fit a GLM with lasso or elasticnet regularization

Fit a generalized linear model via penalized maximum likelihood. The regularization path is computed for the lasso or elasticnet penalty at a grid of values for the regularization parameter lambda. Can deal with all shapes of data, including very large sparse data matrices. Fits linear, logistic and multinomial, poisson, and Cox regression models.

models, regression
glmnet(x, y, family=c("gaussian","binomial","poisson","multinomial","cox","mgaussian"),
    weights, offset=NULL, alpha = 1, nlambda = 100,
    lambda.min.ratio = ifelse(nobs
input matrix, of dimension nobs x nvars; each row is an observation vector. Can be in sparse matrix format (inherit from class "sparseMatrix" as in package Matrix; not yet available for family="cox")
response variable. Quantitative for family="gaussian", or family="poisson" (non-negative counts). For family="binomial" should be either a factor with two levels, or a two-column matrix of counts or proportions
Response type (see above)
observation weights. Can be total counts if responses are proportion matrices. Default is 1 for each observation
A vector of length nobs that is included in the linear predictor (a nobs x nc matrix for the "multinomial" family). Useful for the "poisson" family (e.g. log of exposure time), or for refining a model by
The elasticnet mixing parameter, with $0\le\alpha\le 1$. The penalty is defined as $$(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.$$ alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.
The number of lambda values - default is 100.
Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size nobs relative to th
A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda over
Logical flag for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is standardize=TRUE. If variables are in the same units already, you might not wis
Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE)
Convergence threshold for coordinate descent. Each inner coordinate-descent loop continues until the maximum change in the objective after any coefficient update is less than thresh times the null deviance. Defaults value is 1E-7<
Limit the maximum number of variables in the model. Useful for very large nvars, if a partial path is desired.
Limit the maximum number of variables ever to be nonzero
Indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor (next item).
Separate penalty factors can be applied to each coefficient. This is a number that multiplies lambda to allow differential shrinkage. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in
Vector of lower limits for each coefficient; default -Inf. Each of these must be non-positive. Can be presented as a single value (which will then be replicated), else a vector of length nvars
Vector of upper limits for each coefficient; default Inf. See lower.limits
Maximum number of passes over the data for all lambda values; default is 10^5.
Two algorithm types are supported for (only) family="gaussian". The default when nvar<500< code=""> is type.gaussian="covariance", and saves all inner-products ever computed. This can be much faster than t
If "Newton" then the exact hessian is used (default), while "modified.Newton" uses an upper-bound on the hessian, and can be faster.
This is for the family="mgaussian" family, and allows the user to standardize the response variables
If "grouped" then a grouped lasso penalty is used on the multinomial coefficients for a variable. This ensures they are all in our out together. The default is "ungrouped"

The sequence of models implied by lambda is fit by coordinate descent. For family="gaussian" this is the lasso sequence if alpha=1, else it is the elasticnet sequence. For the other families, this is a lasso or elasticnet regularization path for fitting the generalized linear regression paths, by maximizing the appropriate penalized log-likelihood (partial likelihood for the "cox" model). Sometimes the sequence is truncated before nlambda values of lambda have been used, because of instabilities in the inverse link functions near a saturated fit. glmnet(...,family="binomial") fits a traditional logistic regression model for the log-odds. glmnet(...,family="multinomial") fits a symmetric multinomial model, where each class is represented by a linear model (on the log-scale). The penalties take care of redundancies. A two-class "multinomial" model will produce the same fit as the corresponding "binomial" model, except the pair of coefficient matrices will be equal in magnitude and opposite in sign, and half the "binomial" values. Note that the objective function for "gaussian" is $$1/2 RSS/nobs + \lambda*penalty,$$ and for the other models it is $$-loglik/nobs + \lambda*penalty.$$ Note also that for "gaussian", glmnet standardizes y to have unit variance before computing its lambda sequence (and then unstandardizes the resulting coefficients); if you wish to reproduce/compare results with other software, best to supply a standardized y. The coefficients for any predictor variables with zero variance are set to zero for all values of lambda. The latest two features in glmnet are the family="mgaussian" family and the type.multinomial="grouped" option for multinomial fitting. The former allows a multi-response gaussian model to be fit, using a "group -lasso" penalty on the coefficients for each variable. Tying the responses together like this is called "multi-task" learning in some domains. The grouped multinomial allows the same penalty for the family="multinomial" model, which is also multi-responsed. For both of these the penalty on the coefficient vector for variable j is $$(1-\alpha)/2||\beta_j||_2^2+\alpha||\beta_j||_2.$$ When alpha=1 this is a group-lasso penalty, and otherwise it mixes with quadratic just like elasticnet.


  • An object with S3 class "glmnet","*", where "*" is "elnet", "lognet", "multnet", "fishnet" (poisson), "coxnet" or "mrelnet" for the various types of models.
  • callthe call that produced this object
  • a0Intercept sequence of length length(lambda)
  • betaFor "elnet", "lognet", "fishnet" and "coxnet" models, a nvars x length(lambda) matrix of coefficients, stored in sparse column format ("CsparseMatrix"). For "multnet" and "mgaussian", a list of nc such matrices, one for each class.
  • lambdaThe actual sequence of lambda values used. When alpha=0, the largest lambda reported does not quite give the zero coefficients reported (lambda=inf would in principle). Instead, the largest lambda for alpha=0.001 is used, and the sequence of lambda values is derived from this.
  • dev.ratioThe fraction of (null) deviance explained (for "elnet", this is the R-square). The deviance calculations incorporate weights if present in the model. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). Hence dev.ratio=1-dev/nulldev.
  • nulldevNull deviance (per observation). This is defined to be 2*(loglike_sat -loglike(Null)); The NULL model refers to the intercept model, except for the Cox, where it is the 0 model.
  • dfThe number of nonzero coefficients for each value of lambda. For "multnet", this is the number of variables with a nonzero coefficient for any class.
  • dfmatFor "multnet" and "mrelnet" only. A matrix consisting of the number of nonzero coefficients per class
  • dimdimension of coefficient matrix (ices)
  • nobsnumber of observations
  • npassestotal passes over the data summed over all lambda values
  • offseta logical variable indicating whether an offset was included in the model
  • jerrerror flag, for warnings and errors (largely for internal debugging).


Friedman, J., Hastie, T. and Tibshirani, R. (2008) Regularization Paths for Generalized Linear Models via Coordinate Descent, Journal of Statistical Software, Vol. 33(1), 1-22 Feb 2010 Simon, N., Friedman, J., Hastie, T., Tibshirani, R. (2011) Regularization Paths for Cox's Proportional Hazards Model via Coordinate Descent, Journal of Statistical Software, Vol. 39(5) 1-13 Tibshirani, Robert., Bien, J., Friedman, J.,Hastie, T.,Simon, N.,Taylor, J. and Tibshirani, Ryan. (2012) Strong Rules for Discarding Predictors in Lasso-type Problems, JRSSB vol 74, Stanford Statistics Technical Report Glmnet Vignette

See Also

print, predict, coef and plot methods, and the cv.glmnet function.

  • glmnet
library(glmnet) # Gaussian x=matrix(rnorm(100*20),100,20) y=rnorm(100) fit1=glmnet(x,y) print(fit1) coef(fit1,s=0.01) # extract coefficients at a single value of lambda predict(fit1,newx=x[1:10,],s=c(0.01,0.005)) # make predictions #multivariate gaussian y=matrix(rnorm(100*3),100,3) fit1m=glmnet(x,y,family="mgaussian") plot(fit1m,type.coef="2norm") #binomial g2=sample(1:2,100,replace=TRUE) fit2=glmnet(x,g2,family="binomial") #multinomial g4=sample(1:4,100,replace=TRUE) fit3=glmnet(x,g4,family="multinomial") fit3a=glmnet(x,g4,family="multinomial",type.multinomial="grouped") #poisson N=500; p=20 nzc=5 x=matrix(rnorm(N*p),N,p) beta=rnorm(nzc) f = x[,seq(nzc)]%*%beta mu=exp(f) y=rpois(N,mu) fit=glmnet(x,y,family="poisson") plot(fit) pfit = predict(fit,x,s=0.001,type="response") plot(pfit,y) #Cox set.seed(10101) N=1000;p=30 nzc=p/3 x=matrix(rnorm(N*p),N,p) beta=rnorm(nzc) fx=x[,seq(nzc)]%*%beta/3 hx=exp(fx) ty=rexp(N,hx) tcens=rbinom(n=N,prob=.3,size=1)# censoring indicator y=cbind(time=ty,status=1-tcens) # y=Surv(ty,1-tcens) with library(survival) fit=glmnet(x,y,family="cox") plot(fit) # Sparse n=10000;p=200 nzc=trunc(p/10) x=matrix(rnorm(n*p),n,p) iz=sample(1:(n*p),size=n*p*.85,replace=FALSE) x[iz]=0 sx=Matrix(x,sparse=TRUE) inherits(sx,"sparseMatrix")#confirm that it is sparse beta=rnorm(nzc) fx=x[,seq(nzc)]%*%beta eps=rnorm(n) y=fx+eps px=exp(fx) px=px/(1+px) ly=rbinom(n=length(px),prob=px,size=1) system.time(fit1<-glmnet(sx,y)) system.time(fit2n<-glmnet(x,y))
Documentation reproduced from package glmnet, version 2.0-5, License: GPL-2

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