On The Differential Privacy of Thompson Sampling With Gaussian Prior

TL;DR

This paper proves that Thompson Sampling with Gaussian prior inherently provides differential privacy with low privacy loss, and shows how to control this loss with minimal impact on regret, outperforming previous methods.

Contribution

It demonstrates that standard Gaussian Thompson Sampling is already differentially private and offers a simple way to control privacy loss without complex modifications.

Findings

Thompson Sampling with Gaussian prior has a privacy loss of O(ln^2 T).

Adjusting Gaussian variance controls privacy level with minimal regret increase.

Results outperform previous approaches requiring complex modifications.

Abstract

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 $\epsilon$ level by appropriately increasing the variance of the samples from the Gaussian posterior. And this increases the regret only by a term of $\mathcal{O}(\frac{\ln^2 T}{\epsilon})$. This compares favorably to the previous result for Thompson Sampling in the literature ((Mishra & Thakurta, 2015)) which adds a term of $\mathcal{O}(\frac{K \ln^3 T}{\epsilon^2})$ 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.