2.1 The elastic-net model🔗ℹ

glmnet fits a linear model by minimizing a penalized least-squares objective. For a response y and predictors X (with n observations), it solves

min_{β₀,β} 1/(2n) ‖y − β₀ − Xβ‖² + λ [ α‖β‖₁ + (1−α)/2 ‖β‖₂² ]

over the intercept β₀ and coefficients β. Two knobs control the penalty:

• α (the mixing parameter), in [0,1]. α=0 is the pure ridge (L2) penalty; α=1 is the pure lasso (L1) penalty; values in between blend the two (the elastic net).

• λ (the penalty strength), ≥ 0. Larger λ shrinks coefficients more; λ=0 recovers the unpenalized fit (ordinary least squares).

So the four core models this package exposes are one routine called with different parameters:

2.2 Fitting a model🔗ℹ

elnet-fit is the one entry point behind every model; the four core cases have convenience wrappers. A fit takes the predictor matrix as a list of rows and the response as a list, and returns an elnet-result:

(require glmnet)

(define X '((1.0 2.0) (2.0 1.0) (3.0 4.0) (4.0 3.0) (5.0 6.0)))

(define y '(1.0 4.0 3.0 6.0 5.0))

(define fit (ols X y))

(elnet-result-intercept fit)

(elnet-result-coefficients fit)

ols is just elnet-fit with #:lambda 0.0. The penalized models pass #:alpha and a positive #:lambda, and each has a convenience wrapper: ridge (α = 0), lasso (α = 1), and elastic-net (an explicit #:alpha). See Ordinary least squares (λ = 0), Ridge regression (L2, α = 0), Lasso (L1, α = 1), and Elastic net (0 < α < 1) for worked examples, and API reference for the full keyword list.

2.3 Standardization and intercept🔗ℹ

By default glmnet standardizes each predictor column to unit variance before fitting and reports coefficients on the original scale, and it fits an unpenalized intercept. These match the conventions in the R package’s vignette and are the defaults the bindings expose.

2.4 Classification: the binomial family🔗ℹ

The models above fit a numeric response with the Gaussian elnet solver. For binary classification, the binomial family fits a two-class logistic model with glmnet’s lognet solver instead: the response y is a 0/1 class label, and the model predicts the log-odds of class 1,

log( P(y=1) / P(y=0) ) = β₀ + Xβ,

minimizing the penalized binomial deviance under the same elastic-net penalty. The α and λ knobs mean exactly what they do for the Gaussian models, so #:alpha 1.0 is the sparse (lasso) logistic and #:alpha 0.0 the ridge logistic.

(require glmnet)

(define X '((1.0 5.0) (2.0 4.0) (5.0 1.0) (4.0 2.0)))

(define y '(0 0 1 1))

(define fit (logistic-fit X y #:lambda 0.05))

(logistic-result-coefficients fit)

; class-1 probabilities, then hard 0/1 labels:

(logistic-predict-proba fit X)

(logistic-predict fit X)

logistic-fit returns a logistic-result whose logistic-result-dev-ratio is the fraction of null deviance explained — the logistic analogue of R². logistic-predict-proba maps new rows to class-1 probabilities through the logistic function, and logistic-predict thresholds those (at 0.5 by default) into 0/1 labels. See Binomial logistic regression (classification) for a worked classification example.

2.5 Multiclass classification: the multinomial family🔗ℹ

The binomial family handles two classes; the multinomial family handles K > 2. It is the same lognet solver with the class count set above 1, so the elastic-net penalty is unchanged. The response is an integer class label in {0, …, K−1}, and the fit returns K intercepts and K coefficient vectors (the symmetric parameterization):

(require glmnet)

(define X '((1.0 1.0) (5.0 1.0) (3.0 5.0) (2.0 2.0) (6.0 1.0) (4.0 6.0)))

(define y '(0 1 2 0 1 2))

(define fit (multinomial-fit X y #:lambda 0.05))

; K coefficient vectors, one per class:

(multinomial-result-coefficients fit)

(multinomial-predict-proba fit X)

(multinomial-predict fit X)

Class probabilities are the softmax over the K linear predictors ηk = a0k + x·βk: multinomial-predict-proba returns the per-row distributions (each summing to 1) and multinomial-predict returns the argmax class. As with the other families, #:alpha 1.0 is the sparse (lasso) fit and #:alpha 0.0 the ridge fit. See Multinomial classification (K classes).

2.6 Survival analysis: the Cox family🔗ℹ

The families above all predict from a feature vector to a label or a number. The Cox family instead models survival data: each observation has a follow-up time and a 0/1 event indicator (1 = the event happened, 0 = right-censored). It fits Cox’s proportional-hazards model, in which the hazard is

h(t | x) = h₀(t) · exp(x·β),

so there is no intercept (the baseline hazard h₀ is left unspecified) and a positive coefficient raises the hazard — shortening survival. The elastic-net penalty is unchanged.

(require glmnet)

(define X '((0.5 1.0) (2.0 2.0) (3.5 1.0) (4.0 2.0)))

(define times '(12.0 8.0 4.0 3.0))

; 0 = right-censored, 1 = event observed

(define statuses '(0 1 1 1))

(define fit (cox-fit X times statuses #:lambda 0.1))

(cox-result-coefficients fit)

(cox-relative-risk fit X)

cox-fit returns a cox-result with cox-result-coefficients (no intercept) and cox-result-dev-ratio (the fraction of null partial-likelihood deviance explained). cox-relative-risk gives exp(x·β) — the hazard relative to baseline, for ranking subjects — and cox-linear-predictor the raw x·β. As elsewhere, #:alpha 1.0 is the lasso and #:alpha 0.0 the ridge. See Cox proportional hazards (survival).

2.7 Count data: the Poisson family🔗ℹ

The Poisson family models count responses — non-negative numbers such as events per interval. It uses a log link, so the fitted mean is

μ = exp(β₀ + Xβ),

and a positive coefficient multiplies the expected count. Poisson has an intercept, and the elastic-net penalty is unchanged.

(require glmnet)

(define X '((1.0 2.0) (3.0 1.0) (5.0 2.0) (8.0 1.0)))

; non-negative counts

(define y '(1 2 4 11))

(define fit (poisson-fit X y #:lambda 0.1))

(poisson-result-coefficients fit)

(poisson-predict-mean fit X)

poisson-fit returns a poisson-result with an poisson-result-intercept, poisson-result-coefficients (on the log-mean scale), and poisson-result-dev-ratio. poisson-predict-mean applies the log link to give the fitted rate exp(β₀ + x·β) for new predictors. As elsewhere, #:alpha 1.0 is the lasso and #:alpha 0.0 the ridge. See Poisson regression (counts).

2.8 Multiple responses: the multi-response Gaussian family🔗ℹ

The Gaussian models above fit one numeric response. The multi-response Gaussian family (mgaussian-fit) fits several at once: the response is a matrix Y with one column per response. It uses a grouped lasso across the responses — a predictor is selected for all responses or none — so the fitted coefficient rows share support. Each response gets its own intercept.

(require glmnet)

(define X '((1.0 2.0) (3.0 1.0) (5.0 2.0) (6.0 1.0)))

; two responses, one column each

(define Y '((3.0 9.0) (7.0 7.0) (11.0 5.0) (13.0 4.0)))

(define fit (mgaussian-fit X Y #:lambda 0.1))

(mgaussian-result-coefficients fit)

(mgaussian-predict fit X)

mgaussian-fit returns a mgaussian-result with mgaussian-result-intercepts (one per response), mgaussian-result-coefficients (a vector of per-response coefficient vectors), and mgaussian-result-r-squared. mgaussian-predict returns the per-response predictions a0r + x·βr for each row. As elsewhere, #:alpha 1.0 is the (grouped) lasso and #:alpha 0.0 the ridge. See Multi-response Gaussian (grouped).

2.9 The precision contract🔗ℹ

The vendored Fortran declares its arrays as single-precision real, but the entire elastic-net ABI is treated as double precision: the native library is compiled with -fdefault-real-8 so the default real is 8 bytes. This is what R’s glmnet and Julia’s glmnet_jll do as well. The package asserts the flag took effect at load time (see Getting started).

2.10 Rebuilding the native library🔗ℹ

Using the package needs no Fortran toolchain. To rebuild libglmnetcompat from source (for development or a new platform):