Support methods to calculate AIC and BIC values of biglasso fit

[tomwenseleers]

I was wondering if it would be possible to also support the calculation of AIC and BIC values of a biglasso fit, similar to the way that the ncvreg package provides this (returning a vector of AIC and BIC values for each lambda value used). This would enable selection of the best lambda based on AIC or BIC, which is faster than based on cross validation.

[privefl]

Because the loss (here, RSE) is already returned for the model, I think you can compute the AIC:

set.seed(1)

# Simulating some data
N <- 530
M <- 730
x <- matrix(rnorm(N * M, sd = 5), N)
y <- rowSums(x[, 1:5]) + 8 * rnorm(N)

library(biglasso)
X <- as.big.matrix(x, shared = FALSE)

# Training
ind.train <- sample(N, 400)
mod <- biglasso(X, y, row.idx = ind.train)

lambda <- mod$lambda
n <- length(ind.train)
loss.train <- mod$loss / n

# over-fitting on training, use AIC instead
plot(lambda, loss.train, pch = 20)
k <- ncol(X) - mod$rejections   ## add 1 for the intercept?
## AIC not correct, but works well
aic <- 2 * (k + loss.train)
plot(lambda, aic, pch = 20)

# Check on test set
ind.test <- setdiff(seq_len(N), ind.train)
preds <- predict(mod, X, row.idx = ind.test)
y.test <- y[ind.test]
loss.test <- apply(preds, 2, function(pred) {
  mean((pred - y.test)^2)
})
plot(lambda, loss.train, pch = 20, ylim = c(0, max(aic)))
points(lambda, aic, pch = 20, col = "blue")
points(lambda, loss.test, pch = 20, col = "red")

[tomwenseleers]

Many thanks - yes that will do it, perfect! Could still be nice though to formally define AIC and BIC methods... cheers & thanks again! Ha and would you know if implementing positivity and/or box constraints on the coefficients would be hard to implement in biglasso (see the other issue I posted a while ago)?

[privefl]

Note that I'm not sure at all of the AIC formula.

You can search for the "real" ones in many posts:

• https://stats.stackexchange.com/a/277205/135793
• https://stats.stackexchange.com/a/261422/135793
• https://stats.stackexchange.com/a/87357/135793
• https://www.researchgate.net/post/What_is_the_AIC_formula

None of those works well on my example.

It would be easy to make an AIC method for the class biglasso, but we need to agree on some formula (for the linear and logistic regressions).

Also note that I'm not sure AIC/BIC can be used for this kind of models: https://en.wikipedia.org/wiki/Bayesian_information_criterion#Limitations

[tomwenseleers]

For the gaussian case (together with glmnet) I was using the formulae used here:
https://github.com/gabrielrvsc/HDeconometrics/blob/master/R/ic.glmnet.R
As degrees of freedom it is using the effective degrees of freedom of the LASSO, which is the nr of nonzero coefficients for a given lambda (which is what the slot in glmnet $df is returning), see
https://projecteuclid.org/euclid.aos/1194461726. This would be correct for the LASSO but maybe not for elastic net or ridge (see below).

For the binomial case the recipe would be similar but would then use the log likelihood directly.

The tricky thing I am not sure of myself is how to calculate the effective degree of freedom for the elastic net in general (the ic.glmnet uses effective degrees of the LASSO for everything) - I think the correct formula is given here
https://stats.stackexchange.com/questions/327088/how-can-i-calculate-the-number-of-degrees-of-freedom-in-the-elastic-net-regulari
but I am not sure if this might have been implemented already somewhere or not...

For ridge regression an effective AIC and effective degrees of freedom can be calculated using the rms package, see
http://biostat.mc.vanderbilt.edu/wiki/pub/Main/FHHandouts/iscb98.pdf

If biglasso would return a slot $df with a correct estimate of the effective degrees of freedom for LASSO, ridge or elastic net, then the formulae for AIC or BIC will of course be easy enough... Main thing though would be to get the effective nr of degrees of freedom correct...

[privefl]

The slot $df in glmnet is returning the number of non-zero coefficients (you said non-negative).

And do you know the formula used by the ncvreg package?
If we try to use that, it doesn't work well:

fit2 <- ncvreg(x[ind.train, ], y[ind.train], family = "gaussian",
               penalty = "lasso", alpha = 1)
plot(AIC(fit2))

[tomwenseleers]

Ha yes sorry that's what I meant!
The calculation of the log likelihood in ncvreg is given in function

logLik.ncvreg <- function(object, REML=FALSE, ...) {
  n <- as.numeric(object$n)
  df <- predict(object, type="nvars") + 1
  if (object$family=="gaussian") {
    if (REML) rdf <- n-df
    else rdf <- n
    RSS <- object$loss
    l <- -n/2 * (log(2*pi) + log(RSS) - log(rdf)) - rdf/2
    df <- df + 1
  } else if (object$family=="binomial") {
    l <- -1*object$loss
  } else if (object$family=="poisson") {
    y <- object$y
    ind <- y != 0
    l <- -object$loss + sum(y[ind]*log(y[ind])) - sum(y) - sum(lfactorial(y))
  }
  
  val <- l
  attr(val,"df") <- df
  attr(val,"nobs") <- n
  class(val) <- "logLik"
  val
}

and the degrees of freedom are calculated as in glmnet, ie as the total nr of nonzero coefficients, which in ncvreg is in the predict.R file, line
if (type=="nvars") return(apply(beta!=0,2,sum))
I think using these degrees of freedom in the AIC/BIC calculation would only be correct for the LASSO though, not for elastic net in general or ridge...

[tomwenseleers]

For me BIC in ncvreg for variable selection works fine though:

fit2 <- ncvreg(x, y, family = "gaussian",
               penalty = "lasso", alpha = 1, nlambda=500)
# AIC doesn't work so well for variable selection (though it's not terrible either, just overfits a bit):
plot(AIC(fit2))
optlambda=fit2$lambda[which.min(AIC(fit2))]
plot(coef(fit2, lambda=optlambda),type="h",ylab="Estimated coefficients")
which(coef(fit2, lambda=optlambda)!=0) # not so good

# but BIC does:
plot(BIC(fit2))
optlambda=fit2$lambda[which.min(BIC(fit2))]
plot(coef(fit2, lambda=optlambda),type="h",ylab="Estimated coefficients")
which(coef(fit2, lambda=optlambda)!=0)
# (Intercept)          V1          V2          V3          V4          V5 
#          1           2           3           4           5           6 

To get rid of the downward bias you could probably do adaptive LASSO instead, or just re-estimate the selected covariates using regular least squares... And using BIC with penalty="SCAD" or "MCP" also seems to work and also results in less biased coefficients.

[privefl]

Can you format the code? It is difficult to read your answers right now.

[tomwenseleers]

Done!

[pbreheny]

This is already supported by biglasso:

data(colon)
X <- as.big.matrix(colon$X)
fit <- biglasso(X, colon$y)
head(AIC(fit))
#   0.3022   0.2932   0.2844    0.276   0.2677   0.2598 
# 88.53887 89.06932 87.65349 86.29061 84.97980 83.72012 
> head(BIC(fit))
#   0.3022   0.2932   0.2844    0.276   0.2677   0.2598 
# 92.79314 95.45072 94.03489 92.67201 91.36120 90.10152 

@YaohuiZeng, it seems to me you can close this issue.

[privefl]

The problem is that the path of AIC/BIC

• is not smooth at all
• doesn't always have a minimum

[pbreheny]

Not sure what you mean by "doesn't always have a minimum" (some value is guaranteed to be the smallest, right?). As for it not being smooth, that is certainly true, but I would argue that this is more of a fundamental limitation of AIC/BIC than a problem with the biglasso package.

In my experience, this requires looking at a plot of the AIC/BIC results and using your judgment (see below). I've tried to come up with automated ways of choosing the best AIC/BIC/OtherIC, but it's hard to come up with something foolproof.

data(colon)
X <- as.big.matrix(colon$X)
fit <- biglasso(X, colon$y)
ll <- log(fit$lambda)
plot(ll, BIC(fit), xlim=rev(range(ll)), pch=19, las=1)

Or for something smoother:

ss <- smooth.spline(ll, BIC(fit))
plot(ss, xlim=rev(range(ll)), pch=19, las=1)

[privefl]

What I meant is that for the examples I tested, AIC kept decreasing (so the minimum was the last value).

I'll try to test it again.

[pbreheny]

OK, that's kind of what I thought. But that's a fundamental flaw of AIC -- it completely breaks down in p > n situations, and its use as a model selection criteria for penalized regression is not justified.