Epsilon (ε) privacy is the core parameter of differential privacy — it quantifies the maximum privacy loss that any individual can experience from their data being included in a computation. A smaller epsilon means stronger privacy protection but typically comes at the cost of reduced data utility.

Formal Definition

A mechanism M satisfies ε-differential privacy if for any two neighboring datasets D and D' (differing in one person's data) and any possible output S:

This means the output distribution changes by at most a factor of $e^\varepsilon$ whether or not any individual's data is included.

Interpreting Epsilon

• ε = 0: Perfect privacy — the output reveals absolutely nothing about any individual. But provides no utility.
• ε = 0.1: Very strong privacy — an attacker gains at most ~10% more information from the output.
• ε = 1: Moderate privacy — standard benchmark for "good" differential privacy.
• ε = 10: Weak privacy protection — often considered the upper bound for meaningful privacy.
• ε → ∞: No privacy — output directly reveals the data.

Privacy Budget

• Each query or computation on the data "spends" some epsilon from the privacy budget.
• Composition Theorem: Running k analyses on the same data costs approximately ε × √k total privacy (under advanced composition).
• Once the budget is exhausted, no more queries should be answered to maintain privacy guarantees.

Practical Usage

• Apple: Uses ε = 2–8 for collecting emoji and typing statistics in iOS.
• Google: Uses ε = 2–9 for Chrome usage statistics via RAPPOR.
• US Census: Applied differential privacy with aggregated ε budgets for the 2020 Census.

The Privacy-Utility Trade-Off

Smaller ε requires adding more noise, which reduces the accuracy of results. Choosing ε involves balancing privacy protection against the need for useful, accurate outputs — a fundamental design decision with no universally correct answer.