How does glmnet compute the maximal lambda value?

Question

The glmnet package uses a range of LASSO tuning parameters lambda scaled from the maximal lambda_max under which no predictors are selected. I want to find out how glmnet computes this lambda_max value. For example, in a trivial dataset:

set.seed(1)
library("glmnet")
x <- matrix(rnorm(100*20),100,20)
y <- rnorm(100)
fitGLM <- glmnet(x,y)
max(fitGLM$lambda)
# 0.1975946

The package vignette (http://www.jstatsoft.org/v33/i01/paper) describes in section 2.5 that it computes this value as follows:

sx <- as.matrix(scale(x))
sy <- as.vector(scale(y))
max(abs(colSums(sx*sy)))/100
# 0.1865232

Which clearly is close but not the same value. So, what causes this difference? And in a related question, how could I compute lambda_max for a logistic regression?

Answer 1

To get the same result you need to standardize the variables using a standard deviation with n instead of n-1 denominator.

mysd <- function(y) sqrt(sum((y-mean(y))^2)/length(y))
sx <- scale(x,scale=apply(x, 2, mysd))
sx <- as.matrix(sx, ncol=20, nrow=100)
sy <- as.vector(scale(y, scale=mysd(y)))
max(abs(colSums(sx*sy)))/100
## [1] 0.1758808
fitGLM <- glmnet(sx,sy)
max(fitGLM$lambda)
## [1] 0.1758808