Knowing what problem is solved here
The solved problem is proximal operator of latent group lasso \[ \begin{aligned} &\min_{\beta\in\mathbb R^p}\frac{1}{2}\left\|y - \beta \right\|_2^2 + \lambda *\Omega_{latent}^{\mathcal G}(\beta;w)\\ \text{where }&\Omega_{latent}^{\mathcal G}(\beta;w) = \min_{\substack{ supp(v^{(g)})\subset g, \text{ for }g\in\mathcal G\\\sum_{g\in\mathcal G}v^{(g)}=\beta}} w_g\left\|v^{(g)}\right\|_2 \end{aligned} \] for which the set of groups \(\mathcal G\) are constructed so that a hierarchical sparsity pattern can be achieved in the solution. More specifically, for parameter of interest \(\beta\), we define sets \(s_1, \dots, s_{n.nodes}\subset[1:p]\) such that \(s_i\cap s_j=\emptyset\) for \(i\neq j\) and \(s_i\neq\emptyset\) for all \(i\in[1:n.nodes]\). A directed acyclic graph (DAG) defined on \(\{s_1,\dots,s_{n.nodes}\}\) encodes the desired hierarchical structure in that when we set \(\beta_{s_i}\) to zero, we have \[\beta_{s_i}=0\quad\Rightarrow\quad\beta_{s_j}=0\quad\text{if }s_i\in ancestor(s_j)\] where \(ancestor(s_j)=\{s_k:s_k\text{ is in an ancestor node of }s_j,k\in[1:n.nodes]\}\). The details of \(\mathcal G\) can be found in Section 2.2 of Yan & Bien (2015).
In short, if you have a parameter vector \(\beta\) embedded in a DAG and you desire the support to be parameters embedded in the a top portion of the DAG, solving the problem here can accomplish the goal of parameter selection. In addition, the solver can be used as a core step in a collection of proximal gradient methods including ISTA and FISTA. Please pay attention to the two requirements on DAG specified above:
- No two nodes shall share common parameters (\(s_i\cap s_j=\emptyset\));
- There is no empty node in DAG (\(s_i\neq\emptyset\)).
Making sure the requirements satisfied are essential to the successful run of the package.
Inputs for functions
There are two main functions in the package: hsm and hsm.path. The difference between them is that path solves the problem for a single \(\lambda\) value, while hsm.path does that over a grid of \(\lambda\) values. Besides the inputs for \(\lambda\), these two functions share the same inputs. For the functions to work, you need to provide either map and var, or assign and w.assign. Generally, map and var are easier to define, and if provided they will be used to compute assign and w.assign. Meanwhile, the construction of assign and w.assign are related to the decomposition of DAG, which we will discuss in the next section. The following two examples show how to define map and var based on DAG and \(\beta\).
image
- Example 1: The DAG contains two nodes on the left panel of the above figure. There are no directed edges connecting the two nodes. Suppose \(\beta\in\mathbb R^{10}\) such that \(\beta_1\) through \(\beta_{10}\) are embedded in Node \(1\) and the rest are embedded in Node \(2\). The inputs we have for this DAG is as the following.
map <- matrix(c(1, NA, 2, NA), ncol = 2, byrow = TRUE)
map## [,1] [,2]
## [1,] 1 NA
## [2,] 2 NAvar <- list(1:10, 11:20)
var## [[1]]
## [1] 1 2 3 4 5 6 7 8 9 10
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## [1] 11 12 13 14 15 16 17 18 19 20- Example 2: The DAG is on the right panel of the above figure. Suppose \(\beta\in\mathbb R^8\) and \(\beta_i\) is embedded in Node \(i\) for \(i\in [1:8]\). The inputs we have for this DAG is as the following.
map <- matrix(c(1, 2, 2, 7, 3, 4, 4, 6, 6, 7, 6, 8, 3, 5, 5, 6),
ncol = 2, byrow = TRUE)
map ## [,1] [,2]
## [1,] 1 2
## [2,] 2 7
## [3,] 3 4
## [4,] 4 6
## [5,] 6 7
## [6,] 6 8
## [7,] 3 5
## [8,] 5 6var <- as.list(1:8)
var## [[1]]
## [1] 1
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## [1] 2
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## [1] 8Another argument requiring more attention is the weight vector for groups in \(\mathcal G\). Since groups in \(\mathcal G\) are one-one correspondence to the nodes in DAG, the elements in w should be of the same order as the indices for nodes. Under default value NULL for w, hsm and hsm.path will use \(w_l=\sqrt{\left|g_l\right|}\) as the weight applied to group \(g_l\). If you choose to pass in values for w yourself, you need to make sure \(w_l\) increases with the size of \(g_l\), i.e., if \(\left|g_{l_1}\right | > \left|g_{l_2}\right |\), \(w_{l_1}> w_{l_2}\); otherwise, the functions will fail.
