Let $f$ be a function that takes a database $D$ as input and returns a real number. Assume that $f$ has sensitivity 1: for any databases $D_1$ and $D_2$ differing in a single record, $|f(D_1)-f(D_2)|\le1$.
To protect the results of this function with $(\varepsilon,\delta)$-differential privacy (DP), one can add noise to its result, and output $f(D)+X$, where $X$ is a random variable from a well-chosen distribution $\theta$ with values in $\mathbb{R}$.
Now, suppose that $\theta_1$ and $\theta_2$ are two distributions such that the mechanism above is respectively $(\varepsilon_1,\delta_1)$-DP and $(\varepsilon_2,\delta_2)$-DP. Without any additional assumptions on $\theta_1$ and $\theta_2$, what can we say about the mechanism $M$ defined by:
$$M(D)=f(D)+X_1+X_2$$
where $X_1\sim\theta_1$ and $X_2\sim\theta_2$? By post-processing, we know that it is at most $(\varepsilon_1,\delta_1)$-DP and $(\varepsilon_2,\delta_2)$-DP. Can we do better? (We can certainly find better and even tight bounds for specific distributions, like two Gaussians, but I'm interested in as generic a result as possible.)
The only related result I know of is amplification by iteration, but this seems to assume finite Rényi divergence, and is tailored for many successive noise additions, while I'm interested in a simpler setting with as little assumptions as possible.
Edit 1: Clément Cannone pointed out that Theorem 4.3 of this paper has a result in the case where $\delta_1=\delta_2=0$, which is $\varepsilon=\ln\frac{e^{\varepsilon_1+\varepsilon_2}+1}{e^{\varepsilon_1}+e^{\varepsilon_2}}$. The bound is tight for generic chaining of DP mechanisms, of which this problem is a specific case. This partially solves the question, although 1) the case where $\delta_1>0$ and/or $\delta_2>0$ is open, and 2) maybe additive noise is a specific enough case that better results are available?
Edit 2: Vitaly Feldman pointed out that results related to chaining also appear in this paper (Lemma A.1, Appendix B), which might provide a building block for a generic chaining result?
Edit 3: Ashwin Machanavajjhala suggests that amplification by mixing and diffusion mechanisms is another possible direction, although it requires strong assumptions on the noise distributions.
