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Page 33 of The Algorithmic Foundations of Differential Privacy gives two examples where a composition of mechanisms can be viewed as a vector-values output, histograms, and fixed counting queries, where the privacy bound can be analyzed by considering the sensitivity of the vector-valued output.

I was wondering about a more general statement; when, generally, can a composition can be viewed as a vector-valued output, and when can't it? Is it true that for any set of fixed, arbitrary mechanisms can be viewed as a vector valued output, where the privacy bound can then be analyzed by considering the sensitivity of the vector?

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Yes, it’s always true. A vector is just a formal way as treating an ordered set of elements as an entity in its own right.

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  • $\begingroup$ This was my intuition as well, but it felt too simple somehow. Do you know of any situations where the vector view might be disadvantageous, or misleading? $\endgroup$ Commented Apr 14, 2022 at 18:44
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    $\begingroup$ I think that it might not be helpful when there are dependencies between the components that are hard to untangle. Such cases would be hard to analyse and if a simplifying independence assumption were made, it could be misleading. $\endgroup$
    – Daniel S
    Commented Apr 14, 2022 at 19:29

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