Lernen Laplace Mechanism | Differential Privacy Mechanisms

The Laplace mechanism is one of the foundational tools in differential privacy. It is designed to protect individual privacy by adding carefully calibrated random noise to the output of a function or query. This mechanism is particularly useful when you want to release numeric results, such as counts or sums, while ensuring that the presence or absence of a single individual in the dataset does not significantly affect the output.

The key idea behind the Laplace mechanism is to add noise drawn from the Laplace distribution, which is centered at zero and has a scale parameter determined by the function’s global sensitivity and the desired privacy level, denoted by epsilon (ε). The global sensitivity measures the maximum amount that a single individual's data can change the result of the query. The Laplace mechanism ensures ε-differential privacy by adding noise proportional to this sensitivity.

The formula for the Laplace mechanism is:

Release: f(D) + Lap(S/ε)

where:

$f(D)$ is the true query result on dataset $D$ ;
$Lap(S/ε)$ means noise is drawn from the Laplace distribution with scale $S/ε$ ;
$S$ is the global sensitivity of the function;
$ε$ is the privacy parameter (smaller ε means more privacy, but also more noise).

You use the Laplace mechanism when you need to release aggregate statistics, such as counts or sums, and want strong privacy guarantees for each individual in your dataset. It is especially suitable for queries with bounded sensitivity and when you require pure differential privacy (i.e., delta equals zero).


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import numpy as np
from scipy.stats import laplace

# Example: Adding Laplace noise to a count query

# True count result (e.g., number of users with a certain attribute)
true_count = 42

# Privacy parameter epsilon
epsilon = 1.0

# Global sensitivity for count queries is 1
sensitivity = 1

# Scale parameter for Laplace distribution
scale = sensitivity / epsilon

# Draw Laplace noise
noise = np.random.laplace(loc=0, scale=scale)

# Noisy count (differentially private result)
noisy_count = true_count + noise

print("True count:", true_count)
print("Noisy count (with Laplace noise):", noisy_count)

Study More

The Laplace mechanism is typically used when you require pure differential privacy, and your query's sensitivity is well understood and bounded. The Gaussian mechanism, by contrast, is often used when you can tolerate a small probability of greater privacy loss (approximate differential privacy) or when the query has higher sensitivity. To decide between Laplace and Gaussian, consider your privacy parameters, the required privacy guarantee, and the characteristics of your data and queries.

1. Which property does the Laplace mechanism guarantee when adding noise to a query result?

2. How does increasing the global sensitivity of a query affect the amount of Laplace noise added?

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