Improved regret bounds in tabular RL under local differential privacy

arXiv:2611.03026v1 Announce Type: new Abstract: We study regret minimization for time-homogeneous episodic tabular reinforcement learning with $S$ states and $A$ actions under non-interactive $\epsilon$-local differential privacy (LDP). In each episode $k \in [K]$, the decision-maker sends a policy to a user, the user executes the policy for $H$ steps, and sends an $\epsilon$-locally differentially private message back to the decision-maker. When $H=1$, the problem reduces to an LDP contextual bandit with $S$ contexts, $A$ arms and horizon $K$. We prove that the minimax regret is $\min\bigg(K, \frac{ \sqrt{\max( S, \exp(\epsilon) + 1) S A K (\exp(\epsilon) + 1)}}{\exp(\epsilon) - 1}\bigg)$, up to constants and a $\sqrt{\log(A)}$ factor. The upper bound is achieved by an adaptation of EXP3 and a unary encoding mechanism. When $H > 1$, and for small $\epsilon$, we show a lower bound in $\Omega(HS\frac{\sqrt{A K}}{\epsilon})$ and an upper bound in $\tilde{O}(H^2 S^2 \frac{\sqrt{ AK}}{\epsilon})$. The upper bound is achieved by an algorithm based on UCBVI and a unary encoding mechanism. For small $\epsilon$, these bounds improve the previously known lower bound by a factor $\sqrt{S}$ and the previously known upper bound by a factor of $H \sqrt{A}$.