Algorithm 1.

Robust multi-source confonnal prediction

1:	Input: Training data $𝒟=\left\{{𝒪}_i=\left({\mathit{X}}_i,T_i,R_i,R_i{Y}_i\right),i=1,\dots ,n\right\}$ with number of sites $K>0$, and the target site is indexed by $T=0$; desired coverage probability $1-\alpha$; estimators of nuisance functions $m_k(\theta ,\mathit{X})$, ${\eta }_0(\mathit{X})$, and ${\omega }_{k,0}(\mathit{X})$ for $k=1,\dots ,K-1$; a tuning parameter $\lambda$ (in the optimization step); a testing point $\mathit{X}=\mathit{x}$ from the target site.
2:	Output: A valid prediction set ${\widehat{C}}_{\alpha }(\mathit{X})$.
3:	Split the training data $𝒟$ randomly into ${𝒟}_1$ and ${𝒟}_2$, where ${𝒟}_j=\{{𝒪}_i\in 𝒟,i\in {ℐ}_j\}$ for $j=1,2$ and ${ℐ}_1\cup {ℐ}_2=\{1,2,\dots ,n\}$.
4:	Fit nuisance functions ${\widehat{m}}_k$ and ${\widehat{\omega }}_{k,0}$ using SuperLearner on ${𝒟}_1$ and predict them on ${𝒟}_2$.
5:	For the target site $k=0$, find $\widehat{\theta }={\widehat{r}}_0$ that solves $0=\frac{1}{\left|{ℐ}_2\right|}{\sum }_{i\in {ℐ}_2} {\phi }_0({𝒪}_i;\widehat{\theta },{\widehat{m}}_0,{\widehat{\eta }}_0)$.
6:	For source sites $k\ge 1$, find $\widehat{\theta }={\widehat{r}}_k$ that solves $0=\frac{1}{\left|{ℐ}_2\right|}{\sum }_{i\in {ℐ}_2} {\phi }_k({𝒪}_i;\widehat{\theta },{\widehat{m}}_0,{\widehat{m}}_k,{\widehat{\omega }}_{k,0})$. Compute ${\widehat{\chi }}_k=\left|{\widehat{r}}_0-{\widehat{r}}_k\right|$.
7:	Solve for aggregation weights $\widehat{\mathit{w}}=\left({\widehat{w}}_0,{\widehat{w}}_1,\dots {\widehat{w}}_{K-1}\right)$ that minimize $Q\left(\mathit{w}\right)$ subject to $0\le w_k\le 1$ and ${\sum }_{k=0}^{K-1} w_k=1$.
8:	Compute $\widehat{\theta }={\widehat{r}}_{0,\text{fed}}={\sum }_{k=0}^{K-1} {\widehat{w}}_k{\widehat{r}}_k$.
9:	Return: The prediction set ${\widehat{C}}_{\alpha }(\mathit{X})=\left\{y:S(\mathit{x},y)\le {\widehat{r}}_{0,\text{fed}}\right\}$.