Learn Gaussian Mechanism | Differential Privacy Mechanisms

The Gaussian mechanism is a core method for achieving (ε, δ)-Differential Privacy. Unlike the Laplace mechanism, which is used for pure ε-DP, the Gaussian mechanism is designed for situations where a small probability of failure (δ) is acceptable. This makes it particularly useful in real-world scenarios where perfect privacy cannot be guaranteed, but a strong probabilistic privacy guarantee is still required.

In the Gaussian mechanism, you add noise drawn from a Gaussian (normal) distribution to the output of a query. The amount of noise depends on the global sensitivity of the function (the maximum possible change in the output due to a single individual's data), the privacy parameters $ε$ and $δ$ , and the standard deviation ( $σ$ ) of the noise. The standard deviation is typically chosen to be proportional to the global sensitivity and inversely proportional to $ε$ , with an adjustment for $δ$ .

The Gaussian mechanism is especially suitable for (ε, δ)-DP because the tails of the normal distribution allow for a small probability of larger deviations, which aligns with the $δ$ parameter's interpretation as a "failure probability." This mechanism is widely used in practice, particularly in machine learning and large-scale data analysis, where strict $ε$ -DP may be too restrictive or result in excessive noise.


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import numpy as np
from scipy.stats import norm

# Suppose you have a dataset of user counts
data = np.array([10, 20, 15, 30, 25, 40, 35, 50])

# Define the sum query
true_sum = np.sum(data)

# Set privacy parameters
epsilon = 1.0
delta = 1e-5

# Assume global sensitivity for sum query (e.g., max value a single user can contribute)
global_sensitivity = 50

# Compute required standard deviation (σ) for Gaussian noise
sigma = (global_sensitivity * np.sqrt(2 * np.log(1.25 / delta))) / epsilon

# Add Gaussian noise
noisy_sum = true_sum + np.random.normal(0, sigma)

print("True sum:", true_sum)
print("Noisy sum (Gaussian mechanism):", noisy_sum)
print("Standard deviation of noise (sigma):", sigma)

1. Which of the following statements best describes the main difference between the Laplace and Gaussian mechanisms for differential privacy?

2. In which scenario would you prefer to use the Gaussian mechanism over the Laplace mechanism?

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 3

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              12345678910111213141516171819202122232425
            
import numpy as np
from scipy.stats import norm

# Suppose you have a dataset of user counts
data = np.array([10, 20, 15, 30, 25, 40, 35, 50])

# Define the sum query
true_sum = np.sum(data)

# Set privacy parameters
epsilon = 1.0
delta = 1e-5

# Assume global sensitivity for sum query (e.g., max value a single user can contribute)
global_sensitivity = 50

# Compute required standard deviation (σ) for Gaussian noise
sigma = (global_sensitivity * np.sqrt(2 * np.log(1.25 / delta))) / epsilon

# Add Gaussian noise
noisy_sum = true_sum + np.random.normal(0, sigma)

print("True sum:", true_sum)
print("Noisy sum (Gaussian mechanism):", noisy_sum)
print("Standard deviation of noise (sigma):", sigma)

1. Which of the following statements best describes the main difference between the Laplace and Gaussian mechanisms for differential privacy?

2. In which scenario would you prefer to use the Gaussian mechanism over the Laplace mechanism?

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 3