# ADSIHT in multi-task learning framework An implementation of the sparse group selection in linear regression model via ADSIHT. ```r ADSIHT.ML( x_list, y_list, group_list, s0, kappa = 0.9, ic.type = c("dsic", "loss"), ic.scale = 3, ic.coef = 3, L = 5, weight, coef1 = 1, coef2 = 1, eta = 0.8, max_iter = 20, method = "ols", center = TRUE, scale = 1 ) ``` ## Arguments - `x_list`: The list of input matrix. - `y_list`: The list of response variable. - `group_list`: A vector indicating which group each variable belongs to For variables in the same group, they should be located in adjacent columns of `x` and their corresponding index in `group` should be the same. Denote the first group as `1`, the second `2`, etc. - `s0`: A vector that controls the degrees with group. Default is $d^((l-1)/(L-1))$ : $1 \leq l \leq L$, where d is the maximum group size. - `kappa`: A parameter that controls the rapid of the decrease of threshold. Default is 0.9. - `ic.type`: The type of criterion for choosing the support size. Available options are `"dsic"`, `"loss"`. Default is `"dsic"`. - `ic.scale`: A non-negative value used for multiplying the penalty term in information criterion. Default: `ic.scale = 3`. - `ic.coef`: A non-negative value used for multiplying the penalty term for choosing the optimal stopping time. Default: `ic.coef = 3`. - `L`: The length of the sequence of s0. Default: `L = 5`. - `weight`: The weight of the samples, with the default value set to 1 for each sample. - `coef1`: A positive value to control the sub-optimal stopping time. - `coef2`: A positive value to control the overall stopping time. A small value leads to larger search range. - `eta`: A parameter controls the step size in the gradient descent step. Default: `eta = 0.8`. - `max_iter`: A parameter that controls the maximum number of line search, ignored if `OLS` is employed. - `method`: Whether `ols` (default) or `linesearch` method should be employed. - `center`: A boolean value indicating whether centralization is required. Default: `center = TRUE`. - `scale`: A positive value to control the column-wise L2 norm of each observation matrix. Default: `scale=1`. ## Returns A `list` object comprising: - **beta**: A $p$-by-`length(s0)` matrix of coefficients, stored in column format. - **intercept**: A `length(s0)` vector of intercepts. - **lambda**: A `length(s0)` vector of threshold values - **A_out**: The selected variables given threshold value in `lambda`. - **ic**: The values of the specified criterion for each fitted model given threshold `lamdba`. ## Examples ```r set.seed(1) n <- 200 p <- 100 K <- 4 s <- 5 s0 <- 2 x_list <- lapply(1:K, function(x) matrix(rnorm(n*p, 0, 1), nrow = n)) vec <- rep(0, K * p) non_sparse_groups <- sample(1:p, size = s, replace = FALSE) for (group in non_sparse_groups) { group_indices <- seq(group, K * p, by = p) non_zero_indices <- sample(group_indices, size = s0, replace = FALSE) vec[non_zero_indices] <- rep(2, s0) } y_list <- lapply(1:K, function(i) return( y = x_list[[i]] %*% vec[((i-1)*p+1):(i*p)]+rnorm(n, 0, 0.5)) ) fit <- ADSIHT.ML(x_list, y_list) fit$A_out[, which.min(fit$ic)] ``` ## Author(s) Yanhang Zhang, Zhifan Li, Shixiang Liu, Jianxin Yin.

ADSIHT.ML function