PREM: Privately Answering Statistical Queries with Relative Error
Badih Ghazi, Cristóbal Guzmán, Pritish Kamath, Alexander Knop, Ravi Kumar, Pasin Manurangsi, Sushant Sachdeva
TL;DR
PREM tackles privately answering statistical queries with a relative-error guarantee under DP by introducing a Private Relative Error MWU framework that outputs a synthetic histogram. The core idea combines RangeMonitor, a private threshold-tracking primitive, with a multiplicative-weights update to iteratively build a synthetic dataset that approximates counts up to a multiplicative factor $1\pm\zeta$ and a polylogarithmic additive error $\alpha$. For $(\varepsilon,\delta)$-DP, PREM achieves $\alpha = \tilde{O}\left( \frac{1}{\zeta\varepsilon} \left( \log n \log \frac{1}{\delta} \right)^{\frac{3}{2}} \sqrt{\log |X|} \log \left( \frac{|F|}{\beta} \right) \right)$, while for pure-DP it attains $\alpha = \tilde{O}\left( \sqrt{ \frac{n \log^3 n}{\zeta^2 \varepsilon} \log |X| \log |F| } + \frac{\log n}{\varepsilon} \log \frac{1}{\beta} \right)$. The work also derives near-matching lower bounds for approximate-DP and extends the framework to real-valued queries via a thresholding reduction, outlining open questions about tightening pure-DP gaps and specializing to particular query families. Overall, PREM demonstrates that relative-error privacy-preserving data synthesis can achieve polylogarithmic additive error in key parameters, offering a substantial improvement over additive-only DP mechanisms in many regimes.
Abstract
We introduce $\mathsf{PREM}$ (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a relative error guarantee for statistical queries under $(\varepsilon, δ)$ differential privacy (DP). Namely, for a domain ${\cal X}$, a family ${\cal F}$ of queries $f : {\cal X} \to \{0, 1\}$, and $ζ> 0$, our framework yields a mechanism that on input dataset $D \in {\cal X}^n$ outputs a synthetic dataset $\widehat{D} \in {\cal X}^n$ such that all statistical queries in ${\cal F}$ on $D$, namely $\sum_{x \in D} f(x)$ for $f \in {\cal F}$, are within a $1 \pm ζ$ multiplicative factor of the corresponding value on $\widehat{D}$ up to an additive error that is polynomial in $\log |{\cal F}|$, $\log |{\cal X}|$, $\log n$, $\log(1/δ)$, $1/\varepsilon$, and $1/ζ$. In contrast, any $(\varepsilon, δ)$-DP mechanism is known to require worst-case additive error that is polynomial in at least one of $n, |{\cal F}|$, or $|{\cal X}|$. We complement our algorithm with nearly matching lower bounds.