More adaptive or relaxed: Fitting sparser rule ensembles with relaxed and/or adaptive lasso

Marjolein Fokkema

Introduction

A beneficial property of the lasso penalty is that it shrinks coefficients to zero. Less beneficial is that in the process, the lasso tends to overshrink large coefficients. It has been argued that the lasso should “be considered as a variable screener rather than a model selector” (Su et al., 2017). There appears a trade-off between prediction and selection: The penalty parameter value optimal for variable selection will overshrink the large coefficients, making it suboptimal for prediction. The penalty parameter value optimal for prediction (“lambda.min”) selects too many variables. The relaxed lasso and adaptive lasso have been proposed to improve upon this problem.

When fitting prediction rule ensembles (PREs), the high false-positive selection rate of the lasso may be a nuisance, because often we want to obtain a sparse set of rules and linear terms. This vignette aims to show how the relaxed and adaptive lassos may deliver sparser ensembles, while retaining (relatively) high predictive accuracy.

An excellent discussion of consistency, predictive performance and selection accuracy of the lasso and its variations is provided in the master’s thesis of Kirkland (2014; pages 101-120).

Relaxed lasso

The relaxed lasso was originally proposed by Meinshausen (2007). Investigations of Su et al. (2107) provide insight on why the relaxed lasso is beneficial. Hastie, Tibshirani & Tibshirani (2017) propose a simplified version of the relaxed lasso, which is implemented in package glmnet and can be employed in package pre. Hastie et al. (2017) find that “best subset selection generally performing better in high signal-to-noise (SNR) ratio regimes, and the lasso better in low SNR regimes” and that “the relaxed lasso […] is the overall winner, performing just about as well as the lasso in low SNR scenarios, and as well as best subset selection in high SNR scenarios”. Function pre supports use of the relaxed lasso through passing of argument relax. A short introduction to the relaxed lasso is provided in glmnet vignette “The Relaxed lasso”, accessible in R by typing vignette("relax", "glmnet").

Adaptive lasso

The adaptive lasso has been proposed by Zou (2006). It applies positive weighting factors to the lasso penalty to control the bias through shrinking coefficients with weights inversely proportional to their size. It thus aims to shrink small coefficients more and large coefficients less. It requires an initial estimate of the coefficients, for which OLS (if \(N > p\)) or ridge (if \(N < p\)) estimation is usually employed, to obtain a vector of weights. These weights can then be used to scale the predictor matrix, or to scale the penalty; both approaches have the same effect.

Function pre allows for adaptive lasso estimation through specification of argument ad.alpha. It first uses ridge regression for computing penalty weights. In principle, any elastic net solution can be used, but use of ridge is recommended and recommended. Other solutions can be used by specifying the ad.alpha and ad.penalty arguments. Lasso regression can be used by specifying ad.alpha = 1 and OLS can be used by specifying ad.penalty = 0. Next, the inverse of the absolute values of the estimated coefficients are supplied as penalty factors to the cv.glmnet function. For the initial and final estimates, the same fold assignments are used in the cross validation.

Relaxed adaptive lasso

It should not come as a surprise that a combination has also been proposed: The relaxed adaptive lasso (Zhang et al., 2022). It can easily be employed through specifying both ad.alpha = 0 and relax = TRUE.

Example: Predicting Ozone levels with relaxed and/or adaptive lasso

library("pre")

Relaxed lasso

We fit a PRE to predict Ozone and employ the relaxed lasso by specifying relax = TRUE:

airq <- airquality[complete.cases(airquality), ]
set.seed(42)
airq.ens.rel <- pre(Ozone ~ ., data = airq, relax = TRUE)

If we specify relax = TRUE, the gamma argument (see ?cv.glmnet for documentation on arguments relax and gamma) will by default be set to a range of five values in the interval [0, 1]. This can be overruled by specifying different values for argument gamma in the call to function pre (but the default likely suffices in most applications).

We take a look at the regularization paths for the relaxed fits:

plot(airq.ens.rel$glmnet.fit)

We obtained one regularization path for each value of \(\gamma\). \(gamma\) is a mixing parameter, that determines the weight of the original lasso solution, relative to a solution containing only the selected variables, but with unpenalized coefficient estimates. The path for \(\gamma = 1\) is the default lasso path, which we would also have obtained without specifying relax = TRUE. Lower values of \(\gamma\) ‘unshrink’ the value of the non-zero coefficients of the lasso towards their unpenalized values. We see that for the \(\lambda\) value yielding the minimum MSE (indicated by the left-most vertical dotted line), the value of \(\gamma\) does not make a lot of difference for the MSE, but when \(\lambda\) values increase, higher values of \(\gamma\) tend to improve predictive performance. This is a common pattern for \(\lambda\) and \(\gamma\).

For model selection using the "lambda.min" criterion, by default the \(\lambda\) and \(\gamma\) combination yielding the lowest CV error is returned. For the "lambda.1se" criterion, the \(\lambda\) and \(\gamma\) combination yielding the sparsest solution within 1 standard error of the error criterion of the minimum is returned:

fit <- airq.ens.rel$glmnet.fit$relaxed
mat <- data.frame(lambda.1se = c(fit$lambda.1se, fit$gamma.1se, fit$nzero.1se),
                  lambda.min = c(fit$lambda.min, fit$gamma.min, fit$nzero.min),
                  row.names = c("lamda", "gamma", "# of non-zero terms"))
mat
##                     lambda.1se lambda.min
## lamda                 6.193185   3.382889
## gamma                 0.000000   0.000000
## # of non-zero terms   9.000000  12.000000

Thus, as the dotted vertical lines in the plots already suggest, with the default "lambda.1se" criterion, a final model with 9 terms will be selected, with coefficients obtained using a \(\lambda\) value of 6.193 and a \(\gamma\) value of 0. With the "lambda.min" criterion, we obtain a more complex fit; \(\gamma = 0\) still yields the lowest CV error. Note that use of "lambda.min" increases the likelihood of overfitting, because function pre uses the same data to extract the rules and fit the penalized regression, so in most cases the default "lambda.1se" criterion can be expected to provide a less complex, better generalizable, often more accurate fit.

The default of function pre is to use the "lambda.1se" criterion. When relax = TRUE has been specified in the call to function pre, the default of all functions and S3 methods applied to objects of class pre (print, plot, coef, predict, importance, explain, cvpre, singleplot, pairplot, interact) is to use the solution obtained with "lambda.1se" and the \(\gamma\) value yielding lowest CV error at that value of \(\lambda\). This can be overruled by specifying a different value of \(\lambda\) (penalty.par.val) and/or \(\gamma\) (gamma). Some examples:

summary(airq.ens.rel)
## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  6.193185 
##   gamma =  0
##   number of terms = 9
##   mean cv error (se) = 304.7364 (79.60512)
## 
##   cv error type : Mean-Squared Error
summary(airq.ens.rel, penalty = "lambda.min")
## Final ensemble with minimum cv error: 
## 
##   lambda =  3.382889 
##   gamma =  0
##   number of terms = 12
##   mean cv error (se) = 244.7256 (67.54855)
## 
##   cv error type : Mean-Squared Error
summary(airq.ens.rel, penalty = 8, gamma = 0)
## Final ensemble: 
## 
##   lambda =  7.814913 
##   gamma =  0
##   number of terms = 5
##   mean cv error (se) = 390.2582 (101.9163)
## 
##   cv error type : Mean-Squared Error
summary(airq.ens.rel, penalty = 8, gamma = 1)
## Final ensemble: 
## 
##   lambda =  7.814913 
##   gamma =  1
##   number of terms = 5
##   mean cv error (se) = 682.127 (146.0948)
## 
##   cv error type : Mean-Squared Error

Note how lowest CV error is indeed obtained with the "lambda.min" criterion, while the default "lambda.1se" yields a sparser model, with accuracy within 1 standard error of "lambda.min". If we want to go (much) sparser, we need to specify a lower value for the \(\lambda\) penalty, and a lower value of \(\gamma\) should likely be preferred, to retain good-enough predictive accuracy.

Some rules for specification of \(\lambda\) and \(\gamma\):

Also note that in the code chunk above we refer to the penalty.par.val argument by abbreviating it to penalty; this has the same effect as writing penalty.par.val in full.

Forward stepwise selection with the relaxed lasso

Using \(\gamma = 0\) amounts to a forward stepwise selection approach, with entry order of the variables (rules and linear terms) determined by the lasso. This approach can be useful if we want a rule ensemble with low complexity and high generalizability, and especially when we want to decide a-priori on the number of terms we want to retain. By specifying a high value of \(\lambda\), we can retain a small number of rules, while specifying \(\gamma = 0\) will provide unbiased (unpenalized) coefficients. This avoids the overshrinking of large coefficients. In terms of predictive accuracy, this approach may not perform best, but if low complexity (interpretability) is most important, this is a very useful approach, which does not reduce predictive accuracy too much.

To use forward stepwise regression with variable entry order determined by the lasso, we specify a \(\gamma\) value of 0, and specify the number of variables we want to retain through specification of \(\lambda\) (penalty.par.val). To find the value of \(\lambda\) corresponding to the number of terms one want to retain, check (results not shown for space considerations):

airq.ens.rel$glmnet.fit$glmnet.fit

Function prune_pre is helpful for selecting sparser ensembles. Say, we want to retain an ensemble with only five rules, then prune_pre will return the \(\lambda\) and \(\gamma\) values that yield an ensemble of specified size, with optimal cross-validated predictive accuracy.

Here, we request the optimal parameter values for a five-term ensemble:

opt_pars <- prune_pre(airq.ens.rel, nonzero = 5)
## The best ensemble with 5 non-zero terms is obtained with a lambda value of 6.797013 and a gamma value of 0.
## 
## Overview of performance of ensembles selected with the nearest lambda values:
##       lambda number_of_nonzero_terms optimal_gamma mean_cv_error
## s7  8.985251                       2             0      425.7366
## s8  8.576857                       2             0      414.1260
## s9  8.187026                       4             0      407.1132
## s10 7.814913                       5             0      390.2582
## s11 7.459713                       5             0      378.8175
## s12 7.120658                       5          0.25      384.3582
## s13 6.797013                       5             0      370.8112
## s14 6.488078                       6             0      347.7939
## s15 6.193185                       9             0      304.7364
## s16 5.911695                       9             0      294.8892

Note that the mean_cv_error may be slightly optimistic. Cross validation was performed on the same data that was used the generate the rules. A less optimistic estimate of generalization error can be obtained using function cvpre.

Adaptive lasso

Finally, we fit a PRE with adaptive lasso to predict Ozone levels:

set.seed(42)
airq.ens.ad <- pre(Ozone ~ ., data = airq, ad.alpha = 0)
summary(airq.ens.ad)
## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  16.74798
##   number of terms = 8
##   mean cv error (se) = 318.0806 (91.18799)
## 
##   cv error type : Mean-Squared Error

The adaptive lasso did not provide a sparser ensemble, while the mean CV error suggests better predictive accuracy than the standard, but not the relaxed, lasso. Adaptive lasso settings can further be adjusted by specification of argument ad.penalty.

We can also fit a rule ensemble using the relaxed adaptive lasso:

set.seed(42)
airq.ens.rel.ad <- pre(Ozone ~ ., data = airq, relax = TRUE, ad.alpha = 0)
print(airq.ens.rel.ad)
## 
## Final ensemble with cv error within 1se of minimum: 
## 
##   lambda =  40.53227 
##   gamma =  0.25
##   number of terms = 4
##   mean cv error (se) = 302.991 (91.09976)
## 
##   cv error type : Mean-Squared Error
## 
##          rule  coefficient                   description
##   (Intercept)     75.01177                             1
##       rule191    -21.82831       Wind > 5.7 & Temp <= 87
##       rule173    -18.12641       Wind > 5.7 & Temp <= 82
##       rule204     12.38470  Wind <= 10.3 & Solar.R > 148
##        rule51    -11.81249       Wind > 5.7 & Temp <= 84

The summary suggests that the relaxed adaptive lasso provides highest predictive accuracy compared to the standard, the relaxed and the adaptive lasso, when using the default "lambda.1se" criterion. Note however that the training data have been used to generate the rules, to estimate the weights for the penalty factors using ridge regression and to estimate the final lasso model. Thus, the printed CV error can provide an overly optimistic estimate of predictive accuracy. To obtain an honest estimate of predictive accuracy, it should be computed on a separate test dataset or using an additional layer of cross validation (e.g., using function cvpre or other approach).

Conclusion

Use of the relaxed lasso improves accuracy and sparsity of the final ensemble. Relaxed lasso can be used to obtain an ensemble of pre-specified sparsity, that still provides good predictive performance. Use of the adaptive lasso penalties may further improve predictive accuracy.

References

Hastie, T., Tibshirani, R., & Tibshirani, R. J. (2017). Extended comparisons of best subset selection, forward stepwise selection, and the lasso. arXiv:1707.08692, https://arxiv.org/abs/1707.08692.

Kirklan, L.-A. (2014). LASSO - Simultaneous shrinkage and selection via the L1 norm. Master’s thesis, University of Pretoria. www_dot_researchgate_dot_net/publication/325929497_LASSO_-_Simultaneous_shrinkage_and_selection_via_the_L1_norm (replace dot with a period and there you go, this is just to avoid issues on CRAN submission)

Meinshausen, N. (2007). Relaxed lasso. Computational Statistics & Data Analysis, 52(1), 374-393.

Su, W., Bogdan, M., & Candes, E. (2017). False discoveries occur early on the lasso path. The Annals of Statistics, 45(5), 2133-2150.

Zhang, R., Zhao, T., Lu, Y., & Xu, X. (2022). Relaxed Adaptive Lasso and its asymptotic results. Symmetry, 14(7), 1422.

Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418-1429.

Session info

In case you obtained different results, the results above were obtained using the following:

## R version 4.3.1 (2023-06-16 ucrt)
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## [13] mitml_0.4-5        evaluate_0.23      grid_4.3.1         iterators_1.0.14  
## [17] mvtnorm_1.2-3      fastmap_1.1.1      foreach_1.5.2      jomo_2.7-6        
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## [57] pkgconfig_2.0.3    pillar_1.9.0       bslib_0.5.1        TeachingDemos_2.12
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