From 423b74b6354d5d12156012aa1a3434cb406301df Mon Sep 17 00:00:00 2001 From: Manos Date: Tue, 12 Oct 2021 04:21:46 +0200 Subject: [PATCH] evaluation: lmdk-sel-sol --- text/problem/theotherthing/solution.tex | 102 ++++++++++++------------ 1 file changed, 52 insertions(+), 50 deletions(-) diff --git a/text/problem/theotherthing/solution.tex b/text/problem/theotherthing/solution.tex index 7a7bfa3..54b7e49 100644 --- a/text/problem/theotherthing/solution.tex +++ b/text/problem/theotherthing/solution.tex @@ -2,40 +2,37 @@ \label{subsec:lmdk-sel-sol} 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 $L$. +Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable. +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 $L \subseteq T$. Thus, we create a new set $L'$ such that $L \subset L' \subseteq T$. -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}). -Then (Section~\ref{subsec:lmdk-opt-sel}), 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 ot the options that we created earlier. + +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}). +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}). This process provides an extra layer of privacy protection to {\thethings}, and thus allows the release, and thereafter processing, of {\thething} timestamps. % 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. % 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. -\subsubsection{{\Thething} set options} +\subsubsection{{\Thething} set options generation} \label{subsec:lmdk-set-opts} -This step aims to select a set of candidate {\thething} timestamps options either by randomizing the actual timestamps (Section~\ref{subsec:lmdk-rnd}), or by inserting dummy timestamps (Section~\ref{subsec:lmdk-dum-gen}) to the actual {\thething} timestamps. - - -\paragraph{Dummy {\thething} generation} -\label{subsec:lmdk-dum-gen} - -Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable. -The goal is to select a list of sets with additional timestamps from a series of events at timestamps $\{t_n\}$ for a set of {\thethings} at $\{l_k\} \subseteq \{t_n\}$. Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} approach this problem with an optimal and heuristic methodology, respectively. +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}. +\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. +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 $T$. -Function \calcMetric measures an indicator for the union of $\{l_k\}$ and a timestamp combination from $\{t_n\} \setminus \{l_k\}$. -Function \evalSeq evaluates the result of \calcMetric by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}. -Function \getOpts returns all possible \emph{valid} sets of combinations \opt such that $\{l_{k+i}\} \subset \{l_{k+j}\}, \forall i, j \in [k, n] \mid i < j$, i.e.,~larger options must contain all of the timestamps that are present in smaller ones. -Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n$, where each one of them is a combination of $\{l_k\}$ with $x \in [1, n - k]$ timestamps from $\{t_n\}$. +\paragraph{Optimal} +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. +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} $L$. \begin{algorithm} - \caption{Optimal dummy {\thething} set options selection} + \caption{Optimal dummy {\thething} set options generation} \label{algo:lmdk-sel-opt} \DontPrintSemicolon - \KwData{$\{t_n\}, \{l_k\}$} + \KwData{$T, L$} \SetKwInput{KwData}{Input} @@ -43,11 +40,10 @@ Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n \BlankLine % Evaluate the original - \metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\; - \evalOrig $\leftarrow$ \evalSeq{\metricOrig}\; + \evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\; % Get all possible option combinations - \opts $\leftarrow$ \getOpts{$\{t_n\}, \{l_k\}$}\; + \opts $\leftarrow$ \getOpts{$T, L$}\; % Track the minimum (best) evaluation \diffMin $\leftarrow$ $\infty$\; @@ -55,25 +51,29 @@ Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n % Track the optimal sequence (the one with the best evaluation) \optim $\leftarrow$ $[]$\; - \ForEach{\opt $\in$ \opts}{\label{algo:lmdk-sel-opt-for-each} - \evalSum $\leftarrow 0$\; + \ForEach{\opt $\in$ \opts}{ \label{algo:lmdk-sel-opt-for-each} + \evalCur $\leftarrow 0$\; \ForEach{\opti $\in$ \opt}{ - \metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \opti, \{l_k\}$}\;\label{algo:lmdk-sel-opt-comparison} - \evalSum $\leftarrow$ \evalSum $+$ \evalSeq{\metricCur}\; - - % Compare with current optimal - \diffCur $\leftarrow \left|\evalSum/\#\opt - \evalOrig\right|$\; - \If{\diffCur $<$ \diffMin}{ - \diffMin $\leftarrow$ \diffCur\; - \optim $\leftarrow$ \opt\; - } + \evalCur $\leftarrow$ \evalCur $+$ \evalSeq{$T, \opti, L$}/\#\opt\; \label{algo:lmdk-sel-opt-comparison} } - }\label{algo:lmdk-sel-opt-end} + % Compare with current optimal + \diffCur $\leftarrow \left|\evalCur - \evalOrig\right|$\; + \If{\diffCur $<$ \diffMin}{ + \diffMin $\leftarrow$ \diffCur\; + \optim $\leftarrow$ \opt\; + } + } \label{algo:lmdk-sel-opt-end} \Return{\optim} \end{algorithm} -Algorithm~\ref{algo:lmdk-sel-opt}, in particular, between Lines~{\ref{algo:lmdk-sel-opt-for-each}-\ref{algo:lmdk-sel-opt-end}} evaluates each option in \opts. -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 $\{t_n\}$ with {\thethings} $\{l_k\}$. +Algorithm~\ref{algo:lmdk-sel-opt} guarantees to return the optimal set of dummy {\thethings} with regard to the original set $L$. +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. +Next, we present a heuristic solution with improved time and space requirements. + + +\paragraph{Heuristic} +Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology. +At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $T \setminus L$. \begin{algorithm} \caption{Heuristic dummy {\thething} set options selection} @@ -81,30 +81,28 @@ It finds the option that is the most \emph{similar} to the original (Lines~{\ref \DontPrintSemicolon - \KwData{$\{t_n\}, \{l_k\}$} + \KwData{$T, L$} \KwResult{\optim} \BlankLine % Evaluate the original - \metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\; - \evalOrig $\leftarrow$ \evalSeq{\metricOrig}\; + \evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\; % Get all possible option combinations \optim $\leftarrow$ $[]$\; - $\{l_{k'}\} \leftarrow \{l_k\}$\; + $L' \leftarrow L$\; - \While{$\{l_{k'}\} \neq \{t_n\}$}{\label{algo:lmdk-sel-heur-while} + \While{$L' \neq T$}{\label{algo:lmdk-sel-heur-while} % Track the minimum (best) evaluation \diffMin $\leftarrow$ $\infty$\; - \optimi $\leftarrow$ $0$\; + \optimi $\leftarrow$ Null\; % Find the combinations for one more point - \ForEach{\reg $\in \{t_n\} \setminus \{l_{k'}\}$}{ + \ForEach{\reg $\in T \setminus L'$}{ % Evaluate current - \metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \reg, \{l_{k'}\}$}\;\label{algo:lmdk-sel-heur-comparison} - \evalCur $\leftarrow$ \evalSeq{\metricCur}\; + \evalCur $\leftarrow$ \evalSeq{$T, \reg, L'$}\; \label{algo:lmdk-sel-heur-comparison} % Compare evaluations \diffCur $\leftarrow$ $\left|\evalCur - \evalOrig\right|$\; @@ -116,27 +114,31 @@ It finds the option that is the most \emph{similar} to the original (Lines~{\ref } % Save new point to landmarks - $k' \leftarrow k' + 1$\; - $l_{k'} \leftarrow \optimi$\; + $L'.add(\optimi)$\; % Add new option - \optim.add($\{l_{k'}\} \setminus \{l_k\}$)\; + \optim.append($L' \setminus L$)\; }\label{algo:lmdk-sel-heur-end} \Return{\optim} \end{algorithm} -Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology. -At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $\{t_n\} \setminus \{l_k\}$. Similar to Algorithm~\ref{algo:lmdk-sel-opt}, the selection is done based on a predefined metric (Lines~{\ref{algo:lmdk-sel-heur-comparison}-\ref{algo:lmdk-sel-heur-comparison-end}}). -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.,~$\{l_{k'}\} = \{t_n\}$. +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.,~$L' = T$. -Note that the reverse heuristic approach, i.e.,~starting with $\{t_n\}$ {\thethings} and removing until $\{l_k\}$, performs worse than and occasionally the same with Algorithm~\ref{algo:lmdk-sel-heur}. +In terms of complexity: given $n$ regular events it requires $\mathcal{O}(n^2)$ time and space. +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}. + + + +\mk{WIP: Histograms} \subsubsection{Privacy-preserving option selection} \label{subsec:lmdk-opt-sel} +\mk{WIP} + % Nearby events Events that occur at recent timestamps are more likely to reveal sensitive information regarding the users involved~\cite{kellaris2014differentially}. Thus, taking into account more recent events with respect to {\thethings} can result in less privacy loss and better privacy protection overall.