AI 中文总结
研究在纯差分隐私下，大小为T的全域上k个统计查询能否以平方根速率发布，通过特定构造和论证证明了推测的上界，给出期望误差表达式，机制是信息理论性且有机器检查配套。
AI 中文摘要
Nikolov和Ullman提出问题：在大小为T的全域上，k个统计查询能否在纯差分隐私下以已知下界所暗示的平方根速率发布？我们证明了他们推测的上界。对于每个数据库大小n和隐私参数ε>0，存在一种ε - 差分隐私机制，其期望误差为$O(\min\{1,\sqrt{\log(2T)\log(2k)/(\varepsilon n)}\})$。此结果与标准高维情形下的下界依赖性相匹配。该构造从仅选择私有乘法权重转录本开始，然后用距离惩罚似然包络替换其概率质量函数。为证明修改保持准确性，通过似然水平的Maurey论证，用一小族辅助PMW定律对上界每个汉明球最大值进行界定。Renyi矩界控制附近球，直接混合界控制远处球，在隐私尺度上分组半径可防止误差中出现额外的$1/\varepsilon$因子。该机制是信息理论性且有配套的Lean 4开发进行机器检查。
英文摘要
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.
Comments26 pages; companion Lean 4 formalization included as ancillary material
