In this paper, we study the offline and online settings of reinforcement learning from human feedback (RLHF) with KL-regularization -- a widely used objective function in large language model alignment -- under the $\epsilon$ local differential privacy ($\epsilon$-LDP) model on the label of the human preference. In the offline setting, we design an algorithm based on the principle of pessimism and derive a new suboptimality gap of $\tilde{O}(1/[(e^\epsilon-1)^2 n])$ on the KL-regularized objective under single-policy concentrability. We also prove its optimality by providing a matching lower bound where $n$ is the sample size. In the online setting, we are the first one to theoretically investigate the problem of KL-regularized RLHF with LDP. We design an optimism-based algorithm and derive a logarithmic regret bound of $O(d_{\mathcal{F}}\log (N_{\mathcal{F}}\cdot T) /(e^\epsilon-1)^2 )$, where $T$ is the total time step, $N_{\mathcal{F}}$ is cardinality of the reward function space $\mathcal{F}$ and $d_{\mathcal{F}}$ is a variant of eluder dimension for RLHF. As a by-product of our analysis, our results also imply the first analysis for online KL-regularized RLHF without privacy. We implement our algorithm in the offline setting to verify our theoretical results and release our open source code at: https://github.com/rushil-thareja/PPKL-RLHF-Official.

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