---
title: "The Relaxed Lasso"
author: 
  - Trevor Hastie
  - Balasubramanian Narasimhan
  - Rob Tibshirani
date: "`r format(Sys.time(), '%B %d, %Y')`"
bibliography: assets/glmnet_refs.bib
link-citations: true
output:
  pdf_document:
    fig_caption: yes
    toc: yes
    toc_depth: 3
vignette: >
  %\VignetteIndexEntry{The Relaxed Lasso}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r include=FALSE}
# the code in this chunk enables us to truncate the print output for each
# chunk using the `out.lines` option
# save the built-in output hook
hook_output <- knitr::knit_hooks$get("output")

# set a new output hook to truncate text output
knitr::knit_hooks$set(output = function(x, options) {
  if (!is.null(n <- options$out.lines)) {
    x <- xfun::split_lines(x)
    if (length(x) > n) {
      # truncate the output
      x <- c(head(x, n), "....\n")
    }
    x <- paste(x, collapse = "\n")
  }
  hook_output(x, options)
})
```

## Introduction

In this vignette, we describe how the `glmnet` package can be used to fit the *relaxed lasso*.

The idea of the relaxed lasso is to take a `glmnet` fitted object, and then for each lambda, refit the variables in the active set without any penalization. This gives the "relaxed" fit. (We note that there have been other definitions of a relaxed fit, but this is the one we prefer.) This could of course be done for elastic net fits as well as lasso. However, if the number of variables gets too close to the sample size $N$, the relaxed path will be truncated. Furthermore, for binomial and other nonlinear generalized linear models (GLMs) convergence can be an issue with our current implementation if the number of variables is too large, and perversely if the relaxed fit is too strong.

Suppose the `glmnet` fitted linear predictor at $\lambda$ is $\hat\eta_\lambda(x)$ and the relaxed version is $\tilde\eta_\lambda(x)$. We also allow for shrinkage between the two:
$$\tilde \eta_{\lambda,\gamma}=(1-\gamma)\tilde
\eta_\lambda(x)+\gamma\hat\eta_\lambda(x).$$

$\gamma\in[0,1]$ is an additional tuning parameter which can be selected by cross-validation (CV). The debiasing will potentially improve prediction performance, and CV will typically select a model with a smaller number of variables. 

This procedure is very competitive with forward-stepwise and best-subset regression, and has a considerable speed advantage when the number of variables is large. This is especially true for best-subset, but even so for forward stepwise. The latter has to plod through the variables one-at-a-time, while `glmnet` will just plunge in
and find a good active set.

Further details on this form of relaxed fitting can be found in
@best_subset;  more information on glmnet and elastic-net model in
general is given in @glmnet,
@coxnet, @strongrules, and @block.

## Simple relaxed fitting

We demonstrate the most basic relaxed lasso fit as a first example. We load some pre-generated data and fit the relaxed lasso on it by calling `glmnet` with `relax = TRUE`:
```{r out.lines = 15}
library(glmnet)
data(QuickStartExample)
fit <- glmnet(x, y, relax = TRUE)
print(fit)
```
In addition to the three columns usually printed for `glmnet` objects
(`Df`, `%Dev` and `Lambda`), there is an extra column `%Dev R` (`R`
stands for "relaxed") which is the percent deviance explained by the
relaxed fit. This is always higher than its neighboring column, which
is the percent deviance exaplined for the penalized fit (on the
training data). Notice that when the `Df` stays the same, the `%Dev R`
does not change, since this typically means the active set is the
same. (The code is also smart enough to only fit such models once, so
in the truncated display shown, 9 lasso models are fit, but only 4
relaxed fits are computed).


The fit object is of class `"relaxed"`, which inherits from class `"glmnet"`. Hence, the usual `plot` method for `"glmnet"` objects can be used. The code below demonstrates some additional flexibility that `"relaxed"` objects have for plotting.
```{r}
par(mfrow = c(1, 3), mar=c(4,4,5.5,1))
plot(fit, main = "gamma = 1")
plot(fit, gamma = 0.5, main = "gamma = 0.5")
plot(fit, gamma = 0, main = "gamma = 0")
```

`gamma = 1` is the traditional `glmnet` fit (also `relax = FALSE`, the default), `gamma = 0` is the unpenalized fit, and `gamma = 0.5` is a mixture of the two (at the coefficient level, and hence also the linear predictors).

We can also select `gamma` using `cv.glmnet`, which by default uses the 5 values `c(0, 0.25, 0.5, 0.75, 1)`. This returns an object of class `"cv.relaxed"`.
```{r}
set.seed(1)
cfit <- cv.glmnet(x, y, relax = TRUE)
plot(cfit)
```

To remove the shading of the standard error bands, pass `se.bands = FALSE`:
```{r}
plot(cfit, se.bands = FALSE)
```

As with regular `"cv.glmnet"` objects, you can make predictions from a relaxed CV object. Just as the `s` option (for `lambda`) admits two special strings `"lambda.1se"` and `"lambda.min"` for special values of `lambda`, the `gamma` option admits two special strings `"gamma.1se"` and `"gamma.min"` for special values of `gamma`. For example, the code below makes predictions for `newx` at the `lambda` and `gamma` values that has the smallest CV error:
```{r}
predict(cfit, newx = x[1:5, ], s = "lambda.min", gamma = "gamma.min")
```

Printing class `"cv.relaxed"` objects gives some basic information on the cross-validation:
```{r}
print(cfit)
```

## More details on relaxed fitting

While we only demonstrate relaxed fits for the default Gaussian family, *any* of the families fit by `glmnet` can also be fit with the `relaxed` option.

Although `glmnet` has a `relax` option, you can also fit relaxed lasso models by post-processing a `glmnet` object with the `relax.glmnet` function.
```{r `relaxed`}
fit <- glmnet(x,y)
fitr <- relax.glmnet(fit, x = x, y = y)
```
This will rarely need to be done; one use case is if the original fit took a long time, and the user wants to avoid refitting it. Note that the arguments are named in the call in order for them to be passed correctly via the `...` argument in `relax.glmnet`.

As mentioned, a `"relaxed"` object inherits from class `"glmnet"`. Apart from the class modification, it has an additional component named `relaxed` which is itself a `glmnet` object, but with the relaxed coefficients. The default behavior of extractor functions like `predict` and `coef`, as well as `plot` will be to present results from the `glmnet` fit, unless a value of `gamma` is given different from the default value `gamma = 1` (see the plots above). The `print` method gives additional info on the relaxed fit.

Likewise, a `cv.relaxed` object inherits from class `cv.glmnet`. Here the `predict` method by default uses the optimal relaxed fit; if predictions from the CV-optimal *original* `glmnet` fit are desired, one can directly use `predict.cv.glmnet`. Similarly, use `print` to print information for cross-validation on the relaxed fit, and `print.cv.glmnet` for information on the cross-validation for the original `glmnet` fit.
```{r}
print(cfit)
print.cv.glmnet(cfit)
```

### Possible convergence issues for relaxed fits

`glmnet` itself is used to fit the relaxed fits by using a single value of zero for `lambda`. However, for nonlinear models such as `family = "binomial"`, `family = "multinomial"` and `family="poisson"`, there can be convergence issues. This is because `glmnet` does not do step size optimization, rather relying on
the pathwise fit to stay in the "quadratic" zone of the log-likelihood. We have an optional `path = TRUE` option for `relax.glmnet`, which actually fits a regurized path toward the `lambda = 0` solution, and thus avoids
the issue. The default is `path = FALSE` since this option adds to the computing time.

##  Application to forward stepwise regression

One use case for a relaxed fit is as a faster version of forward stepwise regression. With a large number `p` of variables, forward stepwise regression can be tedious. On the other hand, because the lasso solves a convex problem, it can plunge in and identify good candidate sets of variables over 100 values of `lambda`, even though `p` could be in the tens of thousands. In a case like this, one can have `cv.glmnet` do the selection of variables.
```{r}
fitr <- cv.glmnet(x, y, gamma = 0, relax = TRUE)
plot(fitr)
```

Notice that we only allow `gamma = 0`, so in this case we are not considering the blended fits.

## References
