On The Differential Privacy of Thompson Sampling With Gaussian Prior
This provides a privacy guarantee for a widely used bandit algorithm with minimal performance degradation, benefiting practitioners in privacy-sensitive applications like online advertising or healthcare.
The paper demonstrates that Thompson Sampling with Gaussian Prior is inherently differentially private, achieving a privacy loss of O(ln^2 T) after T rounds, and shows that controlling privacy to any ε level increases regret by only O(ln^2 T/ε), outperforming prior work that required O(K ln^3 T/ε^2) regret increase.
We show that Thompson Sampling with Gaussian Prior as detailed by Algorithm 2 in (Agrawal & Goyal, 2013) is already differentially private. Theorem 1 show that it enjoys a very competitive privacy loss of only $\mathcal{O}(\ln^2 T)$ after T rounds. Finally, Theorem 2 show that one can control the privacy loss to any desirable $ε$ level by appropriately increasing the variance of the samples from the Gaussian posterior. And this increases the regret only by a term of $\mathcal{O}(\frac{\ln^2 T}ε)$. This compares favorably to the previous result for Thompson Sampling in the literature ((Mishra & Thakurta, 2015)) which adds a term of $\mathcal{O}(\frac{K \ln^3 T}{ε^2})$ to the regret in order to achieve the same privacy level. Furthermore, our result use the basic Thompson Sampling with few modifications whereas the result of (Mishra & Thakurta, 2015) required sophisticated constructions.