📊 statistics

Gaussian Differentially Private $e$ -values: Construction, Threshold Calibration, and Multiple Testing

This paper establishes a framework for Gaussian differentially private $e$ -values by introducing an optimal Gaussian noise mechanism and a recursive peeling algorithm for multiple testing, which collectively enable rigorous false discovery rate control while recovering statistical power close to non-private benchmarks.

Original authors: Qi Kuang, Bowen Gang, Yin Xia

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qi Kuang, Bowen Gang, Yin Xia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a massive mystery involving millions of clues. Some clues are real evidence pointing to a culprit, while most are just red herrings (false alarms). Your goal is to find the real evidence without making too many mistakes.

However, there's a catch: the clues contain sensitive information about real people. If you release the clues exactly as they are, you might accidentally reveal private details about an innocent person just by showing which clues you looked at. This is the problem of privacy.

This paper introduces a new toolkit to solve this mystery while keeping everyone's secrets safe. Here is how it works, broken down into simple concepts:

1. The "E-Value" (The Clue Score)

In statistics, instead of just saying "this looks suspicious," researchers use something called an e-value. Think of an e-value as a "suspicion score."

If the score is low, the clue is probably just noise.
If the score is high, it's strong evidence.
Crucially, if the clue is actually fake (a null hypothesis), the average of these scores over many tries must stay low (specifically, less than or equal to 1).

2. The Privacy Problem (The "Blurry Lens")

To protect privacy, you can't show the raw scores. You have to add "noise" (random static) to them, like looking at the clues through a frosted glass.

The Old Way: Usually, people just add random static to the numbers. But this is like trying to add static to a "suspicion score" that can't be negative. If you aren't careful, the static might turn a valid score into a negative number (which makes no sense) or make the average score too high, breaking the rules of the game.
The Paper's Solution: The authors figured out the perfect way to add this static. They discovered that the best kind of "frosted glass" is shaped like a Bell Curve (Gaussian distribution). By using this specific shape, they can add just enough noise to hide the secrets without breaking the mathematical rules of the e-values.

3. The "Smart Threshold" (Adjusting the Magnifying Glass)

Once you add noise, the scores get a bit fuzzy. The old rule was: "If the score is higher than 20, we call it a discovery."

The Flaw: The old rule was too cautious. It was like saying, "Only look through the magnifying glass if the image is very clear," which meant you missed many good clues that were just slightly blurry.
The Fix: The authors recalibrated the magnifying glass. Because they know exactly how the noise is shaped (the Bell Curve), they can lower the threshold slightly. They can say, "Okay, even if it's a bit blurry, if it's above 15, it's still a real discovery."
The Surprise: In some cases (when the data isn't too sensitive), this "smart" blurry method actually finds more real clues than the perfect, non-private method! It's like realizing that a slightly foggy window, if you know exactly how the fog is distributed, lets you see things you would have missed if you were too afraid to look at anything less than crystal clear.

4. The "Peeling" Strategy (The Onion Approach)

Now, imagine you have 1 million clues. If you try to blur all of them at once to protect privacy, the noise becomes so huge that nothing is visible anymore. It's like trying to hide a needle in a haystack by turning the whole haystack into a giant cloud of dust.

The Old Way: Blur everything at once. Result: You find nothing.
The Paper's Solution (Peeling): Instead of blurring everything, you look at the clues one by one (or in small groups).
1. You peek at the top of the pile to see which clues look the most promising.
2. You only blur those top clues.
3. You remove them from the pile and repeat.
The Secret Sauce: To peek at the top without leaking secrets, they use a special trick called Gumbel noise (a specific type of random noise used for ranking). This lets them pick the "winner" without revealing the exact scores of the losers. Then, they only apply the heavy privacy blur to the winners.
Result: They save their "privacy budget" for the clues that actually matter, allowing them to find real signals even in a massive dataset.

5. Real-World Test (The DNA Mystery)

The authors tested this on a real-world dataset involving Genome-Wide Association Studies (GWAS). This is like looking at millions of DNA snippets to find which ones are linked to a disease (Systemic Lupus Erythematosus).

The Result: When they tried to protect privacy by blurring all the DNA data at once, they found zero links.
The Win: Using their new "Peeling" method, they found a huge number of links, almost as many as if they hadn't protected privacy at all, but without risking anyone's personal data.

Summary

This paper builds a better "privacy shield" for statistical detective work.

It finds the perfect shape for privacy noise (Gaussian) so it doesn't break the math.
It creates a smarter rule for deciding what counts as a discovery, recovering power that was previously lost.
It invents a peeling strategy that focuses privacy protection only on the most interesting clues, preventing the "noise" from drowning out the signal in massive datasets.

The result is a way to do large-scale science on sensitive data that is both strictly private and surprisingly powerful.

Technical Summary: Gaussian Differentially Private e-values

Problem Statement

Modern large-scale statistical inference increasingly relies on sensitive individual-level data, necessitating rigorous privacy protection. While Differential Privacy (DP) is the standard framework, the specific challenge addressed in this paper is the construction of valid e-values under Gaussian Differential Privacy (µ-GDP).

Existing methods for privatizing e-values face significant limitations:

Additive Noise Incompatibility: Standard additive noise mechanisms violate the non-negativity and unit-expectation constraints required for valid e-values.
Suboptimal Privacy Accounting: Previous work on private e-values (e.g., Csillag and Mesquita, 2026) operates under $(\epsilon, \delta)$ -DP or Rényi DP, lacking the tight composition properties of µ-GDP, which is the preferred accounting mechanism for multi-stage and adaptive analyses.
Power Loss in Multiple Testing: Applying global privacy budgets to thousands of hypotheses simultaneously (e.g., in Genome-Wide Association Studies) leads to severe noise inflation, obscuring weak signals. Existing adaptive peeling algorithms for p-values (e.g., Xia and Cai, 2023) rely on Gaussian noise for index selection, which the authors argue fails to provide rigorous µ-GDP guarantees due to the concentration of extreme values in Gaussian distributions as the candidate pool grows.

Methodology

The paper proposes a systematic framework for constructing and utilizing µ-GDP e-values, progressing from single-statistic construction to adaptive multiple testing.

1. Canonical Construction of µ-GDP e-values

The authors establish that multiplicative perturbation is the appropriate mechanism for privatizing e-values, corresponding to additive noise in the logarithmic domain.

Optimal Noise Distribution: Under the assumptions of symmetry and log-concavity, the authors prove that the unique noise distribution that fully exhausts the µ-GDP budget while preserving validity is Gaussian. Specifically, if $E$ is a non-private e-value with sensitivity $\Delta = \sup |\log E(D) - \log E(D')|$ , the private e-value is constructed as $E_{DP} = E \cdot e^{-\xi}$ , where $\xi \sim \mathcal{N}(\tau, \Delta^2/\mu^2)$ .
Optimal Mean Parameter: To maximize statistical power and expected log-growth while maintaining the validity constraint ( $E_{H_0}[E_{DP}] \le 1$ ), the mean parameter is set to $\tau = \Delta^2/(2\mu^2)$ . This yields the canonical construction of a µ-GDP e-value.

2. Noise-Calibrated Thresholds

Standard e-value testing rejects when $E \ge 1/\alpha$ based on Markov's inequality. The authors demonstrate that this threshold is conservative when the noise distribution is known.

Sharp Threshold Derivation: By exploiting the known Gaussian distribution of the noise $\xi$ , they derive a globally sharp rejection threshold $c^* < 1/\alpha$ that strictly maintains Type I error control at level $\alpha$ .
Power Analysis: The paper provides an asymptotic analysis showing that in low-sensitivity regimes (small $\Delta$ ), the calibrated private test can achieve a net power gain over the non-private baseline. This counter-intuitive result arises because the random shift introduced by the noise can, with high probability, lower the effective decision boundary for borderline signals.

3. Aggregation and Composition

General Aggregation: For aggregated private e-values under arbitrary dependence, the standard composition theorem applies, resulting in a $\sqrt{K}\mu$ -GDP guarantee for $K$ statistics.
Independent Aggregation: For independent datasets, the authors show that the product of private e-values satisfies a significantly tighter privacy bound, $\mu_{prod} \le \mu$ , avoiding the $\sqrt{K}$ inflation.

4. Adaptive Peeling Algorithm for Multiple Testing

To address the power loss in large-scale multiple testing, the authors introduce a recursive e-Peeling algorithm.

Decoupled Selection: To resolve the privacy leakage inherent in Gaussian arg-max selection (where extreme values concentrate), the algorithm decouples index extraction from value privatization.
- Selection Step: Uses Gumbel noise (via the Gumbel-max trick) to select the index of the most promising hypothesis. Gumbel noise possesses "max-stability," ensuring the privacy cost of selection remains stable regardless of the number of candidates.
- Perturbation Step: Applies the optimal Gaussian mechanism to the selected e-value.
Adaptive Sizing: An adaptive procedure is proposed to determine the peeling size $s$ (the number of hypotheses to select) using a small portion of the privacy budget, balancing the need to capture true signals against the risk of noise inflation.
Downstream Inference: The output is a vector of valid private e-values that can be fed directly into standard e-value multiple testing procedures (e.g., e-BH) to control the False Discovery Rate (FDR) under arbitrary dependence.

Key Results

Theoretical Findings

Optimality: The Gaussian mechanism is proven to be the unique log-concave, symmetric mechanism that exactly exhausts the µ-GDP bound for e-values.
Threshold Calibration: The derived sharp threshold $c^*$ strictly improves power over the standard Markov threshold $1/\alpha$ . Asymptotic analysis confirms that for low sensitivity, the private test outperforms the non-private baseline.
Privacy Accounting: The paper identifies a flaw in previous Gaussian-based selection mechanisms (specifically in Xia and Cai, 2023), showing that they fail to satisfy µ-GDP for large candidate sets due to the concentration of Gaussian extremes. The proposed Gumbel-based selection corrects this, providing rigorous µ-GDP guarantees.

Empirical Performance

Single Hypothesis: Simulations confirm that the calibrated threshold maintains Type I error control while achieving higher power than the standard private threshold. In low-sensitivity regimes, power exceeds the non-private baseline.
Multiple Testing (Independent & Correlated):
- The "All-Noisy" baseline (privatizing all $m$ hypotheses simultaneously) yields negligible power.
- Both fixed and adaptive peeling algorithms significantly outperform the All-Noisy baseline.
- The adaptive peeling algorithm consistently outperforms fixed peeling by optimizing the privacy budget allocation.
- All methods maintain FDR control at the target level ( $\alpha=0.05$ ) under both independent and correlated (one-factor) data structures.
Real-World Application: Applied to a Systemic Lupus Erythematosus (SLE) GWAS dataset ( $m \approx 6.2$ million SNPs), the All-Noisy method found zero discoveries. In contrast, the peeling algorithms recovered a large fraction of the non-private discoveries, with the adaptive method performing comparably to a fixed-size approach without requiring manual tuning.

Significance and Claims

The paper claims to establish a comprehensive framework for differentially private inference based on e-values under µ-GDP. Its primary significance lies in:

Bridging the Gap: It provides the first systematic construction of e-values specifically tailored for the µ-GDP framework, which is increasingly standard for complex, multi-stage analyses.
Power Recovery: It demonstrates that careful calibration of rejection thresholds and the use of adaptive peeling can recover statistical power lost to privacy noise, sometimes even surpassing non-private baselines in low-sensitivity settings.
Rigorous Privacy Guarantees: By identifying the failure of Gaussian noise in selection steps and replacing it with Gumbel noise, the paper offers the first valid µ-GDP guarantees for adaptive selection in multiple testing, correcting a technical gap in prior literature.
Practical Utility: The method is shown to be effective in high-dimensional real-world settings (GWAS), enabling rigorous privacy protection without rendering the data useless for scientific discovery.

The authors note that the optimality of the Gaussian mechanism under relaxed privacy constraints (inequality rather than exact exhaustion) remains an open problem, as the monotonicity intuition of DP does not strictly hold in the low-sensitivity regime where power gains are observed.