SCIENTIA SINICA Informationis, Volume 47, Issue 11: 1493-1509(2017) https://doi.org/10.1360/N112017-00111

## An algorithm for differential privacy streaming data publication based on matrix mechanism under exponential decay mode

• AcceptedJun 13, 2017
• PublishedNov 14, 2017
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### Abstract

At present, many practical applications require the continuous release of statistical streaming data, and the importance of current data is higher than historical data. The solution to this problem is to assign weights to the data and propose a differential privacy data release method under exponential decay. However, existing methods only consider a single query, and cannot effectively use the correlation between queries in the continuous statistical publishing background to further improve the accuracy of the query. In this paper, we present a differential privacy data release algorithm (DMFDA) in exponential decay mode based on a matrix mechanism, which uses the advantages of the matrix to deal with relevant queries. Firstly, we use the construction method to generate the matrix decomposition strategy to meet the real-time requirements of streaming data. Secondly, the diagonal matrix is used to adjust the structure of the constructed strategy matrix so as to improve the release accuracy. Finally, according to the substructure of the constructed strategy matrix, a fast method of solving the diagonal matrix is proposed. The experiment is designed to compare DMFDA and similar algorithms for streaming data release in exponential decay. Experimental results show that the DMFDA algorithm is effective and feasible.

### References

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• Figure 1

Noise accumulation

• Figure 2

Comparison of query distortion with different moments (Search Logs). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 3

Comparison of query distortion with different moments (Nettrace). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 4

Comparison of query distortion with different moments (WorldCup98). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 5

Comparison of query distortion with different decay factors (Search Logs). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 6

Comparison of query distortion with different decay factors (Nettrace). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 7

Comparison of query distortion with different decay factors (WorldCup98). (a) $\epsilon=1.0$; (b) $\epsilon=0.1$; (c) $\epsilon=0.01$

• Figure 8

Comparison of different algorithm efficiency

• Figure 9

Run time of diagonal optimization algorithm

• Table 1   Data sets
 Data set Search Logs NetTrace WorldCup98 Size 32768 65536 7518579
•

Algorithm 1 策略矩阵构造算法BuildL

Require:数据规模$N$,衰减因子$p$;

Output:策略矩阵${\boldsymbol~L}$;

初始化策略矩阵${\boldsymbol~L}$,将所有元素置为0;

for $j=1$ to $N$

$i=j$;

while $i<N$ do

更新矩阵元素$\textbf{{L}}[i][j]=p^{i-j}$;

$i\leftarrow~i+\mathrm{lowbit}(i)$;

end while

end for

返回矩阵${\boldsymbol~L}$.

•

Algorithm 2 策略矩阵构造算法BuildB

Require:数据规模$N$,衰减因子$p$;

Output:还原矩阵${\boldsymbol~B}$;

初始化策略矩阵${\boldsymbol~B}$,将所有元素置为0;

for $i=1$ to $N$

$j=i$;

while $j>0$ do

更新矩阵元素$\textbf{{B}}[i][j]=p^{i-j}$;

$j\leftarrow~j-\mathrm{lowbit}(j)$;

end while

end for

返回矩阵${\boldsymbol~B}$.

•

Algorithm 3 指数衰减模式下差分隐私流数据发布DM

Require:预设时刻上限$T$,衰减因子$p$;

Output:每一时刻的发布结果$s_{t}$;

for $t=1$ to $T$

更新实际统计量$\Phi~_{\mathrm{lowbit}(t)}\leftarrow~D_{t}+\sum_{j=0}^{\mathrm{lowbit}(t)-1}\Phi~_{j}$;

添加噪声$\overset{\sim~}{\Phi~_{\mathrm{lowbit}(t)}}\leftarrow~\Phi~_{\mathrm{lowbit}(t)}+\mathrm{Lap}(\frac{\Delta~\textbf{{L}}}{\epsilon~})$;

$k\leftarrow~t,~s_{t}\leftarrow~0$;

while $k>0$ do

$s_{t}\leftarrow~s_{t}+(\overset{\sim~}{\Phi~_{\mathrm{lowbit}(k)}}\times~p^{i-k})$;

$k\leftarrow~k-\mathrm{lowbit}(k)$;

end while

发布隐私数据$s_{t}$.

end for

•

Algorithm 4 对角阵系数求解算法getLambda

Require:时刻上限$T$, 下标$k$, 衰减因子$p$;

Output:对角阵系数$\lambda~_{k}$;

初始化$\lambda~_{k}$为1, 根据式(32)计算出所需的系数$\delta~_{1}~\sim~\delta~_{\mathrm{log}_{2}(T)+1}$;

$kt\leftarrow~k,~m\leftarrow~\mathrm{log}_{2}(T)+1,~\mathrm{div}\leftarrow~2^{m-1}$;

while $\mathrm{div}\neq~kt$ do

if $kt<\mathrm{div}$ then

$\lambda~_{k}\leftarrow~\lambda~_{k}\times~\delta~_{m}$;

else

$kt\leftarrow~kt-\mathrm{div}$;

end if

$\mathrm{div}~\leftarrow~\frac{\mathrm{div}}{2},~m\leftarrow~m-1$;

end while

$\lambda~_{k}\leftarrow~\frac{\lambda~_{k}\times~(1-\delta~_{m})}{p}$;

返回对角阵系数$\lambda~_{k}$.

•

Algorithm 5 指数衰减模式下的基于对角矩阵优化差分隐私流数据发布算法DMFDA

Require:时刻上限$T$, 衰减因子$p$;

Output:每一时刻的发布结果$s_{t}$;

for $t=1$ to $T$

更新实际统计量$\Phi~_{\mathrm{lowbit}(t)}\leftarrow~D_{t}+\sum_{j=0}^{\mathrm{lowbit}(t)-1}\Phi~_{j}$;

$\lambda~_{t}\leftarrow~\mathrm{getLambda}(T,t,p)$;

添加噪声$\overset{\sim~}{\Phi~_{\mathrm{lowbit}(t)}}\leftarrow~\lambda~_{t}\times~\Phi~_{\mathrm{lowbit}(t)}+\mathrm{Lap}(\frac{1}{\epsilon~})$;

$k\leftarrow~t,~s_{t}\leftarrow~0$;

while $k>0$ do

$s_{t}\leftarrow~s_{t}+\frac{(\overset{\sim~}{\Phi~_{\mathrm{lowbit}(k)}}\times~p^{i-k})}{\lambda~_{k}}$;

$k\leftarrow~k-\mathrm{lowbit}(k)$;

end while

发布隐私数据$s_{t}$.

end for

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