Learning discrete distributions: user vs item-level privacy
This addresses the challenge of providing stronger privacy guarantees in applications like federated learning, where protecting all items per user is crucial, offering a near-optimal solution with concrete improvements over existing methods.
The paper tackles the problem of learning discrete distributions under user-level differential privacy, showing that standard mechanisms require a number of users scaling as O(k/(mα^2) + k/εα) to achieve an ℓ₁ distance of α, and proposes a new mechanism that reduces this to Õ(k/(mα^2) + k/(√m εα)), achieving a √m times smaller privacy penalty in certain settings.
Much of the literature on differential privacy focuses on item-level privacy, where loosely speaking, the goal is to provide privacy per item or training example. However, recently many practical applications such as federated learning require preserving privacy for all items of a single user, which is much harder to achieve. Therefore understanding the theoretical limit of user-level privacy becomes crucial. We study the fundamental problem of learning discrete distributions over $k$ symbols with user-level differential privacy. If each user has $m$ samples, we show that straightforward applications of Laplace or Gaussian mechanisms require the number of users to be $\mathcal{O}(k/(mα^2) + k/εα)$ to achieve an $\ell_1$ distance of $α$ between the true and estimated distributions, with the privacy-induced penalty $k/εα$ independent of the number of samples per user $m$. Moreover, we show that any mechanism that only operates on the final aggregate counts should require a user complexity of the same order. We then propose a mechanism such that the number of users scales as $\tilde{\mathcal{O}}(k/(mα^2) + k/\sqrt{m}εα)$ and hence the privacy penalty is $\tildeΘ(\sqrt{m})$ times smaller compared to the standard mechanisms in certain settings of interest. We further show that the proposed mechanism is nearly-optimal under certain regimes. We also propose general techniques for obtaining lower bounds on restricted differentially private estimators and a lower bound on the total variation between binomial distributions, both of which might be of independent interest.