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problem: Additional remarks in lmdk-sel-sol

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Manos Katsomallos 2021-10-12 13:00:26 +02:00
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text/problem/theotherthing/solution.tex
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@ -186,7 +186,7 @@ For every possible histogram version, the \getDiff function finds the difference
\Return{\opts}
\end{algorithm}
Between Lines~{\ref{algo:lmdk-sel-hist-cmp-start}-\ref{algo:lmdk-sel-hist-cmp-end}} we check every possible histogram version by incrementing each bin by $1$ and comparing it to the original (Line~\ref{algo:lmdk-sel-hist-cmp}).
In Lines~{\ref{algo:lmdk-sel-hist-cmp-start}-\ref{algo:lmdk-sel-hist-cmp-end}} we check every possible histogram version by incrementing each bin by $1$ and comparing it to the original (Line~\ref{algo:lmdk-sel-hist-cmp}).
In the end of the process, we return \opts which contains all the versions of \hist that are closest to \hist for all possible sizes of \hist.
@ -195,6 +195,9 @@ In the end of the process, we return \opts which contains all the versions of \h
The Algorithms of Section~\ref{subsec:lmdk-set-opts} return a set of possible versions of the original {\thething} set $L$ by adding extra timestamps in it from the series of events at timestamps $T \supseteq L$.
In the next step of the process, we randomly select a set by utilizing the exponential mechanism (Section~\ref{subsec:prv-mech}).
\paragraph{Score function}
Prior to selecting a set, the exponential mechanism evaluates each set using a score function.
One way evaluate each set is by taking into account the temporal position the events in the sequence.
@ -215,3 +218,9 @@ On the one hand, sets with more dummy {\thethings} may render actual {\thethings
That is due to the fact that, it is harder for an adversary to pick a {\thething} when the ratio of {\thethings} to the size of the set gets lower.
On the other hand, more dummy {\thethings} lead to distributing the privacy budget to more events, and therefore investing less at each timestamp.
Thus, providing a better level of privacy protection.
\paragraph{Option release}
The options that Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} generate contain actual timestamps which can be utilized directly by the {\thething} privacy mechanisms that we presented in Section~\ref{subsec:lmdk-mechs}.
However, Algorithm~\ref{algo:lmdk-sel-hist} returns histograms instead of timestamps.
Therefore, we need to process the result of the exponential mechanism further by creating a sample from the true {\thethings} and populating it with the remaining amount of choices, i.e.,~$\left|L'\right| - \left|L\right|$ by performing sampling without replacement from the resulting option $L$.