Description Usage Arguments Value Note See Also Examples
Adjust a linear model penalized by a mixture of a (possibly weighted) linfinitynorm (bounding the magnitude of the parameters) and a (possibly structured) l2norm (ridgelike). The solution path is computed at a grid of values for the infinitypenalty, fixing the amount of l2 regularization. See details for the criterion optimized.
1 2 3 4 5  bounded.reg(x, y, lambda1 = NULL, lambda2 = 0.01, penscale = rep(1, p),
struct = NULL, intercept = TRUE, normalize = TRUE, naive = FALSE,
nlambda1 = ifelse(is.null(lambda1), 100, length(lambda1)),
min.ratio = ifelse(n <= p, 0.01, 0.001), max.feat = ifelse(lambda2 < 0.01,
min(n, p), min(4 * n, p)), control = list(), checkargs = TRUE)

x 
matrix of features, possibly sparsely encoded
(experimental). Do NOT include intercept. When normalized os

y 
response vector. 
lambda1 
sequence of decreasing linfinity
penalty levels. If 
lambda2 
real scalar; tunes the l2penalty in the bounded regression. Default is 0.01. Set to 0 to regularize only by the infinity norm (be careful regarding numerical stability in that case, particularly in the high dimensional setting). 
penscale 
vector with real positive values that weight the infinity norm of each feature. Default set all weights to 1. See details below. 
struct 
matrix structuring the coefficients. Must be at
least positive semidefinite (this is checked internally if the

intercept 
logical; indicates if an intercept should be
included in the model. Default is 
normalize 
logical; indicates if variables should be
normalized to have unit L2 norm before fitting. Default is

naive 
logical; Compute either 'naive' of 'classic' bounded
regression: mimicking the Elasticnet, the vector of parameters is
rescaled by a coefficient 
nlambda1 
integer that indicates the number of values to put
in the 
min.ratio 
minimal value of infinitypart of the penalty
that will be tried, as a fraction of the maximal 
max.feat 
integer; limits the number of features ever to
enter the model: in our implementation of the bounded regression,
it corresponds to the variables which have left the boundary along
the path. The algorithm stops if this number is exceeded and

control 
list of argument controlling low level options of the algorithm –use with care and at your own risk– :

checkargs 
logical; should arguments be checked to
(hopefully) avoid internal crashes? Default is

an object with class quadrupen
, see the
documentation page quadrupen
for details.
The optimized criterion is
penscale
argument. The l2
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
adopted by the 'bulletproof'
mode, that will send a warning
while switching the method to 'fista'
and keep on
optimizing on the remainder of the path. When bulletproof
is set to FALSE
, the algorithm stops at an early stage of
the path of solutions. Hence, users should be careful when
manipulating the resulting 'quadrupen'
object, as it will
not have the size expected regarding the dimension of the
lambda1
argument.
Singularity of the system can also be avoided with a larger
l2regularization, via lambda2
, or a
"nottoosmall" linfinity regularization, via
a larger 'min.ratio'
argument.
See also quadrupen
,
plot,quadrupenmethod
and crossval
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  ## 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:(251))) ## Toeplitz correlation for irrelevant variables
Sww < matrix(cor,10,10) ## bloc correlation between active variables
Sigma < 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

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