🤖 AI Summary
This paper investigates the fundamental statistical estimation limits under user-level local differential privacy (LDP) in the multi-observation setting: each of $n$ users holds $T$ independent observations. The authors establish the first general information-theoretic lower bound for user-level LDP, revealing a $T$-driven phase transition in both mean estimation and nonparametric density estimation—namely, a critical threshold exists below which estimation risk does not vanish with $n$, and above which consistent estimation becomes possible. They further demonstrate that high-dimensional sparse mean estimation is feasible under user-level LDP but impossible under standard item-level LDP. Tight (up to logarithmic factors) minimax upper and lower bounds are derived for univariate/multivariate mean estimation, sparse mean estimation, and density estimation, explicitly characterizing the critical interplay among $T$, dimension $d$, and sparsity $s$. These results provide novel feasibility criteria for statistical inference under user-level privacy constraints.
📝 Abstract
Most of the literature on differential privacy considers the item-level case where each user has a single observation, but a growing field of interest is that of user-level privacy where each of the $n$ users holds $T$ observations and wishes to maintain the privacy of their entire collection. In this paper, we derive a general minimax lower bound, which shows that, for locally private user-level estimation problems, the risk cannot, in general, be made to vanish for a fixed number of users even when each user holds an arbitrarily large number of observations. We then derive matching, up to logarithmic factors, lower and upper bounds for univariate and multidimensional mean estimation, sparse mean estimation and non-parametric density estimation. In particular, with other model parameters held fixed, we observe phase transition phenomena in the minimax rates as $T$ the number of observations each user holds varies. In the case of (non-sparse) mean estimation and density estimation, we see that, for $T$ below a phase transition boundary, the rate is the same as having $nT$ users in the item-level setting. Different behaviour is however observed in the case of $s$-sparse $d$-dimensional mean estimation, wherein consistent estimation is impossible when $d$ exceeds the number of observations in the item-level setting, but is possible in the user-level setting when $T gtrsim s log (d)$, up to logarithmic factors. This may be of independent interest for applications as an example of a high-dimensional problem that is feasible under local privacy constraints.