For epsilon-global-DP Bernoulli bandits, the paper proves a tighter lower bound and matching upper bounds (up to a factor alpha that can approach 1) using a new quantity d_epsilon and a new DP-Chernoff concentration inequality.
Concentrated differential privacy for bandits
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Optimal Regret of Bernoulli Bandits under Global Differential Privacy
For epsilon-global-DP Bernoulli bandits, the paper proves a tighter lower bound and matching upper bounds (up to a factor alpha that can approach 1) using a new quantity d_epsilon and a new DP-Chernoff concentration inequality.