Local Differential Privacy (LDP) addresses significant privacy concerns in sensitive data collection. In this work, we focus on numerical data collection under LDP, targeting a significant gap in the literature: existing LDP mechanisms are optimized for either a very small ($|\Omega| \in \{ 2, 3\}$) or infinite output spaces. However, no generalized method for constructing an optimal mechanism for an arbitrary output size $N$ exists. To fill this gap, we propose the \textbf{ N-output mechanism}, a generalized framework that maps numerical data to one of $N$ discrete outputs. We formulate the mechanism's design as an optimization problem to minimize estimation variance for any given $N \geq 2$ and develop both numerical and analytical solutions. This results in a mechanism that is highly accurate and adaptive, as its design is determined by solving an optimization problem for any chosen $N$. Furthermore, we extend our framework and existing mechanisms to the task of distribution estimation. Empirical evaluations show that the N-output mechanism achieves state-of-the-art accuracy for mean, variance, and distribution estimation with small communication costs.
Categories: cs.CR