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InfTDA: A Simple TopDown Mechanism for Hierarchical Differentially Private Counting Queries

Fabrizio Boninsegna

TL;DR

This work generalizes InfTDA to datasets with $d$ categorical features and provides a DP synthetic dataset $\tilde{D}$ that accurately answers all $k$-hierarchical queries with a max error of $\tilde{O}(\sqrt{k^3 d})$. It introduces InfTDA, a TopDown mechanism that injects per-level Gaussian noise and then optimizes via Chebyshev distance to enforce hierarchical consistency and non-negativity, yielding a full contingency table with non-negative integer counts. The method allocates privacy budget $\rho/d$ per level under zCDP and employs the integer-optimal procedure IntOpt to stay in the integer domain, achieving scalability through per-branch independence and $O(\max_i |\,\mathcal{X}_i|)$ per-level time. A key open question remains whether the same non-negative, integer-continuous DP table can reach $\tilde{O}(\sqrt{k d})$ accuracy as in some hierarchical mechanisms. Overall, the approach provides a practical, utility-backed framework for private, general-purpose synthetic data capable of answering hierarchical marginals on complex categorical datasets.

Abstract

This paper extends $\texttt{InfTDA}$, a mechanism proposed in (Boninsegna, Silvestri, PETS 2025) for mobility datasets with origin and destination trips, in a general setting. The algorithm presented in this paper works for any dataset of $d$ categorical features and produces a differentially private synthetic dataset that answers all hierarchical queries, a special case of marginals, each with bounded maximum absolute error. The algorithm builds upon the TopDown mechanism developed for the 2020 US Census.

InfTDA: A Simple TopDown Mechanism for Hierarchical Differentially Private Counting Queries

TL;DR

This work generalizes InfTDA to datasets with categorical features and provides a DP synthetic dataset that accurately answers all -hierarchical queries with a max error of . It introduces InfTDA, a TopDown mechanism that injects per-level Gaussian noise and then optimizes via Chebyshev distance to enforce hierarchical consistency and non-negativity, yielding a full contingency table with non-negative integer counts. The method allocates privacy budget per level under zCDP and employs the integer-optimal procedure IntOpt to stay in the integer domain, achieving scalability through per-branch independence and per-level time. A key open question remains whether the same non-negative, integer-continuous DP table can reach accuracy as in some hierarchical mechanisms. Overall, the approach provides a practical, utility-backed framework for private, general-purpose synthetic data capable of answering hierarchical marginals on complex categorical datasets.

Abstract

This paper extends , a mechanism proposed in (Boninsegna, Silvestri, PETS 2025) for mobility datasets with origin and destination trips, in a general setting. The algorithm presented in this paper works for any dataset of categorical features and produces a differentially private synthetic dataset that answers all hierarchical queries, a special case of marginals, each with bounded maximum absolute error. The algorithm builds upon the TopDown mechanism developed for the 2020 US Census.
Paper Structure (12 sections, 2 theorems, 15 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 2 theorems, 15 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Preliminaries
  5. Non-Negative Hierarchical Tree
  6. Hierarchical Queries
  7. Differential Privacy
  8. TopDown Mechanism with Chebyshev Distance
  9. InfTDA
  10. IntOpt
  11. Conclusions and Open Questions
  12. Utility of InfTDA

Key Result

Theorem 1

Let $D = (\mathcal{X}_1\times \mathcal{X}_d)^n$ be a dataset containing $n$ points of $d$ categorical values. Let $\tilde{D}$ be the dataset returned by InfTDA. Then, $|\tilde{D}| = n$ and for any $k\in [1, \dots, d]$ and $\beta \in (0,1)$ we have that Additionally, $\texttt{InfTDA}$ satisfies $\rho$-zCDP.

Figures (1)

  • Figure 1: An example of the non-negative hierarchical tree defined by hierarchical queries for a data universe $\mathcal{X} = \{y_1^1, y_1^2\} \times \{y_2^1, y_2^2\}$. The hierarchical consistency assures that the tree is hierarchical, so that child nodes' attributes sum to their father's attribute.

Theorems & Definitions (8)

  • Definition 1: Non-Negative Hierarchical Tree boninsegna2025
  • Definition 2: Counting Query
  • Definition 3: $k$-way marginal query
  • Definition 4: $k$-hierarchical query
  • Theorem 1: Utility and Privacy of InfTDA
  • proof
  • Corollary 1: Corollary 9 Canonne_Kamath_Steinke_2022
  • proof : Proof of utility in Theorem \ref{['theorem: utility']}