This R package solves the sparse-group lasso problem using second-order information via the Semismooth Newton Augmented Lagrangian method from Zhang et al. (2020).
For a vector $x$ partitioned into $g$ non-overlapping groups, nonnegative weights ${w_i}$, and mixing parameter $\alpha \in [0, 1]$, define the penalty function
\Phi(x) = \alpha\lambda\lVert x \rVert_1 + (1-\alpha)\lambda\sum_{i=1}^g w_i \lVert x_{(i)} \rVert_2.
The sparse-group lasso problem has the form
\text{min}_x\ \frac{1}{2} \lVert Ax - b\rVert_2^2 + \Phi(x) \qquad \text{(P)}
while the dual problem is given by
\begin{matrix}\max_{y,z}\; -\langle b, y \rangle - \frac{1}{2}\lVert y\rVert_2^2 - \Phi^\ast(z) \\ \mathrm{s.t.}\; A^\top y + z = 0 \qquad \text{(D)} \end{matrix}
where $\Phi^\ast$ denotes the convex conjugate. Unlike first-order descent based methods which focus on $\mathrm{P}$, the SSNAL method uses second-order techniques to solve $\mathrm{D}$.
This package is designed for high-dimensional data with a (non-overlapping) group structure. The timing and accuracy of this method seems to be better than first-order descent methods in a large variety of problems.
The code was initially ported directly from the original Matlab implementation by Yangjing Zhang, Ning Zhang, Defeng Sun, and Kim-Chuan Toh, the authors of the paper this package implements.
install.packages("devtools")
# devtools::install_github("roobnloo/sglssnal")
install.packages("sglssnal", repos = c('https://roobnloo.r-universe.dev', 'https://cloud.r-project.org'))
set.seed(11234)
n <- 100
p <- 200
bstar <- c(rnorm(20), rep(0, p - 20))
A <- Matrix::rsparsematrix(n, p, density = 0.5, rand.x = rnorm)
ystar <- as.numeric(A %*% bstar + rnorm(n, sd = 0.1))
# group[j] is the group id of column j of A; here columns 1:20, 21:100,
# and 101:200 form three groups.
group <- rep(1:3, times = c(20, 80, 100))
result <- sglssnal::sglssnal(A, ystar, group, lambda = 2, alpha = 0.5)
print(result$info)
print(coef(result))