# Solve the TSLA optimization problem Find the solutions with a Smoothing Proximal Gradient (SPG) algorithm for a sequence of $\alpha$ and $\lambda$ values. ```r TSLA.fit( y, X_1 = NULL, X_2, treemat, family = c("ls", "logit"), penalty = c("CL2", "RFS-Sum"), gamma.init = NULL, weight = NULL, group.weight = NULL, feature.weight = NULL, control = list(), modstr = list() ) ``` ## Arguments - `y`: Response in matrix form, continuous for `family = "ls"` and binary (0/1) for `family = "logit"`. - `X_1`: Design matrix for unpenalized features (excluding intercept). Need to be in the matrix form. - `X_2`: Expanded design matrix for `penalty = "CL2"`; Original design matrix for `penalty = "RFS-Sum"`. Need to be in the matrix form. - `treemat`: Expanded tree structure in matrix form for `penalty = "CL2"`; Original structure for `penalty = "RFS-Sum"`. - `family`: Two options. Use "ls" for least square problems and "logit" for logistic regression problems. - `penalty`: Two options for group penalty on $\gamma$, "CL2" or "RFS-Sum". - `gamma.init`: Initial value for the optimization. Default is a zero vector. The length should equal to 1+`ncol(X_1)`+`ncol(A)`. See details of A in `get_tree_obj()`. - `weight`: A vector of length two and it is used for logistic regression only. The first element corresponds to weight of y=1 and the second element corresponds to weight of y=0. - `group.weight`: User-defined weights for group penalty. Need to be a vector and the length equals to the number of groups. - `feature.weight`: User-defined weights for each predictor after expansion. - `control`: A list of parameters controlling algorithm convergence. Default values: `tol = 1e-5`, convergence tolerance; `maxit = 10000`, maximum number of iterations; `mu = 1e-3`, smoothness parameter in SPG. - `modstr`: A list of parameters controlling tuning parameters. Default values: `lambda = NULL`. If lambda is not provided, the package will give a default lambda sequence; `lambda.min.ratio = 1e-04`, smallest value for lambda as a fraction of lambda.max (given by default when lambda is NULL); `nlambda = 50`, number of lambda values (equal spacing on log scale) used when lambda is NULL; `alpha = seq(0, 1, length.out = 10)`, sequence of alpha. Here, alpha is tuning parameter for generalized lasso penalty and 1-alpha is the tuning parameter for group lasso penalty. ## Returns A list of model fitting results. - **gammacoef**: Estimation for $\gamma$. - **groupnorm**: Weighted norms for each group. - **lambda.seq**: Sequence of $\lambda$ values. - **alpha.seq**: Tuning parameter sequence for the generalized lasso penalty. - **rmid**: Column index for all zero features. - **family**: Option of `family`. - **cov.name**: Names for unpenalized features. - **bin.name**: Names for binary feautres. - **tree.object**: Outputs from `get_tree_obj()`. ## Details We adopt the warm start technique to speed up the calculation. The warm start is applied with a fixed value of $\alpha$ and a descending sequence of $\lambda$. The objective function for "ls" is $$ 1/2 RSS+\lambda(\alpha P(\beta)+(1-\alpha) P(\gamma)), $$ subject to $\beta=A\gamma$. The objective function for "logit" is $$ -loglik + \lambda(\alpha P(\beta)+(1-\alpha) P(\gamma)), $$ subject to $\beta=A\gamma$. Note that, in this package, the input parameter "alpha" is the tuning parameter for the generalized lasso penalty. Details for "penalty" option: For `penalty = "CL2"`, see details for the "Child-l2" penalty in the main paper. For `penalty = "RFS-Sum"`, the theoretical optimal weights are used. Please check the details in paper "Rare feature selection in high dimensions". ## Examples ```r # Load the synthetic data data(RegressionExample) tree.org <- RegressionExample$tree.org # original tree structure x2.org <- RegressionExample$x.org # original design matrix y <- RegressionExample$y # response # Do the tree-guided expansion expand.data <- getetmat(tree.org, x2.org) x2 <- expand.data$x.expand # expanded design matrix tree.expand <- expand.data$tree.expand # expanded tree structure # specify some model parameters set.seed(100) control <- list(maxit = 100, mu = 1e-3, tol = 1e-5, verbose = FALSE) # fit model with a pair of lambda and alpha modstr <- list(lambda = 1, alpha = 0.1) x1 <- NULL fit1 <- TSLA.fit(y, x1, x2, tree.expand, family = 'ls', penalty = 'CL2', gamma.init = NULL, weight = NULL, group.weight = NULL, feature.weight = NULL, control, modstr) # get group norms from fit1 fit1$groupnorm ``` ## References Chen, J., Aseltine, R. H., Wang, F., & Chen, K. (2024). **Tree-Guided Rare Feature Selection and Logic Aggregation with Electronic Health Records Data. Journal of the American Statistical Association 119(547), 1765-1777**, tools:::Rd_expr_doi("10.1080/01621459.2024.2326621") . Chen, X., Q. Lin, S. Kim, J. G. Carbonell, and E. P. Xing (2012). **Smoothing proximal gradient method for general structured sparse regression. The Annals of Applied Statistics 6(2), 719–752**, tools:::Rd_expr_doi("10.1214/11-AOAS514") . Yan, X. and J. Bien (2021). **Rare feature selection in high dimensions. Journal of the American Statistical Association 116(534), 887–900**, tools:::Rd_expr_doi("10.1080/01621459.2020.1796677") .

TSLA.fit function