy-regret trade-offs with theoretical analysis.

Findings

Achieves a privacy-regret trade-off controlled by parameter α

Provides regret bounds of O(K log^{α+1}(T)/Δ)

Establishes links between exploration mechanisms in Thompson Sampling and UCB

Abstract

We address differentially private stochastic bandit problems from the angles of exploring the deep connections among Thompson Sampling with Gaussian priors, Gaussian mechanisms, and Gaussian differential privacy (GDP). We propose DP-TS-UCB, a novel parametrized private bandit algorithm that enables to trade off privacy and regret. DP-TS-UCB satisfies $ \tilde{O} \left(T^{0.25(1-\alpha)}\right)$-GDP and enjoys an $O \left(K\ln^{\alpha+1}(T)/\Delta \right)$ regret bound, where $\alpha \in [0,1]$ controls the trade-off between privacy and regret. Theoretically, our DP-TS-UCB relies on anti-concentration bounds of Gaussian distributions and links exploration mechanisms in Thompson Sampling-based algorithms and Upper Confidence Bound-based algorithms, which may be of independent interest.