218 lines
11 KiB
TeX
218 lines
11 KiB
TeX
\subsection{Protecting {\thethings}}
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\label{subsec:lmdk-sel-sol}
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The main idea of the privacy-preserving {\thething} selection component is to privately select extra {\thething} event timestamps, i.e.,~dummy {\thethings}, from the set of timestamps $T /\ L$ of the time series $S_T$ and add them to the original {\thething} set $L$.
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Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable.
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The goal is to select a list of sets with additional timestamps from a series of events at timestamps $T$ for a set of {\thethings} at $L \subseteq T$.
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Thus, we create a new set $L'$ such that $L \subset L' \subseteq T$.
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First, we generate a set of dummy {\thething} set options by adding regular event timestamps from $T /\ L$ to $L$ (Section~\ref{subsec:lmdk-set-opts}).
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Then, we utilize the exponential mechanism, with a utility function that calculates an indicator for each of the options in the set based on how much it differs from the original {\thething} set $L$, and randomly select one of the options (Section~\ref{subsec:lmdk-opt-sel}).
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This process provides an extra layer of privacy protection to {\thethings}, and thus allows the release, and thereafter processing, of {\thething} timestamps.
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% We utilize the exponential mechanism with a utility function that calculates an indicator for each of the options in the set that we selected in the previous step.
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% The utility depends on the positioning of the {\thething} timestamps of an option in the series, e.g.,~the distance from the previous/next {\thething}, the distance from the start/end of the series, etc.
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\subsubsection{{\Thething} set options generation}
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\label{subsec:lmdk-set-opts}
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Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} approach this problem with an optimal and heuristic methodology, respectively.
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Function \evalSeq evaluates the result of the union of $L$ and a timestamp combination from $T \setminus L$ by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}.
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\getOpts returns all the possible \emph{valid} sets of combinations \opt such that larger options contain all of the timestamps that are present in smaller ones.
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Each combination contains a set of timestamps with sizes $\left|L\right| + 1, \left|L\right| + 2, \dots, \left|T\right|$, where each one of them is a combination of $L$ with $x \in [1, \left|T\right| - \left|L\right|]$ timestamps from $T$.
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\paragraph{Optimal}
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Algorithm~\ref{algo:lmdk-sel-opt}, between Lines~{\ref{algo:lmdk-sel-opt-for-each}--\ref{algo:lmdk-sel-opt-end}} evaluates each option in \opts.
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It finds the option that is the most \emph{similar} to the original (Lines~{\ref{algo:lmdk-sel-opt-comparison}-\ref{algo:lmdk-sel-opt-end}}), i.e.,~the option that has an evaluation that differs the least from that of the sequence $T$ with {\thethings} $L$.
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\begin{algorithm}
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\caption{Optimal dummy {\thething} set options generation}
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\label{algo:lmdk-sel-opt}
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\DontPrintSemicolon
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\KwData{$T, L$}
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\SetKwInput{KwData}{Input}
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\KwResult{\optim}
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\BlankLine
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% Evaluate the original
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\evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\;
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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% Track the optimal sequence (the one with the best evaluation)
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\opts $\leftarrow$ $[]$\;
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\ForEach{\opt $\in$ \getOpts{$T, L$}}{ \label{algo:lmdk-sel-opt-for-each}
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\evalCur $\leftarrow 0$\;
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\ForEach{\opti $\in$ \opt}{
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\evalCur $\leftarrow$ \evalCur $+$ \evalSeq{$T, \opti, L$}/\#\opt\; \label{algo:lmdk-sel-opt-comparison}
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}
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% Compare with current optimal
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\diffCur $\leftarrow \left|\evalCur - \evalOrig\right|$\;
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\If{\diffCur $<$ \diffMin}{
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\diffMin $\leftarrow$ \diffCur\;
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\opts $\leftarrow$ \opt\;
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}
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} \label{algo:lmdk-sel-opt-end}
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\Return{\opts}
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\end{algorithm}
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Algorithm~\ref{algo:lmdk-sel-opt} guarantees to return the optimal set of dummy {\thethings} with regard to the original set $L$.
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However, it is rather costly in terms of complexity: given $n$ regular events and a combination of size $r$, it requires $\mathcal{O}(C(n, r) + 2^C(n, r))$ time and $\mathcal{O}(r*C(n, r))$ space.
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Next, we present a heuristic solution with improved time and space requirements.
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\paragraph{Heuristic}
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Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology.
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At each step it selects a new timestamp, that corresponds to a regular ({non-\thething}) event from $T \setminus L$, to create an option.
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\begin{algorithm}
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\caption{Heuristic dummy {\thething} set options selection}
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\label{algo:lmdk-sel-heur}
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\DontPrintSemicolon
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\KwData{$T, L$}
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\KwResult{\opts}
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\BlankLine
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% Evaluate the original
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\evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\;
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% Get all possible option combinations
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\opts $\leftarrow$ $[]$\;
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$L' \leftarrow L$\;
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\While{$L' \neq T$}{\label{algo:lmdk-sel-heur-while}
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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\optimi $\leftarrow$ Null\;
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% Find the combinations for one more point
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\ForEach{\reg $\in T \setminus L'$}{
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% Evaluate current
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\evalCur $\leftarrow$ \evalSeq{$T, \reg, L'$}\; \label{algo:lmdk-sel-heur-comparison}
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% Compare evaluations
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\diffCur $\leftarrow$ $\left|\evalCur - \evalOrig\right|$\;
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\If{\diffCur $<$ \diffMin}{
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\diffMin $\leftarrow$ \diffCur\;
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\optimi $\leftarrow$ \reg\;
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}\label{algo:lmdk-sel-heur-cmp-end}
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}
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% Save new point to landmarks
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$L'$.add(\optimi)\;
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% Add new option
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\opts.append($L' \setminus L$)\;
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}\label{algo:lmdk-sel-heur-end}
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\Return{\opts}
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\end{algorithm}
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Similar to Algorithm~\ref{algo:lmdk-sel-opt}, it selects new options based on a predefined metric (Lines~{\ref{algo:lmdk-sel-heur-comparison}-\ref{algo:lmdk-sel-heur-cmp-end}}).
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This process (Lines~{\ref{algo:lmdk-sel-heur-while}-\ref{algo:lmdk-sel-heur-end}}) goes on until we select a set that is equal to the size of the series of events, i.e.,~$L' = T$.
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In terms of complexity, given $n$ regular events it requires $\mathcal{O}(n^2)$ time and space.
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Note that the reverse heuristic approach, i.e.,~starting with $T$ {\thethings} and removing until $L$, performs similarly with Algorithm~\ref{algo:lmdk-sel-heur}.
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\paragraph{Partitioned}
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We improve the complexity of Algorithm~\ref{algo:lmdk-sel-opt} by partitioning the {\thething} timestamp sequence $L$.
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Algorithm~\ref{algo:lmdk-sel-hist}, \getHist generates a histogram from $L$ with bins of size \h.
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We find \h by using the Freedman–Diaconis rule which is resilient to outliers and takes into account the data variability and data size~\cite{meshgi2015expanding}.
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For every possible histogram version, the \getDiff function finds the difference between two histograms; for this operation we utilize the Euclidean distance~(see Section~\ref{subsec:sel-utl} for more details).
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\begin{algorithm}
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\caption{Partitioned dummy {\thething} set options selection}
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\label{algo:lmdk-sel-hist}
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\DontPrintSemicolon
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\KwData{$T, L$}
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\KwResult{\opts}
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\BlankLine
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\hist, \h $\leftarrow$ \getHist{$T, L$}\;
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\histCur $\leftarrow$ hist\;
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\opts $\leftarrow$ $[]$\;
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\While{sum($L'$) $\neq$ len($T$)}{ \label{algo:lmdk-sel-hist-while}
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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% The candidate option
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\opt $\leftarrow$ \histCur\;
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% Check every possibility
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\ForEach{\hi \reg $L'$}{ \label{algo:lmdk-sel-hist-cmp-start}
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% Can we add one more point?
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\If{\hi $+$ $1$ $\leq$ \h}{
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\histTmp $\leftarrow$ \histCur\;
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\histTmp$[i]$ $\leftarrow$ \histTmp$[i]$ $+$ $1$\;
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% Find difference from original
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\diffCur $\leftarrow$ \getDiff{\hist, \histTmp}\;
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% Remember if it is the best that you've seen
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\If{\diffCur $<$ \diffMin}{ \label{algo:lmdk-sel-hist-cmp}
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\diffMin $\leftarrow$ \diffCur\;
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\opt $\leftarrow$ \histTmp\;
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}
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}
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} \label{algo:lmdk-sel-hist-cmp-end}
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% Update current histogram
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\histCur $\leftarrow$ \opt\;
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% Add current best to options
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\opts $\leftarrow$ \opt\;
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} \label{algo:lmdk-sel-hist-end}
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\Return{\opts}
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\end{algorithm}
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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}).
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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.
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\subsubsection{Privacy-preserving option selection}
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\label{subsec:lmdk-opt-sel}
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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 $T \supseteq L$.
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In the next step of the process, we randomly select a set by utilizing the exponential mechanism (Section~\ref{subsec:prv-mech}).
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Prior to selecting a set, the exponential mechanism evaluates each set using a score function.
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One way evaluate each set is by taking into account the temporal position the events in the sequence.
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% Nearby events
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Events that occur at recent timestamps are more likely to reveal sensitive information regarding the users involved~\cite{kellaris2014differentially}.
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Thus, taking into account more recent events with respect to {\thethings} can result in less privacy loss and better privacy protection overall.
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This leads to worse data utility.
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% Depending on the {\thething} discovery technique
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The values of events near a {\thething} are usually similar to that of the latter.
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Therefore, privacy-preserving mechanisms are likely to approximate their values based on the nearest {\thething} instead of investing extra privacy budget to perturb their actual values; thus, spending less privacy budget.
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Saving privacy budget for releasing perturbed versions of actual event values can bring about better data utility.
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% Distant events
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However, indicating the existence of dummy {\thethings} nearby actual {\thethings} can increase the adversarial confidence regarding the location of the latter within a series of events.
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Hence, choosing dummy {\thethings} far from the actual {\thethings} (and thus less relevant) can limit the final privacy loss.
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Another approach for the score function is to consider the number of events in each set.
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On the one hand, sets with more dummy {\thethings} may render actual {\thethings} more indistinguishable probabilistically.
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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.
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On the other hand, more dummy {\thethings} lead to distributing the privacy budget to more events, and therefore investing less at each timestamp.
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Thus, providing a better level of privacy protection.
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