Summary the iterations during ADMM algorithm

This function is designed to summary and improve the resutls during the iteration of ADMM algorithm.

RSAVS_Summary_Iteration(
  y_vec,
  x_mat,
  beta_vec,
  mu_vec,
  s_vec,
  w_vec,
  loss_type,
  loss_param,
  phi
)

Arguments

y_vec

numerical response vector. n = length(y_vec) is the number of observations.

x_mat

numerical covariate matrix. p = ncol(x_mat) is the number of covariates.

beta_vec

covariate effect vector during the ADMM algorithm.

mu_vec

subgroup effect vector during the ADMM algorithm

s_vec

augmented vector for pair-wise difference of mu_vec in ADMM algorithm.

w_vec

augmented vector for beta_vec in ADMM algorithm.

loss_type

character string indicating type of loss function.

loss_param

numerical vector for necessary parameters in loss function.

phi

a parameter needed in mBIC. It controls how strong mBIC penalizes the complexity of the candidate model.

Value

a list, containing:

• bic: the bic value.

• mu_vec: the improved mu vector.

• group_num: number of subgroups in the improved mu_vec.

• active_num: number of active covariates in the beta_vec.

Details

This function has two purposes:

• Determine and improve beta_vec and mu_vec, if possible.

• Compute BIC.

Since for large scale data set, especially with big number of observations, it's impossible to first save all the variables during the ADMM algorithm over the lam1_length * lam2_length grid points of lambdas, then pick a best solution with mBIC. For s_vec alone, this means we have to save a matrix with n * (n - 1) / 2 rows and lam1_length * lam2_length columns, which is hard for a single computer. Instead, we summarise each iteration during the algorithm. Then there's no need for storing so many data.

In the ADMM algorithm, it will take many iterations to reach a sharp tolerance. But one can stop the algorithm early stage by setting a small max_iter. This is equivalent to setting a loose tolerance. Then the mu_vec and beta_vec will not be close to their augmented counterparts s_vec and w_vec. But these counterparts actually provides the sparsity information during the algorithm, therefore

• w_vec will provide the estimate of covariate effect while beta_vec is just a intermediate variable.

• mu_vec is also the intermediate variable. Improvement is needed like forming a reasonalbe subgroup structure. One possible solution is to utilize s_vec to improve mu_vec. Another is to apply some cluster methods on mu_vec. See RSAVS_Determine_Mu for more details.