# Three composition theorems for differential privacy

This is a brief post, bringing together three composition theorems for differential privacy.

1. The composition of an ε1-differentially private algorithm and an ε2-differentially private algorithm is an (ε12)-differentially private algorithm.
2. The composition of an (ε1, δ1)-differentially private algorithm and an (ε2, δ2)-differentially private algorithm is an (ε12, δ12)-differentially private algorithm.

The three composition rules can be summarized briefly as follows:

ε1 ∘ ε2 → (ε1 + ε2)
1, δ1) ∘ (ε2, δ2) → (ε12, δ12)
(α, ε1) ∘ (α, ε2) → (α, ε12)

What is the significance of these composition theorems? In short, ε-differential privacy and Rényi differential privacy compose as one would hope, but (ε, δ)-differential privacy does not.

The first form of differential privacy proposed was ε-differential privacy. It is relatively easy to interpret, composes nicely, but can be too rigid.

If you have Gaussian noise, for example, you are lead naturally to (ε, δ)-differential privacy. The δ term is hard to interpret. Roughly speaking you could think  it as the probability that ε-differential privacy fails to hold. Unfortunately with (ε, δ)-differential privacy the epsilons add and so do the deltas. We would prefer that δ didn’t grow with composition.

Rényi differential privacy is a generalization of ε-differential privacy that uses a family of information measures indexed by α to measure the impact of a single row being or not being in a database. The case of α = ∞ corresponds to ε-differential privacy, but finite values of α tend to be less pessimistic. The nice thing about the composition theorem for Rényi differential privacy is that the α parameter doesn’t change, unlike the δ parameter in (ε, δ)-differential privacy.