# Variance-Covariance Matrix for $l_1$ regularized binomial logistic regression

Given design matrix $X \in \mathbb{R}^{n \times p}$ and response vector $y \in \{ 0,1 \}^n$, I want to find the variance-covariance matrix of the coefficients $\hat{\beta}$ from an $l_1$-regularized logistic regression with regularization parameter $\lambda > 0$.

If I understand correctly, there is no closed-form solution for this as the penalized log-likelihood function $l(\beta \vert X, y, \lambda)$ is not differentiable. Approximations have been proposed in Tibshirani (1996), and Li and Fan (2001), but these only apply to the non-zero coefficients in $\hat{\beta}$. Bootstrap and Bayesian lasso (2010) are other methods to compute standard errors, but I think they would be too computationally intensive for my purposes. There is a question here with an answer that only applies to $l_2$-regularized logistic regression.

Are there any other fast and accurate alternatives for variance-covariance matrix computation of the coefficients in frequentist lasso logistic regression?

This is a very hard question to give an good answer to. Even in the non-penalized case, the covariance estimate for the parameters is based on a normal approximation. When you start penalizing, you also enter the realm of "post-selection inference" which is an active area of research. The work on post-selection inference for GLMs (including logistic regression) is in its infancy, but see [1] for a recent reference on it. I believe the method described in this paper is implemented in the selectiveInference R package [2].