Differentially Private Stochastic Linear Bandits: (Almost) for Free

In this paper, we propose differentially private algorithms for the problem of stochastic linear bandits in the central, local and shuffled models. In the central model, we achieve almost the same regret as the optimal non-private algorithms, which means we get privacy for free. In particular, we achieve a regret of $\tilde{O}(\sqrt{T}+\frac{1}{\epsilon})$ matching the known lower bound for private linear bandits, while the best previously known algorithm achieves $\tilde{O}(\frac{1}{\epsilon}\sqrt{T})$. In the local case, we achieve a regret of $\tilde{O}(\frac{1}{\epsilon}{\sqrt{T}})$ which matches the non-private regret for constant $\epsilon$, but suffers a regret penalty when $\epsilon$ is small. In the shuffled model, we also achieve regret of $\tilde{O}(\sqrt{T}+\frac{1}{\epsilon})$ %for small $\epsilon$ as in the central case, while the best previously known algorithm suffers a regret of $\tilde{O}(\frac{1}{\epsilon}{T^{3/5}})$. Our numerical evaluation validates our theoretical results.