Learn Intuition Behind Epsilon-DP and (Epsilon, Delta)-DP

Understanding the intuition behind the privacy parameters epsilon (ε) and delta (δ) is essential for grasping the guarantees provided by differential privacy. The parameter epsilon (ε) is often called the privacy loss parameter. It quantifies the maximum change in the probability of any output when a single individual's data is either included or excluded from a dataset. A smaller ε means that the presence or absence of any one person has little effect on the output, providing stronger privacy. Conversely, a larger ε allows for greater influence of any individual's data, resulting in weaker privacy guarantees.

Delta (δ) comes into play in the more general (ε, δ)-differential privacy definition. While ε controls the direct privacy loss, δ represents the probability of failure—the chance that the privacy guarantee could be violated by more than ε. In practice, δ is chosen to be a very small number, such as 10⁻⁶ or less, ensuring that the chance of a significant privacy breach is negligible. Together, (ε, δ)-DP allows for a tiny probability of a larger privacy loss, making certain mechanisms practical while still keeping risk very low.


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import numpy as np
import matplotlib.pyplot as plt

def simulate_dp_output(epsilon, n_simulations=10000):
    # Simulate two probabilities: with and without a person
    p_without = 0.5
    p_with = np.minimum(np.exp(epsilon) * p_without, 1.0)
    outputs_without = np.random.binomial(1, p_without, n_simulations)
    outputs_with = np.random.binomial(1, p_with, n_simulations)
    return outputs_without, outputs_with

epsilons = [0.01, 0.1, 0.5, 1.0]
plt.figure(figsize=(10, 6))

for epsilon in epsilons:
    out_wo, out_w = simulate_dp_output(epsilon)
    diff = np.abs(np.mean(out_wo) - np.mean(out_w))
    plt.bar(str(epsilon), diff, label=f"ε={epsilon}")

plt.ylabel("Difference in Output Probability")
plt.xlabel("Epsilon Value")
plt.title("Effect of Epsilon on Output Probability Difference")
plt.show()

1. Which statement best describes the interpretation of epsilon (ε) in differential privacy?

2. What is the role of delta (δ) in (ε, δ)-differential privacy?

Everything was clear?

Thanks for your feedback!

Section 1. Chapter 4

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              1234567891011121314151617181920212223
            
import numpy as np
import matplotlib.pyplot as plt

def simulate_dp_output(epsilon, n_simulations=10000):
    # Simulate two probabilities: with and without a person
    p_without = 0.5
    p_with = np.minimum(np.exp(epsilon) * p_without, 1.0)
    outputs_without = np.random.binomial(1, p_without, n_simulations)
    outputs_with = np.random.binomial(1, p_with, n_simulations)
    return outputs_without, outputs_with

epsilons = [0.01, 0.1, 0.5, 1.0]
plt.figure(figsize=(10, 6))

for epsilon in epsilons:
    out_wo, out_w = simulate_dp_output(epsilon)
    diff = np.abs(np.mean(out_wo) - np.mean(out_w))
    plt.bar(str(epsilon), diff, label=f"ε={epsilon}")

plt.ylabel("Difference in Output Probability")
plt.xlabel("Epsilon Value")
plt.title("Effect of Epsilon on Output Probability Difference")
plt.show()

1. Which statement best describes the interpretation of epsilon (ε) in differential privacy?

2. What is the role of delta (δ) in (ε, δ)-differential privacy?

Everything was clear?

Thanks for your feedback!

Section 1. Chapter 4