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A statistical framework for differential privacy

Larry Wasserman, Shuheng Zhou

arXiv:0811.2501v2math.ST

TL;DR

The paper studies how to release data while protecting individual privacy and preserving information content. It analyzes differential-privacy mechanisms through convergence rates, finding that privacy schemes generally do not achieve minimax rates and linking the exponential mechanism to small-ball probabilities.

  • Problem

    A challenge is finding a data-release mechanism that satisfies differential privacy while still yielding informative datasets.

  • Method

    The paper studies differential-privacy mechanisms and evaluates distributions and densities constructed from released data, including orthogonal series density estimators.

  • Results

    In general, privacy schemes do not yield minimax rates, while perturbation methods under L2 loss are exceptions that do.

  • Takeaways & Limitations

    The paper identifies a previously unobserved connection between differential privacy and small-ball probabilities.

  • Takeaways & Limitations

    The paper leaves some differential-privacy theory as open work and notes that risk can nonetheless be quite large.

Abstract

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One goal of statistical privacy research is to construct a data release mechanism that protects individual privacy while preserving information content. An example is a {\em random mechanism} that takes an input database $X$ and outputs a random database $Z$ according to a distribution $Q_n(\cdot|X)$. {\em Differential privacy} is a particular privacy requirement developed by computer scientists in which $Q_n(\cdot |X)$ is required to be insensitive to changes in one data point in $X$. This makes it difficult to infer from $Z$ whether a given individual is in the original database $X$. We consider differential privacy from a statistical perspective. We consider several data release mechanisms that satisfy the differential privacy requirement. We show that it is useful to compare these schemes by computing the rate of convergence of distributions and densities constructed from the released data. We study a general privacy method, called the exponential mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy of this method is intimately linked to the rate at which the probability that the empirical distribution concentrates in a small ball around the true distribution.

1 Introduction

The paper examines differential privacy statistically, comparing privacy mechanisms through convergence rates for distributions and densities based on released data. It focuses on the exponential mechanism and links its accuracy to empirical-distribution small ball probabilities.

  • The paper studies mechanisms that transform database X into released database Z while protecting individual privacy and preserving information content.
  • Differential privacy requires the release distribution to be insensitive to changes in one data point, making individual membership difficult to infer from Z.
  • The paper compares release mechanisms by evaluating how quickly distances between the true distribution and distributions or densities estimated from Z converge to zero.
  • For the exponential mechanism, accuracy is linked to the rate at which the empirical distribution concentrates in a small ball around the true distribution.
  • The paper identifies the connection between differential privacy and small ball probabilities as previously unobserved, while leaving whether slower rates are intrinsic as an open question.
  • The paper focuses on statistical properties and does not compare differential privacy with other privacy approaches or address computational efficiency.
  • The analysis considers KS distance and squared L2 distance, alongside minimax rates for density estimation under stated smoothness assumptions.
  • In general, privacy schemes do not achieve minimax rates, although perturbation methods evaluated under L2 loss are exceptions.

2 Differential Privacy

Differential privacy requires released-data distributions to remain insensitive to changing one database entry, limiting inference about individual membership. The section defines the framework, motivates probability-ratio protection, and records consequences for testing and post-processing.

  • Mechanisms: A data release mechanism maps an input database X to a sanitized database Z through a conditional distribution Q_n(·|X).The released database may have a different size from the input, with k controlling a privacy–accuracy tradeoff.
  • Privacy definition: α-differential privacy bounds output-probability ratios for databases differing in one entry, with smaller α providing a stronger privacy guarantee.The definition uses measurable output sets and interprets the ratio as 1 when numerator and denominator are both zero.
  • Motivation: Probability ratios are used because they protect rare cases with small probability and make individual membership difficult to infer from released data.The privacy condition limits how much changing one record can alter the output distribution.
  • Privacy consequences: Any level γ test based on Z, P, and Q_n for whether an individual is present has power at most γe^α.The corresponding Bayes factor is bounded between e^-2α and e^2α.
  • General facts: The exponential mechanism satisfies α-differential privacy, and measurable post-processing preserves differential privacy.The latter also supports generating iid released observations from a privacy-preserving density.

3 Informative Mechanisms

The paper evaluates whether privacy-preserving release mechanisms retain enough distributional information for statistical inference. It formalizes informativeness through convergence of the released empirical distribution to the true distribution and identifies conditions required for consistency.

  • Motivation: A central challenge is designing differentially private mechanisms that produce datasets retaining enough information for precise inference.The statistical goal is to infer P or functionals of P from sanitized data.
  • Evaluation framework: Informativeness is evaluated by whether ρ(F, bF_Z) converges to zero, where F is the true distribution function and bF_Z is the released empirical distribution.The distance ρ can measure distributional or density accuracy.
  • Evaluation framework: Consistency requires ρ(F, bF_Z)=O_P(ε_n) for a sequence ε_n tending to zero.The framework also considers closeness between the original and released empirical distributions.
  • Consistency: If ρ satisfies the triangle inequality and the original empirical distribution is consistent at rate ε_n, released-data accuracy inherits that rate when the two empirical distributions are comparably close.Both ρ(F, bF_Z)=O_P(ε_n) and ρ(bF, bF_Z)=O_P(ε_n) imply the corresponding true-distribution rate under the stated condition.
  • Scope: The analyst is assumed to observe sanitized data Z but not Q_n, while using knowledge of the release mechanism to improve inference is left for future work.The paper notes that assumptions on F are necessary for consistent schemes.
  • Consistency limits: Under differential privacy, a point-mass population distribution yields an inconsistent released empirical distribution in Kolmogorov–Smirnov distance.There is a δ>0 such that the probability of error exceeding δ remains bounded away from zero asymptotically.

4 Sampling From a Histogram

This section develops two histogram-based private release schemes: sampling from smoothed histograms and from perturbed histograms. Their accuracy depends on histogram construction and the loss measure, with perturbation achieving the minimax L2 rate but not the KS rate.

  • Methods: The two schemes release random samples from a smoothed histogram or a randomly perturbed histogram while preserving differential privacy.Histograms are used for familiarity, simplicity, and relevance to differential-privacy applications.
  • Methods: The histogram must be carefully constructed to ensure differential privacy, and accuracy depends on both construction and the chosen accuracy measure.The section compares the schemes through rates of inference from released data.
  • Smoothed histogram: For the smoothed-histogram scheme, KS accuracy is consistent but converges more slowly than the minimax rate n^-1/2.The theorem chooses m ≍ n^r/(6+r), k ≍ n^4/(6+r), and δ=(mk/n^α) under the stated constraint.
  • Smoothed histogram: The smoothed-histogram scheme is differentially private under its stated parameter condition, whereas sampling from the ordinary histogram with δ=0 is not private.The privacy constraint creates a tradeoff among the number of bins m, released sample size k, and smoothing parameter δ.
  • Perturbed histogram: For the perturbed-histogram scheme, squared L2 accuracy achieves the minimax rate O(n^-2/(2+r)) when m ≍ n^r/(2+r) and k≥n.The release is built by perturbing histogram counts with independent Laplace noise and then sampling a released database.
  • Comparison: The perturbation method does not achieve the minimax KS rate, while the exponential-mechanism method achieves a better KS rate.The paper therefore describes perturbation as preferable for L2 distance, not uniformly across losses.

5 Exponential Mechanism

The exponential mechanism provides a general private release method whose accuracy can be analyzed through small-ball probabilities. Choosing the released sample size is delicate: larger releases harm privacy, while smaller releases harm accuracy, and the KS rate remains below optimal in the stated example.

  • Overview: The section derives general accuracy results for the exponential mechanism and applies them to means and nonparametric density estimation.It also addresses how to choose the released database size k.
  • Sample-size choice: Choosing k is delicate because taking k too large compromises privacy, whereas taking k too small compromises accuracy.The tradeoff governs how the exponential mechanism can remain both private and consistent.
  • Small-ball analysis: The exponential mechanism’s accuracy is bounded by a formula containing the probability that a size-k empirical distribution lies in a small ball around the true distribution.This quantity is called the small ball probability associated with ρ.
  • Consistency: If k is chosen so the theorem’s bound tends to zero, the exponential mechanism is consistent under the stated bounded-density and metric assumptions.The framework requires a sequence ε_n→0 and a distance satisfying the triangle inequality.
  • KS example: For KS distance, sanitization yields rate n^-1/3, slower than the optimal n^-1/2 rate for the unsanitized empirical distribution.The paper leaves open whether the sanitized rate can be improved.
  • Mean example: In a sample-mean example, the released data converge at the same rate as the original data, and preserving one piece of information requires k=1.The example links the released database size to the amount of information being preserved.

6 Orthogonal Series Density Estimation

The section develops exponential and perturbation approaches for orthogonal-series density estimation under differential privacy. The perturbation technique achieves the minimax convergence rate, while the exponential mechanism appears less accurate, though its bounds may not be tight.

  • Mechanisms and estimators: The analysis compares exponential and perturbation privacy mechanisms using orthogonal-series density estimators and L2 convergence rates.The perturbation approach samples from a perturbed histogram, whereas the exponential scheme is developed through density estimation.
  • Benchmark rate: The nonprivate minimax L2 rate for the considered smoothness class is n^-2γ/(2γ+1).The section states that this rate is achieved by an orthogonal-series estimator with m_n = n^1/(2γ+1).
  • Exponential mechanism: The sanitized estimator converges at a slower rate than the minimax rate.This result concerns the exponential-mechanism analysis under the section’s assumptions.
  • Perturbation approach: The perturbation technique achieves the minimax rate of convergence in L2 distance.The result is presented for the perturbation approach under the stated theorem conditions.
  • Comparison: The perturbation approach appears superior to the exponential mechanism, but the authors do not know whether the difference is inherent or due to non-tight bounds.The comparison is therefore qualified by uncertainty about the exponential mechanism’s theoretical bounds.
  • Simulation illustration: Figure 1 compares mean integrated squared error for original and perturbed histograms across sample sizes and settings.The top plots use n = 100 and the bottom plots use n = 1,000; the perturbed histogram is the upper line.

7 Example

The section uses simulations to examine how perturbing histograms affects accuracy. Across sample sizes, privacy levels, bin counts, and two density shapes, smaller privacy parameters increase error and sanitization makes smoothing choices more consequential.

  • Simulation design: The simulation studies perturbed histograms for Beta(10,10) data with n = 100 or 1,000 and α = 0.1 or 0.01.Squared error is used, and results are based on 1,000 simulations across different numbers of bins.
  • Privacy effect: Smaller α values induce greater information loss and larger mean squared error.The simulations compare privacy levels through squared-error risk.
  • Accuracy: Although the perturbed histogram achieves the minimax rate, perturbation substantially inflates its error.The result shows that risk constants matter in addition to asymptotic convergence rates.
  • Smoothing choice: The sanitized histogram’s risk is more sensitive to the number of cells than the original histogram’s risk.Thus, the smoothing parameter has a larger observed effect after sanitization.
  • Robustness across densities: Simulations with an equal mixture of Beta(10,3) and Beta(3,10) densities produced results nearly identical to the initial density experiment.The additional experiment examines whether the observed pattern changes for a bimodal density.

8 Conclusion

The conclusion frames differential privacy statistically by comparing released-data distributions and densities and by analyzing the exponential mechanism through small-ball probabilities. It finds differing minimax behavior across methods and identifies unresolved questions about rates, adaptivity, risk minimization, and sanitization methods.

  • Statistical comparison: Distribution- and density-based losses provide a way to compare differential-privacy mechanisms.The paper presents this as part of its statistical-language treatment of privacy.
  • Histogram mechanisms: Sampling from a histogram preserves differential privacy when the histogram is shifted from zero or its cells are appropriately perturbed.The latter perturbation method achieves a faster L2 convergence rate.
  • Finite-sample behavior: Despite faster convergence, simulations show that the perturbation method can have substantial risk, motivating more precise finite-sample risk bounds.The sanitized histogram is also more sensitive to the smoothing parameter than the original histogram.
  • Exponential mechanism: The paper derives an accuracy formula for the exponential mechanism involving small-ball probabilities.The authors state that this connection between differential privacy and small-ball probabilities had not previously been observed, to their knowledge.
  • Open theory: The paper does not yet provide a complete minimax theory for differential privacy.The authors identify this as the focus of current work.
  • Open questions: Open questions concern preserving minimax rates, retaining adaptivity after sanitization, and understanding sanitization’s effect on risk minimization.The conclusion also asks whether better sanitization methods exist.

9 Proofs

The proofs establish privacy and consistency-related properties of the release mechanisms and derive convergence bounds for perturbed histograms. They combine density-ratio arguments, empirical-distribution bounds, and parameter choices for the histogram construction.

  • Privacy arguments: The privacy proof bounds the likelihood ratio between outputs generated from databases differing in one coordinate.The resulting testing argument gives power at most γe^α under the differential-privacy relation.
  • Mechanism properties: Post-processing preserves differential privacy, and the released sample is conditionally independent of the original database given the privatized statistic.These properties are used to transfer privacy from an intermediate mechanism to released outputs.
  • Consistency argument: A consistency argument constructs a sequence of neighboring databases and uses differential privacy to propagate zero-probability output events.The argument reaches a contradiction with concentration of the released empirical distribution.
  • Histogram privacy: For perturbed histograms, neighboring databases change each cell proportion by less than 1/n.The proof bounds the resulting density ratio by considering the affected bin and the possible direction of the change.
  • Worst-case ratio: The density-ratio analysis treats separately zero and positive neighboring cell proportions and may reverse the roles of the two databases.This identifies the worst-case ratio within a single bin.
  • Convergence bounds: The convergence proof bounds the empirical distribution function by decomposing boundary regions into histogram cubes and applying a Lipschitz condition.A VC bound controls the distributional component of the error.
  • Analytic setup: The released observations are modeled as iid draws from the perturbed histogram density, while the underlying histogram and its expectation define the comparison quantities.The basis-function analysis assumes uniformly bounded basis functions and uses an orthonormal basis for L2(0,1).

9.6 Proof of Theorem 4.3

The proof decomposes the released histogram’s error into the original histogram risk and perturbation terms, then chooses the bin count to recover the usual rate.

  • The released histogram error is decomposed into approximation, estimation, and perturbation contributions.
  • The usual histogram risk is O(m−2/r + m/n).
  • The perturbation analysis bounds the maximum noise magnitude by O(log m) almost surely for large n.
  • The perturbed histogram’s bin-probability discrepancies satisfy O(log m/n) almost surely for all large n.
  • Choosing m ≍ n^r/(2+r) yields risk n−2/(2+r) for all sufficiently large n.

9.8 Proof of Theorem 5.3

The proof establishes concentration bounds for empirical distributions under several distances, using triangle inequalities, VC bounds, and small-ball estimates.

  • The event A_n controls the empirical distribution’s distance from the true distribution by ϵ_n/16.
  • Triangle inequalities relate distances between the empirical distributions based on released samples, original samples, and the true distribution.
  • Changing one data entry changes the empirical cdf by at most 1/n at every threshold, yielding the KS-distance conclusion.
  • A VC-dimension bound is applied to lower-orthant sets, with dimension r, to establish the required empirical-distribution control.
  • The relevant lower-tail analysis uses empirical-process convergence to Brownian bridges and associated small-ball probabilities.
  • For the density estimator, Parseval’s relation and bounded covariance eigenvalues support the corresponding small-ball calculation.

9.12 Proof of Theorem 6.3

The proof compares the privately released estimator with an intermediate estimator and shows that the added release error is negligible when the auxiliary sample is sufficiently large.

  • Truncation effects are negligible in the comparison between the target estimator and the privately released estimator.
  • The released estimator differs from an intermediate estimator by OP(m/k), which is negligible when k ≥ n.
  • The intermediate estimator’s error splits into the usual estimation error and the discrepancy between the original and intermediate estimators.
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