# Approximate differential privacy: avoiding composition in vector-queries

Assume we have an $$n$$-dimensional real-valued function $$f$$ whose $$\ell_1$$ sensitivity is equal to $$GS(f) = 1$$. We can also assume the sensitivity of each dimension is also $$\Delta f = 1$$.

For pure differential privacy, we have the Laplace Mechanism, whose magnitude is parametrized by the privacy level $$\epsilon > 0$$ and the $$\ell_1$$ sensitivity. If we were to add noise to each $$n$$ dimension by using sequential/basic-composition, we were going to need to scale the Laplace distribution with $$\propto \Delta f / (\epsilon / n) = n / \epsilon$$. However, since $$GS(f) = 1$$, we do not need to rely on basic proposition, rather, we can immediately add noise with magnitude $$\propto GS(f) /\epsilon = 1 / \epsilon$$. In summary, when we have $$\Delta f = GS(f)$$, we can sample $$n$$ iid noise from Laplace distribution with $$\propto \Delta f/\epsilon$$, and the $$n$$-dimensional result is still $$\epsilon$$-DP (privacy does not decay).

I am wondering whether we can extend this to approximate differential privacy; not for a specific noise mechanism like the Gaussian, but in general.

Question: Assume there exists a mechanism $$\mathcal{M}$$ that satisfies $$(\epsilon, \delta)$$-DP for any single-dimensional query with sensitivity $$\Delta f = 1$$. If I use this for each $$n$$ component of $$f$$, do I still have $$(\epsilon, \delta)$$-DP? Again, if we had $$GS(f) = n \cdot \Delta f$$, then we would have decayed to $$(n\epsilon, n\delta)$$-DP, but now I am wondering if when $$GS(f) = \Delta f$$ we can claim a similar result as in the Laplace noise case explained above.

If the mechanism can do anything, then here is a counterexample with pure DP. Consider the mechanism $$M$$ that rounds up its (real-valued) input then applies Laplace noise of scale 1. Then, for any single-dimensional query with sensitivity $$1$$, applying $$M$$ to the result will give ε-DP with ε=1. But now, consider a function $$f$$ that outputs [0, ..., 0] on dataset $$D_1$$ and [0.1, ..., 0.1] (with $$n=10$$) on $$D_2$$. That function has global sensitivity 1, but applying $$M$$ to each component gives ε=10, not ε=1.