  Numerical vector aggregation plays a crucial role in privacy-sensitive
applications, such as distributed gradient estimation in federated learning and
statistical analysis of key-value data. In the context of local differential
privacy, this study provides a tight minimax error bound of
$O(\frac{ds}{n\epsilon^2})$, where $d$ represents the dimension of the
numerical vector and $s$ denotes the number of non-zero entries. By converting
the conditional/unconditional numerical mean estimation problem into a
frequency estimation problem, we develop an optimal and efficient mechanism
called Collision. In contrast, existing methods exhibit sub-optimal error rates
of $O(\frac{d^2}{n\epsilon^2})$ or $O(\frac{ds^2}{n\epsilon^2})$. Specifically,
for unconditional mean estimation, we leverage the negative correlation between
two frequencies in each dimension and propose the CoCo mechanism, which further
reduces estimation errors for mean values compared to Collision. Moreover, to
surpass the error barrier in local privacy, we examine privacy amplification in
the shuffle model for the proposed mechanisms and derive precisely tight
amplification bounds. Our experiments validate and compare our mechanisms with
existing approaches, demonstrating significant error reductions for frequency
estimation and mean estimation on numerical vectors.
