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Minimax optimal differentially private synthetic data for smooth queries

Rundong Ding, Yiyun He, Yizhe Zhu

TL;DR

The paper addresses generating $(\varepsilon,\delta)$-DP synthetic data that preserves utility for $k$-smooth queries on $[-1,1]^d$. It introduces a Chebyshev-moment matching approach, extending the Chebyshev framework to $k$-smooth function classes and leveraging a multivariate Jackson-type approximation to bound IPMs $d_k$ between the original and synthetic data distributions. A grid-based algorithm with Gaussian noise added to Chebyshev moments achieves minimax-optimal utility up to a $\log(n)$ factor, with a clear phase transition at $k=d$ where further smoothness no longer improves the rate. The work also provides a minimax lower bound under $(\varepsilon,\delta)$-DP, matching the upper bounds up to dimension-dependent factors, thereby clarifying the role of smoothness and dimension in private data synthesis. Collectively, the results yield improved utility guarantees for a broad class of practical, smooth statistics, informing both theory and practice in private data sharing and analysis.

Abstract

Differentially private synthetic data enables the sharing and analysis of sensitive datasets while providing rigorous privacy guarantees for individual contributors. A central challenge is to achieve strong utility guarantees for meaningful downstream analysis. Many existing methods ensure uniform accuracy over broad query classes, such as all Lipschitz functions, but this level of generality often leads to suboptimal rates for statistics of practical interest. Since many common data analysis queries exhibit smoothness beyond what worst-case Lipschitz bounds capture, we ask whether exploiting this additional structure can yield improved utility. We study the problem of generating $(\varepsilon,δ)$-differentially private synthetic data from a dataset of size $n$ supported on the hypercube $[-1,1]^d$, with utility guarantees uniformly for all smooth queries having bounded derivatives up to order $k$. We propose a polynomial-time algorithm that achieves a minimax error rate of $n^{-\min \{1, \frac{k}{d}\}}$, up to a $\log(n)$ factor. This characterization uncovers a phase transition at $k=d$. Our results generalize the Chebyshev moment matching framework of (Musco et al., 2025; Wang et al., 2016) and strictly improve the error rates for $k$-smooth queries established in (Wang et al., 2016). Moreover, we establish the first minimax lower bound for the utility of $(\varepsilon,δ)$-differentially private synthetic data with respect to $k$-smooth queries, extending the Wasserstein lower bound for $\varepsilon$-differential privacy in (Boedihardjo et al., 2024).

Minimax optimal differentially private synthetic data for smooth queries

TL;DR

The paper addresses generating -DP synthetic data that preserves utility for -smooth queries on . It introduces a Chebyshev-moment matching approach, extending the Chebyshev framework to -smooth function classes and leveraging a multivariate Jackson-type approximation to bound IPMs between the original and synthetic data distributions. A grid-based algorithm with Gaussian noise added to Chebyshev moments achieves minimax-optimal utility up to a factor, with a clear phase transition at where further smoothness no longer improves the rate. The work also provides a minimax lower bound under -DP, matching the upper bounds up to dimension-dependent factors, thereby clarifying the role of smoothness and dimension in private data synthesis. Collectively, the results yield improved utility guarantees for a broad class of practical, smooth statistics, informing both theory and practice in private data sharing and analysis.

Abstract

Differentially private synthetic data enables the sharing and analysis of sensitive datasets while providing rigorous privacy guarantees for individual contributors. A central challenge is to achieve strong utility guarantees for meaningful downstream analysis. Many existing methods ensure uniform accuracy over broad query classes, such as all Lipschitz functions, but this level of generality often leads to suboptimal rates for statistics of practical interest. Since many common data analysis queries exhibit smoothness beyond what worst-case Lipschitz bounds capture, we ask whether exploiting this additional structure can yield improved utility. We study the problem of generating -differentially private synthetic data from a dataset of size supported on the hypercube , with utility guarantees uniformly for all smooth queries having bounded derivatives up to order . We propose a polynomial-time algorithm that achieves a minimax error rate of , up to a factor. This characterization uncovers a phase transition at . Our results generalize the Chebyshev moment matching framework of (Musco et al., 2025; Wang et al., 2016) and strictly improve the error rates for -smooth queries established in (Wang et al., 2016). Moreover, we establish the first minimax lower bound for the utility of -differentially private synthetic data with respect to -smooth queries, extending the Wasserstein lower bound for -differential privacy in (Boedihardjo et al., 2024).
Paper Structure (21 sections, 18 theorems, 130 equations, 1 algorithm)

This paper contains 21 sections, 18 theorems, 130 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Differentially private synthetic data
  3. Main results
  4. Technical overview
  5. Preliminaries
  6. Differential privacy
  7. Normalized multivariate Chebyshev polynomials
  8. Integral probability metric and Chebyshev moments
  9. DP synthetic data generation for $k$-smooth queries
  10. Privacy
  11. Accuracy
  12. Minimax lower bound
  13. Total variance distance bound
  14. Construction of a family of test functions
  15. Construction of input datasets
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\varepsilon \in (0,1)$. There exists a polynomial-time $(\varepsilon,\delta)$-DP algorithm that outputs a synthetic dataset $Y\subseteq \Omega$ such that where $C_k\leq (Ck)^k$ with some absolute constant $C$.

Theorems & Definitions (32)

  • Definition 1: $(\varepsilon,\delta)$-Differential Privacy
  • Theorem 1: DP synthetic data generation for $k$-smooth queries
  • Theorem 2: Minimax lower bound
  • Lemma 1: Gaussian Mechanism dwork2014algorithmicballe2018improvingvinterbo2021closedformscalebound
  • Lemma 2: Isometry to cosine variables and orthonormal basis
  • Theorem 3: Order-$k$ coefficient decay
  • Lemma 3: Jackson's Theorem for $d$-dimensional $k$-smooth functions
  • Theorem 4: $d_k$-distance from Chebyshev moments
  • proof
  • Lemma 4: TV bound from approximate likelihood ratio
  • ...and 22 more