Effective-Sensitivity Correction for Correlated Multi-Output Gaussian Releases

We prove an exact reduction formula for the ε-hockey-stick divergence when a Gaussian mechanism releases multiple correlated scalar outputs jointly observable by an adversary.

For a k-output mechanism M(D) = (f1(D) + Z1, . . . , fk(D) + Zk) with independent noise Zi ∼ N (0, σ2 ) and linear coupling fj(D) = cjf1(D), the log-likelihood ratio depends on the observations only through the scalar projection S(x) = c ⊤x/∥c∥2, which is a sufficient statistic for distinguishing adjacent datasets. This yields an isometric reduction: the privacy- loss distribution of M reduces to that of a scalar Gaussian mechanism (via sufficient statistic projection) with effective sensitivity ∆eff = ∆ ∥c∥2: Hε(M) = Hscalar ε (σ, ∆eff).

The result is exact within the model (linear coupling, independent noise, full-vector observa- tion) and requires no new accountant code. We verify the formula against the dp accounting.pld library (Google), confirm the baseline match at c = 0, and report gaps up to 2.26× at full coupling. We connect this to the GCI Sign Theorem of Paper #1, which provides the com- plementary result for correlated noise via sign(∂R/∂ρ|0) = sign(m1m2).