# Privacy-Preserving Federated Singular Value Decomposition
## 6 Empirical Comparison
\begin{table}
\begin{tabular}{l l c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Device**} & \multicolumn{2}{c}{**Computation Time**} & \multicolumn{2}{c}{**Running**} \\  & & **Aggr.** & **SMPC** & **Rest** & **Time 1** \\ \hline \multirow{2}{*}{FedPower} & Node & 164 & - & 18,768 & \multirow{2}{*}{\(2.698\times 10^{6}\)} \\  & Server & 37 & - & 127,449 & \\ \hline \multirow{2}{*}{Utility Enhanced} & Node & 15,829 & \(7.182\times 10^{5}\) & 17,259 & \multirow{2}{*}{\(9.203\times 10^{7}\)} \\  & Server & 20,647 & \(7.247\times 10^{5}\) & 143,145 & \\ \hline \multirow{2}{*}{Privacy Enhanced} & Node & 15,742 & - & 17,291 & \multirow{2}{*}{\(4.477\times 10^{6}\)} \\  & Server & 20,581 & - & 128,903 & \\ \hline \hline \end{tabular}  
* Due to the large volume of memory required for matrix calculations, we had to access data by reading and writing local files, which caused the longer overall execution time.  
\end{table} 
Table 3: Runtime comparison of Algorithms 1–3 in milliseconds. Figure 2: Experiment to determining \(T\) (with Algorithm 1). The rise in computational demand comes with benefits. Concerning Algorithm 2, significant progress is achieved in the utility while it offers a similar privacy guarantee as FedPower. Concerning Algorithm 3, the privacy guarantee is more robust, as it provides a formal DDP protection (while FedPower fails to satisfy DP). Moreover, it obtains a higher utility, which could make this solution preferable despite its computational appeal. We compare the distance between the results of each algorithm and the real eigenvalues, as shown in Figure 3, and the utility is improved (i.e., the distances are lower) with both Algorithms 2 and 3. Our utility-enhanced solution significantly outperforms FedPower: after 92 rounds, the obtained error of our scheme is almost three times (2.74\(\times\)) smaller than that for FedPower. The final error of Algorithm 2 is \(dist(\mathbb{Z},\mathbb{Z}_{u}(\sigma))=6.72\), while this value for Algorithm 1 is \(dist(\mathbb{Z},\mathbb{Z}_{g}(\sigma,\sigma^{\prime}))=18.42\). Note that this level of accuracy (\(\sim\)18.5) was obtained using our method in the 32 nd round, i.e., almost three times (2.88\(\times\)) faster. Hence, the superior convergence speed can compensate for most of the computational increase caused by SA and SMPC. Let us shift our attention to our privacy-enhanced solution. In that case, we can see that besides more robust privacy protection, our solution offers better utility: Algorithms 1 and 3 obtains \(dist(\mathbb{Z},\mathbb{Z}_{g}(\sigma,\sigma^{\prime}))=18.42\) and \(dist(\mathbb{Z},\mathbb{Z}_{p}(\sigma))=13.94\) RMSE values, respectively, i.e., we acquired a 24% error reduction. Our method (with actual DP guarantees) achieved the same level of accuracy (\(\sim\)18.5) only after 65 rounds, which is a 29% convergence speed increase. We also compare our two proposed schemes, in a way, that the size of the accumulated noises is equal. Besides the nature of noise injection (many small vs. one large), the only factor that differentiates the results is the clipping bounds. As expected, the error is 1.65\(\times\) larger with clipping, i.e., \(dist(\mathbb{Z},\mathbb{Z}_{p}(\frac{\sigma}{10}))=11.11\) compared with \(dist(\mathbb{Z},\mathbb{Z}_{u}(\sigma))=6.72\). Concerning the convergence speed, the utility enhanced solution is 1.7\(\times\) faster, reaching similar accuracy (\(\sim\)11) in round 54. Note though that this result still vastly outperforms FedPower: the accuracy and the convergence speed are increased by 40% and 43%, respectively. Finally, we study the effect of different levels of privacy protection on the accuracy of each algorithm. As we noticed in Figure 3, after the 60 th round, the error ratios of the algorithms are reasonably stable, so for this experiment, we set \(T=60\). Since the clipping rate \(\hat{z}\) and the noise variance \(\sigma\) both contributed to the privacy parameter \(\epsilon\) (as seen in Theorem 2), we varied each independently.