Pure-DP Statistical Query Release at the Conjectured Square-Root Rate

2026-07-22

AI Summary

This work addresses the problem of releasing $k$ statistical queries under pure differential privacy while achieving, in the worst coordinate, an expected error rate of $\sqrt{\log k / (\varepsilon n)}$, matching the known theoretical lower bound. To this end, we enhance the private multiplicative weights mechanism by replacing the original probability mass function with a distance-penalized likelihood envelope and integrating Maurey’s sparsification technique at the likelihood level with a grouping-radius strategy. This approach yields the first proof of Nikolov and Ullman’s conjectured upper bound on square-root error rates, circumventing the extraneous $1/\varepsilon$ factor inherent in prior methods. Our mechanism attains optimal error $O(\min\{1, \sqrt{\log(2T)\log(2k)/(\varepsilon n)}\})$ without additional assumptions and is formally verified for correctness and privacy guarantees in Lean 4.

Abstract

Nikolov and Ullman asked whether k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds. We prove their conjectured upper bound. For every database size n and privacy parameter $\varepsilon>0$, there is an $\varepsilon$-differentially private mechanism with expected error $O(\min\{1,\sqrt{\log(2T)\log(2k)/(\varepsilon n)}\})$. This matches the lower-bound dependence in the standard high-dimensional regimes where those bounds apply; the shifted logarithms and outer minimum make the upper bound valid without additional parameter assumptions. The construction starts from a selection-only private multiplicative weights transcript, then replaces its probability mass function by a distance-penalized likelihood envelope. To prove that the modification preserves accuracy, a likelihood-level Maurey argument upper-bounds each Hamming-ball maximum by a small family of auxiliary PMW laws. Renyi moment bounds control nearby balls, a direct mixture bound controls distant balls, and grouping radii at the privacy scale prevents an additional $1/\varepsilon$ factor in the error. The mechanism is information-theoretic. A companion Lean 4 development machine-checks the finite construction, pure privacy after deterministic decoding, and the displayed all-regimes upper bound.

Problem

• differential privacy
• statistical query release
• square-root rate
• expected error
• privacy-preserving

Innovation

• pure differential privacy
• statistical query release
• square-root error rate
• multiplicative weights
• formal verification