# Privacy-Preserving Byzantine-Resilient Swarm Learning for E-Healthcare
## 3 Preliminaries
### Swarm Learning
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & **Privacy** & **Poisoning** & **Central Server** & **Robust to** & **Lightweight** \\  & **Protection** & **Defense** & **or Blockchain** & **Off-Line** & **Computation** \\ \hline \multirow{2}{*}{\begin{tabular}{c} SecProbe \\ [20] \\ \end{tabular} } & \multirow{2}{*}{DP-based} & \multirow{2}{*}{\(\times\)} & Central server & \multirow{2}{*}{\(\surd\)} & \multirow{2}{*}{\(\surd\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} PPDL \\ [31] \\ \end{tabular} } & \multirow{2}{*}{HE-based} & \multirow{2}{*}{\(\times\)} & Central server & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} PPML \\ [27] \\ \end{tabular} } & \multirow{2}{*}{MPC-based} & \multirow{2}{*}{\(\times\)} & Central server & \multirow{2}{*}{\(\surd\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} Krum \\ [16] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \(\surd\) & Central server & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\surd\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} Trim-mean \\ [39] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \(\surd\) & Central server & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\surd\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} FoolsGold \\ [35] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \(\surd\) & Central server & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\surd\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} PEFL \\ [41] \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} HE-based \\ Paillier \\ \end{tabular} } & \multirow{2}{*}{\(\surd\)} & Central server & \multirow{2}{*}{\(\surd\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} BAFFLE \\ [44] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} & Blockchain & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} BlockFL \\ [46] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{Blockchain} & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} Swarm Learning \\ [46] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{Blockchain} & \multirow{2}{*}{\(\times\)} & \multirow{2}{*}{\(\times\)} \\  & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} BFLC \\ [45] \\ \end{tabular} } & \multirow{2}{*}{\(\times\)} & \(\surd\) & Blockchain & \multirow{2}{*}{\(\times\)} & \(\surd\) \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} PBSL \\ CKKS \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} HE-based \\ CKKS \\ \end{tabular} } & \multirow{2}{*}{\(\surd\)} & Blockchain & \(\surd\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A comparative summary between our scheme and previous schemes. \[F(\mathbf{x}_{i},\mathbf{w},y_{i})=\min_{\mathbf{w}}\mathbb{E}_{(\mathbf{x}_{i},y_{ i})\sim\bar{D}}L(f(\mathbf{x}_{i};\mathbf{w}),y_{i}) \tag{1}\]  
where \((\mathbf{x}_{i},y_{i})\) is the training sample, \(L(\cdot,\cdot)\) is the empirical error function, and \(\bar{D}\) is the training sample distribution. We adopt stochastic gradient descent (SGD) to solve this minimization problem.