bounded_reg: Fit a linear model with infinity-norm plus ridge-like...

In quadrupen: Sparse and Group Sparse Linear Models

Fit a linear model with infinity-norm plus ridge-like regularization

Description

Adjust a linear model penalized by a (possibly weighted) \ell_\infty-norm (bounding the magnitude of the parameters) and a (possibly structured) \ell_2-norm (ridge-like). The solution path is computed at a grid of values for the infinity-penalty, fixing the amount of \ell_2 regularization. See details for the criterion optimized.

Usage

bounded_reg(
  x,
  y,
  lambda1 = NULL,
  lambda2 = 0.01,
  penscale = rep(1, ncol(x)),
  struct = Matrix::Diagonal(ncol(x), 1),
  intercept = TRUE,
  normalize = TRUE,
  nlambda1 = ifelse(is.null(lambda1), 100, length(lambda1)),
  minratio = ifelse(nrow(x) <= ncol(x), 0.01, 1e-04),
  maxfeat = ifelse(lambda2 < 0.01, min(nrow(x), ncol(x)), min(4 * nrow(x), ncol(x))),
  control = list()
)

bounded.reg(
  x,
  y,
  lambda1 = NULL,
  lambda2 = 0.01,
  penscale = rep(1, ncol(x)),
  struct = Matrix::Diagonal(ncol(x), 1),
  intercept = TRUE,
  normalize = TRUE,
  nlambda1 = ifelse(is.null(lambda1), 100, length(lambda1)),
  minratio = ifelse(nrow(x) <= ncol(x), 0.01, 1e-04),
  maxfeat = ifelse(lambda2 < 0.01, min(nrow(x), ncol(x)), min(4 * nrow(x), ncol(x))),
  control = list()
)

Arguments

x	matrix of features, possibly sparsely encoded (experimental). Do NOT include intercept. When normalized is TRUE, coefficients will then be rescaled to the original scale.
y	response vector.
lambda1	sequence of decreasing \ell_1-penalty levels. If NULL (the default), a vector is generated with nlambda1 entries, starting from a guessed level lambda1.max where only the intercept is included, then shrunken to minratio*lambda1.max.
lambda2	real scalar; tunes the \ell_2 penalty in the Elastic-net. Default is 0.01. Set to 0 to recover the Lasso.
penscale	vector with real positive values that weight the penalty of each feature. Default sets all weights to 1.
struct	matrix structuring the coefficients, possibly sparsely encoded. Must be at least positive semidefinite (this is checked internally). If NULL (the default), the identity matrix is used. See details below.
intercept	logical; indicates if an intercept should be included in the model. Default is TRUE.
normalize	logical; indicates if variables should be normalized to have unit L2 norm before fitting. Default is TRUE.
nlambda1	integer that indicates the number of values to put in the lambda1 vector. Ignored if lambda1 is provided.
minratio	minimal value of \ell_1-part of the penalty that will be tried, as a fraction of the maximal lambda1 value. A too small value might lead to instability at the end of the solution path corresponding to small lambda1 combined with \lambda_2=0. The default value tries to avoid this, adapting to the 'n<p' context. Ignored if lambda1 is provided.
maxfeat	integer; limits the number of features ever to enter the model; i.e., non-zero coefficients for the Elastic-net: the algorithm stops if this number is exceeded and lambda1 is cut at the corresponding level. Default is min(nrow(x),ncol(x)) for small lambda2 (<0.01) and min(4*nrow(x),ncol(x)) otherwise. Use with care, as it considerably changes the computation time.
control	list of argument controlling low level options of the algorithm –use with care and at your own risk– : verbose: integer; activate verbose mode –this one is not too risky!– set to 0 for no output; 1 for warnings only, and 2 for tracing the whole progression. Default is 1. Automatically set to 0 when the method is embedded within cross-validation or stability selection. timer: logical; use to record the timing of the algorithm. Default is FALSE. maxiter the maximal number of iteration used in the active set algorithm to solve the problem for a given value of lambda1 . Default is 50. maxadd the maximal number of variables (or groups, for group-sparse models) activated at once in the active set algorithm, chosen among the largest violations of the optimality conditions. Default is 10 (5 for group-sparse models). method a string for the underlying solver used. Either "quadra", "fista" or "pgd". Default is "quadra". factmat Boolean indicating if matrix factorization should be used to solve the sub-system. If TRUE (the default), a Cholesky decomposition is maintained along the path. If FALSE, the sub-system are solved with a conjugate gradient algorithm. threshold a threshold for convergence. The algorithm stops when the optimality conditions are fulfill up to this threshold. Default is 1e-6. monitor indicates if a monitoring of the convergence should be recorded, by computing a lower bound between the current solution and the optimum: when '0' (the default), no monitoring is provided; when '1', the bound derived in Grandvalet et al. is computed; when '>1', the Fenchel duality gap is computed along the algorithm.

Details

The optimized criterion is the following: β^hat _λ1,λ2 = argmin_β 1/2 RSS(β) + λ₁ | D β |_∞ + λ/2 ₂ β^T S β, where D is a diagonal matrix, whose diagonal terms are provided as a vector by the penscale argument. The \ell_2 structuring matrix S is provided via the struct argument, a positive semidefinite matrix (possibly of class Matrix).

Note that the quadratic algorithm for the bounded regression may become unstable along the path because of singularity of the underlying problem, e.g. when there are too much correlation or when the size of the problem is close to or smaller than the sample size. In such cases, it might be a good idea to switch to the proximal solver, slower yet more robust. This is the strategy automatically adopted in code, that will send a warning in verbose mode while switching the method to 'fista' and keep on optimizing on the remainder of the path.

Singularity of the system can also be avoided with a larger \ell_2-regularization, via lambda2, or a "not-too-small" \ell_\infty regularization, via a larger 'minratio' argument.

Value

an object with class QuadrupenFit.

an object with class BoundedRegressionFit, inheriting from QuadrupenFit.

Examples

## Simulating multivariate Gaussian with blockwise correlation
## and piecewise constant vector of parameters
beta <- rep(c(0,1,0,-1,0), c(25,10,25,10,25))
cor <- 0.75
Soo <- toeplitz(cor^(0:(25-1))) ## Toeplitz correlation for irrelevant variables
Sww  <- matrix(cor,10,10) ## bloc correlation between active variables
Sigma <- Matrix::bdiag(Soo,Sww,Soo,Sww,Soo)
diag(Sigma) <- 1
n <- 50
x <- as.matrix(matrix(rnorm(95*n),n,95) %*% chol(Sigma))
y <- 10 + x %*% beta + rnorm(n,0,10)

## Infinity norm without/with an additional l2 regularization term
## and with structuring prior
labels <- rep("irrelevant", length(beta))
labels[beta != 0] <- "relevant"
plot(bounded_reg(x,y,lambda2=0) , label=labels) ## a mess
plot(bounded_reg(x,y,lambda2=10), label=labels) ## good guys are at the boundaries
plot(bounded_reg(x,y,lambda2=10,struct=solve(Sigma)), label=labels) ## even better

quadrupen documentation built on Sept. 18, 2026, 1:06 a.m.