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arxiv: 2606.29835 · v1 · pith:PSVQW6OQnew · submitted 2026-06-29 · 💻 cs.CR · cs.DS· cs.NA· math.NA· stat.AP· stat.ML

A Sieve-Accelerated Quadrature Method for Exact Privacy Accounting in the 2020 U.S. Decennial Census

Buxin Su , Weijie Su , Chendi Wang This is my paper

Pith reviewed 2026-06-30 05:43 UTC · model grok-4.3

classification 💻 cs.CR cs.DScs.NAmath.NAstat.APstat.ML
keywords differential privacydiscrete Gaussian mechanismprivacy accountingquadrature methodsieve algorithm2020 US Censuscharacteristic functioncomposition theorem
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The pith

A sieve-accelerated quadrature method delivers the first exact, assumption-free privacy accounting for the 2020 Census data releases at 1824 times prior speed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to exactly compute the privacy guarantees provided by the discrete Gaussian noise added to 2020 Census tabulations. This is done by turning the problem of finding tail probabilities of convoluted random variables into a high-dimensional numerical integration task solved via the discrete Fourier transform. The approach exploits the analytic properties of the characteristic functions to achieve rapid convergence with the trapezoidal rule and introduces a sieve to discard most quadrature points that contribute negligibly. If successful, this allows the Census Bureau to use the smallest possible noise levels that still meet privacy targets, which would improve the accuracy of the released data for applications like allocating federal funds and drawing political districts.

Core claim

By recasting exact privacy accounting as a numerical integration problem via the discrete Fourier transform and applying a sieve algorithm to prune negligible quadrature nodes, the method evaluates the tail probabilities of high-dimensional convolutions of heterogeneous discrete Gaussian distributions to within 10^{-35} error, achieving the first assumption-free exact privacy accounting for the 2020 Census Demographic and Housing Characteristics File at 1,824 times the speed of prior techniques.

What carries the argument

the sieve-accelerated DFT-based trapezoidal quadrature for tail probabilities of discrete Gaussian convolutions

If this is right

  • The Census Bureau can determine the absolute minimum noise needed for a given privacy budget.
  • Excess noise injection is avoided, increasing the statistical utility of the published data.
  • The method applies directly to the composition of heterogeneous discrete Gaussian mechanisms used in the 2020 releases.
  • Error tolerances mandated by the census are maintained throughout the computation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar quadrature techniques could accelerate privacy accounting in other large-scale data releases beyond the census.
  • The approach might extend to evaluating privacy profiles for mechanisms other than discrete Gaussians if their characteristic functions share the same analytic properties.
  • Reduced computation time could enable real-time or iterative privacy budget allocation during data processing.

Load-bearing premise

The characteristic functions of the discrete Gaussian distributions must be complex analytic and periodic for the trapezoidal rule to achieve the required exponential convergence after sieve pruning.

What would settle it

Computing the privacy loss for a low-dimensional case with known closed-form solution or brute-force enumeration and verifying that the quadrature result matches it to within the stated tolerance.

Figures

Figures reproduced from arXiv: 2606.29835 by Buxin Su, Chendi Wang, Weijie Su.

Figure 1
Figure 1. Figure 1: Probability mass function of the convolution of two discrete Gaussian distributions and [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The f-DP curves and (ε, δ)-DP curve for the 2020 Census DHC File computed using our method (blue) and the Census Bureau’s accounting method (red). 5.1 f-DP curves The left panel of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The f-DP curves for the composed mechanisms [Mf0,Mfl ] for all l, where Mf0 corresponds to the privacy budget allocation in [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The (ε, δ)-DP curves for the composed mechanisms [Mf0,Mfl ] for all l, where Mf0 cor￾responds to the privacy budget allocation in [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Percentage difference between the privacy level of the composed mechanism [ [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

In 2020, the U.S. Census Bureau adopted differential privacy for the Decennial Census by injecting integer-valued Gaussian noise into published census tabulations. Exactly evaluating the privacy guarantees of these data releases would enable the Bureau to determine the absolute minimum noise required to satisfy a given privacy budget, preventing the injection of unnecessary excess noise and thereby substantially enhancing the statistical utility of the data for downstream applications such as federal funding allocation and political redistricting. In this paper, we introduce a computationally efficient and mathematically rigorous quadrature method to evaluate the exact privacy profile of practical, large-scale census releases under the composition of heterogeneous discrete Gaussian mechanisms. Mathematically, this problem reduces to evaluating the tail probabilities of high-dimensional convolutions of integer-valued random variables sampled from heterogeneous discrete Gaussian distributions under exceptionally stringent numerical error tolerances (e.g., $10^{-35}$). By recasting the exact privacy accounting as a numerical integration problem via the discrete Fourier transform, we explicitly exploit the exponential convergence of the trapezoidal rule for complex analytic, periodic characteristic functions. Furthermore, to overcome the computational bottleneck of evaluating highly oscillatory integrands in high dimensions, we develop a sieve algorithm that identifies and prunes negligible quadrature nodes, accelerating the computation by three orders of magnitude. Taken together, these numerical innovations enable the first exact, assumption-free privacy accounting for the 2020 Census Demographic and Housing Characteristics File, achieving a 1,824-fold speedup over prior methods while maintaining census-mandated error tolerances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a quadrature method for exact privacy accounting of compositions of heterogeneous discrete Gaussian mechanisms, recast as DFT-based numerical integration of periodic characteristic functions. A sieve algorithm prunes negligible nodes to accelerate evaluation of high-dimensional tail probabilities to 10^{-35} tolerances. The method is applied to the 2020 Census DHC File, claiming the first assumption-free exact accounting and a 1,824-fold speedup over prior approaches.

Significance. If the convergence and error bounds hold, the work would enable tighter, precisely calibrated noise injection in census releases, directly improving statistical utility for applications such as redistricting and funding allocation. The explicit use of periodicity and analyticity for exponential trapezoidal convergence, combined with the sieve pruning, is a concrete algorithmic contribution that could generalize to other privacy-accounting settings with discrete mechanisms.

major comments (1)
  1. [Quadrature error analysis and sieve pruning] The central claim of meeting 10^{-35} error after sieve pruning rests on exponential convergence of the trapezoidal rule. No explicit bound is given relating the sieve threshold to the quadrature truncation error when the analytic strip width is governed by the smallest variance in a heterogeneous high-dimensional composition (see the discussion of Poisson summation and strip width in the quadrature section).
minor comments (1)
  1. [Abstract] The abstract states both 'three orders of magnitude' and the specific 1,824-fold figure; ensure these are reconciled in the main text and that the baseline method for the speedup comparison is clearly identified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. The feedback highlights an important aspect of the error analysis that we will strengthen in the revision. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Quadrature error analysis and sieve pruning] The central claim of meeting 10^{-35} error after sieve pruning rests on exponential convergence of the trapezoidal rule. No explicit bound is given relating the sieve threshold to the quadrature truncation error when the analytic strip width is governed by the smallest variance in a heterogeneous high-dimensional composition (see the discussion of Poisson summation and strip width in the quadrature section).

    Authors: We agree that the manuscript relies on the known exponential convergence rate of the trapezoidal rule for periodic analytic functions (via the Poisson summation formula and the width of the analytic strip in the complex plane), but does not supply a fully explicit quantitative relation between the chosen sieve threshold and the resulting quadrature truncation error in the heterogeneous case. The current analysis notes that the strip width is controlled by the smallest variance parameter, yet stops short of a closed-form bound that directly incorporates the sieve pruning threshold. In the revised manuscript we will add a dedicated lemma and corollary in Section 3 that derives such a bound: we show that for a composition of heterogeneous discrete Gaussians the total error after sieve pruning is at most C·exp(−2π·δ·N) + ε_sieve, where δ is the minimal strip width, N the number of quadrature nodes, and ε_sieve is controlled by the threshold; the constant C is made explicit in terms of the variance vector. This will also include a short proof sketch confirming that the bound remains valid under the heterogeneity present in the 2020 Census DHC mechanisms. revision: yes

Circularity Check

0 steps flagged

No circularity: standard numerical quadrature on DFT integrands with pruning optimization

full rationale

The paper derives a quadrature-based algorithm for tail probabilities of discrete-Gaussian convolutions by recasting the problem as integration of the DFT of the characteristic function and applying the known exponential convergence of the trapezoidal rule on 2π-periodic analytic functions, followed by a sieve to drop negligible nodes. This chain rests on external Fourier-analysis facts (Poisson summation, trapezoidal error bounds for strip-analytic periodic integrands) rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The 2020 Census application is a concrete instance of the algorithm; no target privacy quantity is defined in terms of the output or obtained by fitting. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard properties of the discrete Fourier transform and convergence results for the trapezoidal rule applied to analytic periodic functions; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Characteristic functions of discrete Gaussian distributions are complex analytic and periodic
    Invoked to justify exponential convergence of the trapezoidal rule after DFT recasting.

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