  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.
