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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} \Return{\opts}
\end{algorithm} \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. 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 $T \supseteq L$. 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}). 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. 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. 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. 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. 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. 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 $L$.