Differentially Private Debiased Estimator

• The paper introduces a debiased estimator using a sample-and-aggregate method with bias-corrected block estimators and additive Laplace noise to ensure differential privacy.
• It achieves asymptotic unbiasedness and efficiency, matching the MLE's variance bound by carefully balancing block size and noise scale.
• The approach supports valid statistical inference by preserving confidence interval accuracy and is adaptable to various parametric models, including the exponential family.

A differentially private debiased estimator is a statistical inference method that achieves formal differential privacy guarantees while simultaneously correcting or eliminating the bias typically induced by noise injection procedures. Under classical privacy mechanisms, additive noise—required to limit sensitivity to any single individual’s data—can severely degrade the accuracy and validity of point estimation and downstream inference. The notion of a “debiased” estimator in this context refers to an approach that preserves unbiasedness and/or efficiency (for example, matching the properties of a maximum likelihood estimator) despite privatization, often enabling valid confidence regions or hypothesis testing analogous to the nonprivate regime.

1. Sample-and-Aggregate Construction for Debiased Estimation

The foundational construction for an efficient differentially private debiased estimator, as introduced in "Efficient, Differentially Private Point Estimators" (0809.4794), is based on the sample-and-aggregate paradigm. Suppose $x = (x_1, ..., x_n)$ are $n$ observations from a parametric family with parameter $\theta$ in a parameter space of diameter $\Lambda$. The construction proceeds as follows:

• Partitioning and Block Estimation: Split the data into $k$ non-overlapping blocks (each of size $t = n/k$). On each block $B_j$, compute a bias-corrected maximum likelihood estimator, denoted $T_\mathrm{bc}(B_j)$. The bias correction ensures that the estimator for each block satisfies a bias rate faster than $1/\sqrt{n}$, which is necessary when aggregating over blocks.
• Averaging: Form an intermediate, nonprivate estimator by averaging the block-wise bias-corrected estimators:

$$\overline{T} = \frac{1}{k} \sum_{j=1}^k T_\mathrm{bc}(B_j)$$

• Noise Injection and Privacy: To ensure differential privacy, add independent Laplace noise $R$ with scale parameter $\Lambda/k$ to the average:

$$T^*(x) = \overline{T} + R = \frac{1}{k} \sum_{j=1}^k T_\mathrm{bc}(B_j) + \mathrm{Lap}(\Lambda/k)$$

This scale ensures that, for any neighboring datasets $x$ and $x'$, the transition densities are within multiplicative $\exp(\varepsilon)$ factors in likelihood, i.e., the standard definition of $\varepsilon$-differential privacy for statistics.

• Privacy Guarantee: Because any single individual can affect at most one block, the sensitivity of $\overline{T}$ is $\Lambda/k$, and thus the Laplace mechanism is sharp.

2. Asymptotic Efficiency, Unbiasedness, and Distributional Properties

The estimator $T^*$ has the following statistical properties, under standard regularity and model assumptions:

• Asymptotic Mean Squared Error: For suitable $k$ (specifically $k = o(n^{2/3})$), the contribution from both the added noise and any bias introduced by block-wise estimation diminishes with $n$, yielding

$$J_{T^*}(\theta) = \frac{1}{n}\frac{1 + o(1)}{I_f(\theta)}$$

where $I_f(\theta)$ denotes the Fisher information of the model, and $J_{T^*}(\theta)$ is the estimator’s mean squared error (MSE).

More precisely:

$$J_{T^*}(\theta) = \frac{1}{n} \left[ \frac{1}{I_f(\theta)} + O\left( \frac{\Lambda^{6/5}}{n^{1/5}} \right) + O\left( \frac{k^3}{n^2} \right) \right]$$

By choosing $k \approx \lceil n^{3/5} \Lambda^{2/5} \rceil$, both the bias and variance terms are controlled and vanish as $n \to \infty$.

• Asymptotic Unbiasedness: Any residual bias from blocking and noise is negligible compared to the statistical uncertainty, resulting in asymptotic unbiasedness.
• Limit Distribution: Under appropriate conditions,

$$\sqrt{n}(T^*(X) - \theta) \xrightarrow{d} N\left( 0, 1/I_f(\theta) \right)$$

guaranteeing that the limiting distribution of the estimator, after normalization, matches that of the MLE.

• Inference Validity: Because the debiased estimator $T^*$ is both private and asymptotically equivalent to the MLE, it supports classical inference procedures. Confidence intervals and hypothesis tests constructed in the same way as for the MLE remain valid (with error terms that vanish as $n$ grows).

3. Sensitivity, Block Size Trade-offs, and Parameter Constraints

The design of the estimator involves balancing bias, variance, and privacy-induced noise:

• Block Size ($k$): If $k$ is too small, the added Laplace noise is large; if $k$ is too large, block-wise bias increases. The optimal $k$ scales as $n^{3/5} \Lambda^{2/5}$ for models with parameter space diameter $\Lambda$.
• Parameter Space Diameter ($\Lambda$): The amount of noise and the convergence rates depend on the boundedness of the parameter space. This is essential, since differentially private mechanisms fundamentally depend on the global sensitivity of the estimator—if $\Lambda$ is unbounded, differential privacy cannot be enforced at finite privacy budget without massive distortion.
• Bias Correction: The block-wise estimator $T_\mathrm{bc}(B_j)$ must be constructed so that its bias decays faster than $1/\sqrt{n}$. This requires a higher-order bias correction which is model-specific but essential for achieving overall efficiency and unbiasedness when aggregating across blocks before noise addition.

4. Implications for Valid Statistical Inference

The central implication is that rigorous privacy constraints need not require practitioners to compromise valid statistical inference:

• Optimality: The estimator asymptotically achieves the Cramér–Rao lower bound for variance, meaning that, despite privacy, the estimator is statistically optimal in large samples.
• Plug-in Inference: For practical sample sizes, provided the privacy parameter (noise scale) decays at a suitable rate relative to $n$, plug-in inference methods—such as Wald intervals—retain their nominal coverage up to negligible error.
• Simultaneity of Privacy and Validity: The result demonstrates the consistency of differential privacy as a requirement with conventional statistical validity: the trade-off, with proper design, is asymptotically negligible.

5. Extensions, Related Methods, and General Applicability

While the original construction was for parametric models and MLE-type estimators, the principles extend broadly:

• Exponential Family and Beyond: The methodology adapts to exponential family models and more general settings, including generalized linear models and any scenario where efficient bias-corrected blockwise estimators can be defined.
• Extensions in the Literature: Measurement-error or “denoising” models, such as those for private $\beta$-model estimation in networks (Karwa et al., 2012), utilize similar two-step logic: privately release noisy sufficient statistics, then “debias” the statistic via projection and correct estimation to match the nonprivate estimator’s asymptotic properties.
• Robustness Requirements: For functions of statistics or under broader classes of estimators, boundedness, gross-error sensitivity, and robust influence are critical: as established in (Chaudhuri et al., 2012), the convergence rates for differentially private estimators are governed by these notions.
• Implications for Algorithm Design: The approach provides a blueprint for achieving high-fidelity inference in practical scenarios where privacy is mandated, and inspires follow-on work in privacy-preserving inference, such as Bayesian and high-dimensional extensions, synthetic data postprocessing, and adaptive partitioning for scalable computation.

6. Summary Table: Core Steps and Properties

Step	Description	Mathematical Guarantee
Partition	Break data into $k$ blocks	Bias per block $O(1/t)$
Per-block Estimation	Compute bias-corrected MLE $T_{\mathrm{bc}}(B_j)$	Block estimator bias decays faster than $1/\sqrt{n}$
Aggregate	Compute $\overline{T}$, the average of block estimators	Asymptotic variance $1/[(n)I_f(\theta)]$ (MLE match)
Noise Injection	Add $\mathrm{Lap}(\Lambda/k)$ noise for privacy	$\exp(\varepsilon)$-DP, sensitivity $\Lambda/k$
Debiased Estimator $T^*$	$\overline{T} + \mathrm{Lap}(\Lambda/k)$	Asymptotically unbiased and efficient

7. Context, Limitations, and Implications

The differentially private debiased estimator approach pioneered in (0809.4794) is the precedent for essentially all subsequent differentially private statistical inference targeting optimality and unbiasedness. Its application is limited when parameter space is unbounded or when sufficient statistics do not allow sharp bias correction, but the paradigm is foundational for current research where privacy loss must be formally controlled without sacrificing the scientific validity of statistical conclusions. With appropriate model and estimator choices, debiased estimation under differential privacy is achievable and operationally useful for statistical practice in privacy-critical domains.