The Elastic Net with the Simulator

In simulator: An Engine for Running Simulations

library(knitr)
code <- file.path("elastic-net",
                  c("model_functions.R", 
                    "method_functions.R",
                    "eval_functions.R", 
                    "main.R"))
code_lastmodified <- max(file.info(code)$mtime)
sapply(code, read_chunk)

In this vignette, we perform a simulation with the elastic net to demonstrate the use of the simulator in the case where one is interested in a sequence of methods that are identical except for a parameter that varies. The elastic net is the solution $\hat\beta_{\lambda,\alpha}$ to the following convex optimization problem: $$ \min_{\beta\in\mathbb R^p}\frac1{2}\|y-X\beta\|_2^2+\lambda(1-\alpha)\|\beta\|^2_2+\lambda\alpha\|\beta\|_1. $$

Here, $\lambda\ge0$ controls the overall amount of regularization whereas $\alpha\in[0,1]$ controls the tradeoff between the lasso and ridge penalties. While sometimes one performs a two-dimensional cross-validation over $(\lambda,\alpha)$ pairs, in some simulations one might wish instead to view each fixed $\alpha$ as corresponding to a separate version of the elastic net (each solved along a grid of $\lambda$ values). Such a view is useful for understanding the effect $\alpha$.

Main simulation

Understanding the effect of the elastic net's $\alpha$ parameter

We begin with a simulation showing the best-case performance of the elastic net for several values of $\alpha$.

library(simulator)

<<models>>
<<methods>>
<<cv>>
<<metrics>>

<<init>>
<<main>>

<<init>>
sim_lastmodified <- file.info(sprintf("files/sim-%s.Rdata",
                              name_of_simulation))$mtime
if (is.na(sim_lastmodified) || code_lastmodified > sim_lastmodified) {
  <<main>>
  <<maincv>>
} else{
  sim <- load_simulation(name_of_simulation)
  sim_cv <- load_simulation("elastic-net-cv")
}

In the above code, we consider a sequence of models in which we vary the correlation rho among the features. For each model, we fit a sequence of elastic net methods (varying the tuning parameter $\alpha$). For each method, we compute the best-case mean-squared error. By best-case, we mean $\min_{\lambda\ge0}\frac1{p}\|\hat\beta_{\lambda,\alpha}-\beta\|_2^2$, which imagines we have an oracle-like ability to choose the best $\lambda$ for minimizing the MSE.

We provide below all the code for the problem-specific components. We use the R package glmnet to fit the elastic net. The most distinctive feature of this particular vignette is how the list of methods list_of_elastic_nets was created. This is shown in the Methods section.

plot_evals(sim, "nnz", "sqr_err")

The first plot shows the MSE versus sparsity level for each method (parameterized by $\lambda$). As expected, we see that when $\alpha=1$ (pure ridge regression), there is no sparsity. We see that the performance of the methods with $\alpha<1$ degrades as the correlation among features increases, especially when a lot of features are included in the fitted model.

It is informative to look at how the height of the minimum of each of the above curves varies with $\rho$.

plot_eval_by(sim, "best_sqr_err", varying = "rho", include_zero = TRUE)

We see that when the correlation between features is low, the methods with some $\ell_1$ penalty do better than ridge regression. However, as the features become increasingly correlated, a pure ridge penalty becomes better. Of course, none of the methods are doing as well in the high correlation regime (which is reminiscent of the "bet on sparsity principle").

A side note: the simulator automatically records the computing time of each method as an additional metric:

plot_eval(sim, "time", include_zero = TRUE)

Results for Cross-Validated Elastic Net

We might be reluctant to draw conclusions about the methods based on the oracle-like version that we used above (in which each method on each random draw gets to pick the best possible $\lambda$ value). We might therefore look at the performance of the methods using cross-validation to select $\lambda$.

<<maincv>>

Reassuringly, the relative performance of these methods is largely the same (though we see that all methods' MSEs are higher).

<<plotscv>>

Components

The most distinctive component in this vignette is in the Methods section. Rather than directly creating a Method object, we write a function that creates a Method object. This allows us to easily create a sequence of elastic net methods that differ only in their setting of the $\alpha$ parameter.

Models

<<models>>

Methods

<<methods>>

The function make_elastic_net takes a value of $\alpha$ and creates a Method object corresponding to the elastic net with that value of $\alpha$.

In the second set of simulations, we studied cross-validated versions of each elastic net method. To do this, we wrote list_of_elastic_nets + cv. This required writing the following MethodExtension object cv. The vignette on the lasso has more about writing method extensions.

<<cv>>

Metrics

<<metrics>>

Conclusion

To cite the simulator, please use

citation("simulator")

unlink("files", recursive = TRUE)

simulator documentation built on Feb. 16, 2023, 9:34 p.m.