Compute value of objective function

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

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

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

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)

Note

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

Arguments

Value

Details

Note

See also