Incorrect intercept returned for logistic regression

[ff1201]

Using fit_sgs() with intercept=TRUE and type="logistic" yields incorrect estimates for the intercept, despite correct parameter estimates. Reproducible example, comparing it with the built-in glm() function:

set.seed(3)
n=200
p=10
X <- as.matrix(rnorm_multi(n=n,vars=p,mu=0,sd=1,r=0))
y <- 1/(1+exp(-(X %*%rnorm(p,mean=0,sd=sqrt(10)) + rnorm(n,mean=0,sd=4))))
y <- ifelse(y>0.5,1,0)

groups = 1:p
glm_fit <- glm(y ~ as.matrix(X),family="binomial")
sgs = fit_sgs(X=X,y=y, groups=groups, type="logistic", lambda=0, alpha=1, vFDR=0.1, gFDR=0.1,intercept=TRUE,standardise="none")
cbind(glm_fit$coefficients,sgs$beta)

expect_equivalent(coef(glm_fit)[-1],
                  as.matrix(sgs$beta)[-1],
                  tol = 1e-3
)

Compared to glmnet(), the intercept is closer but still slightly off (although the parameter estimates are again correct):

lasso <- glmnet(X, y, lambda = 0.1, standardize = FALSE,family="binomial",intercept=TRUE)
sgs_2 = fit_sgs(X=X,y=y, groups=groups, type="logistic", lambda=1, alpha=1, vFDR=0.1, gFDR=0.1,standardise="none",intercept=TRUE,w_weights = rep(0,p),v_weights = rep(0.1,p),tol=1e-5)

cbind(as.vector(sgs_2$beta),as.vector(rbind(lasso$a0, lasso$beta)))

Currently, the intercept is fitted alongside the other parameters but has no penalisation applied to it. In contrast to the Gaussian case, where the fitting process takes place without the intercept and it is calculated at the end.

Additionally, slower convergence is observed when using an intercept for logistic regression.