RSAVS_Compute_Loss_Value: Compute value of objective function

In fenguoerbian/RSAVS: Robust Subgroup Analysis and Variable Selection

Compute value of objective function

Description

This function computes objective function's value for the ADMM algorithm.

Usage

RSAVS_Compute_Loss_Value(
  y_vec,
  x_mat,
  l_type = "L1",
  l_param = NULL,
  p1_type = "S",
  p1_param = c(2, 3.7),
  p2_type = "S",
  p2_param = c(2, 3.7),
  const_r123,
  const_abc,
  beta_vec,
  mu_vec,
  z_vec,
  s_vec,
  w_vec,
  q1_vec,
  q2_vec,
  q3_vec
)

Arguments

y_vec	numerical vector of response. n = length(y_vec) is the number of observations.
x_mat	numerical matrix of covariates. Each row for one observation and p = ncol(x_mat) is the number of covariates.
l_type	character string, type of loss function. "L1": l-1 loss(absolute value loss) "L2": l-2 loss(squared error loss) "Huber": Huber loss. Its parameter is given in l_param. Default value is "L1".
l_param	numeric vector containing necessary parameters of the corresponding loss function. The default value is NULL.
p1_type, p2_type	a character indicating the penalty types for subgroup identification and variable selection. "S": SCAD "M": MCP "L": Lasso Default values for both parameters are "S".
p1_param, p2_param	numerical vectors providing necessary parameters for the corresponding penalties. For Lasso, lam = p_param[1] For SCAD and MCP, lam = p_param[1], gamma = p_param[2] Default values for both parameters are c(2, 3.7).
const_r123	a length-3 numerical vector, providing the scalars needed in the augmented lagrangian part of the ADMM algorithm
const_abc	a length-3 numeric vector, providing the scalars to adjust weight of regression function, penalty for subgroup identification and penalty for variable selection in the overall objective function. Defaults to c(1, 1, 1).
beta_vec, mu_vec, z_vec, s_vec, w_vec, q1_vec, q2_vec, q3_vec	variables needed in the objective function

Details

The augmented lagrangian objective function for the ADMM algorithm contains

• regression part, 1 / const_a * sum(rho(z_vec)), where rho is the loss function. Refer to loss_function for more details.

• subgroup analysis part, const_b * sum(P_1(s_vec)).

• variable selection part, const_c * sum(P_2(w_vec)).

• augmented part1: r_1 / 2 * norm(y_vec - mu_vec - x_mat * beta - z_vec) ^ 2 + inner_product(y_vec - mu_vec - x_mat * beta - z_vec, q1_vec)

• augmented part1: r_2 / 2 * norm(D_mat * mu_vec - s_vec) ^ 2 + inner_product(D_mat * mu_vec - s_vec, q2_vec)

• augmented part1: r_3 / 2 * norm(beta_vec - w_vec) ^ 2 + inner_product(beta_vec - w_vec, q3_vec)

Value

a list containing

• loss: overall loss value

• loss_part1: loss value from regression part

• loss_part2: loss value from subgroup analysis part

• loss_part3: loss value from variable selection part

• loss_aug1: loss value from augmented part1

• loss_aug2: loss value from augmented part2

• loss_aug3: loss value from augmented part3

• diff_z: difference between z_vec and y_vec - mu_vec - x_mat %*% beta_vec

• diff_s: difference between s_vec and d_mat %*% mu_vec

• diff_w: difference between w_vec and beta_vec

Note

for "L2" loss, the z_vec could be eliminated, but currently this is not implemented.

See Also

loss_function

fenguoerbian/RSAVS documentation built on Oct. 25, 2024, 3:16 p.m.