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Having read this link, I know that it is important to fix the CV sample using the foldid argument when tuning the $\alpha$ parameter incv.glmnet.

Question: How does one define an optimal $\alpha$?

Question: And how is the optimal $\alpha$ selected ?

Taking into account of the foldid issue, I ensured I used the same CV sample throughout. (See sample code below). Then I tried fixing the value of $\alpha$ first and then found the optimal both $\lambda_{min}$ and the optimal $\lambda_{`1SE}$ and then looped through the remaining values of $\alpha$ and then selected the model with optimal $\lambda_{min}$ and $\lambda_{1SE}$? This approach seems to give different results for different CV samples. And hence , unstable whether I choose to $\lambda_{min}$ or $\lambda_{1SE}$.

l               = 25
foldid          = sample(1:k_fold, size=length(y_train),replace=T)
alpha_list      = seq(0,1,length.out=l)
en              = lapply(alpha_list, function(aa,cv_id){
                                      res <- cv.glmnet(foldid=cv_id,alpha = aa, 
                                                       x=x_train,y=y_train, 
                                                       intercept=T,
                                                       ,nfolds=10,
                                                       family=c("gaussian"),type.measure="mse")
                                      print(head(foldid))
                                      return(res$cvm[which(res$lambda ==res$lambda.min)])
                                      },cv_id=foldid)

res             = matrix(NA,nrow=l,ncol=2)
res[,1]         = alpha_list
res[,2]         = unlist(en)
plot(x=res[,1],y=res[,2],type="l",col="red",xlab="Alpha",ylab="CV ESPE")

enter image description here

I know that caret does it all but honestly, I couldn't find a proper documentation of how caret specifically tunes these 2 parameters for the glmnet and based on what criteria it does the tuning?

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