Mean Estimation of Numerical Data Under (ϵ,δ) -Utility-Optimized Local Differential Privacy

Utility-optimized local differential privacy (ULDP) considers input domain including non-sensitive values which reduces utility loss by leaking some non-sensitive values, without lowering protection to any sensitive one compared with local differential privacy (LDP). The existing ULDP mechanisms are designed under $\epsilon $ -ULDP which preserve sensitive values under $\epsilon $ -LDP. Nevertheless, it is still challenging to achieve $(\epsilon ,\delta)$ -ULDP. In this paper, we consider mean aggregation in $(\epsilon ,\delta)$ -ULDP, where sensitive values are protected under $(\epsilon ,\delta)$ -LDP. Specifically, we first propose One-Bit perturbation Mechanism (OBM) for unbiased mean estimation under $(\epsilon ,\delta)$ -LDP and then obtain optimal OBM by minimizing its worst-case error. In OBM, each output is a 1-bit value, and it thus is highly communication-efficient. Second, based on OBM, we design an unbiased mean estimation mechanism in $(\epsilon ,\delta)$ -ULDP, Utility-optimized OBM (UOBM), where sensitive values are strictly protected under $(\epsilon ,\delta)$ -LDP while non-sensitive ones could be disclosed to achieve higher utility. Further, we extend UOBM to the personalized scene where each user has specific privacy budget and sensitive range. Additionally, we theoretically and experimentally compare the proposed mechanisms with existing ones. The results show OBM outperforms existing mechanisms in utility, though its output is just a 1-bit value. UOBM can dramatically decrease the estimation error, compared with OBM.