Stationarity (KKT) residual of an L1-penalized GLM fit

Dependency-free correctness certificate for a nodewise lasso, analogous to glasso_kkt() for the graphical lasso. With standardized predictors X and fitted mean mu (identity link for gaussian, logistic for binomial), the subgradient conditions are \(n^{-1} X_j^\top (y - \mu) = \lambda\, \mathrm{sign}(\beta_j)\) for active coordinates and \(|n^{-1} X_j^\top (y - \mu)| \le \lambda\) otherwise. Near-zero certifies the penalized-likelihood optimum.

Usage

glm_lasso_kkt(
  X,
  y,
  b0,
  beta,
  lambda,
  family = "gaussian",
  weights = NULL,
  active_tol = 1e-08
)

Arguments

X

Standardized predictor matrix (mean 0, unit variance columns).

y

Response.

b0

Fitted intercept.

beta

Fitted (standardized) coefficients.

lambda

Penalty.

family

"gaussian" or "binomial".

weights

Optional observation weights (NULL = unweighted).

active_tol

Magnitude above which a coefficient is "active".

Value

Maximum absolute stationarity violation (scalar). Near-zero certifies the fit is at the penalized-likelihood optimum.

Examples

set.seed(1)
x <- scale(matrix(stats::rnorm(200 * 3), 200, 3))
y <- as.numeric(x %*% c(0.5, 0, -0.3) + stats::rnorm(200))
fit <- stats::lm.fit(cbind(1, x), y)
glm_lasso_kkt(x, y, fit$coefficients[1], fit$coefficients[-1], lambda = 0)
#> [1] 1.053324e-16