SLOPE: Sorted L-One Penalized Estimation

In jolars/SLOPE: Sorted L1 Penalized Estimation

Sorted L-One Penalized Estimation

Description

Fit a generalized linear model regularized with the sorted L1 norm, which applies a non-increasing regularization sequence to the coefficient vector (\beta) after having sorted it in decreasing order according to its absolute values.

Usage

SLOPE(
  x,
  y,
  family = c("gaussian", "binomial", "multinomial", "poisson"),
  intercept = TRUE,
  center = c("mean", "min", "none"),
  scale = c("sd", "l1", "l2", "max_abs", "none"),
  alpha = c("path", "estimate"),
  lambda = c("bh", "gaussian", "oscar", "lasso"),
  alpha_min_ratio = if (NROW(x) < NCOL(x)) 0.01 else 1e-04,
  path_length = 100,
  q = 0.1,
  theta1 = 1,
  theta2 = 0.5,
  tol_dev_change = 1e-05,
  tol_dev_ratio = 0.999,
  max_variables = NROW(x) + 1,
  solver = c("auto", "hybrid", "pgd", "fista", "admm"),
  max_passes = 1e+06,
  tol = 1e-04,
  threads = NULL,
  diagnostics = FALSE,
  patterns = FALSE,
  gamma = 1,
  tol_abs,
  tol_rel,
  tol_rel_gap,
  tol_infeas,
  tol_rel_coef_change,
  prox_method,
  screen,
  verbosity,
  screen_alg
)

Arguments

x	the design matrix, which can be either a dense matrix of the standard matrix class, or a sparse matrix inheriting from Matrix::sparseMatrix. Data frames will be converted to matrices internally.
y	the response, which for family = "gaussian" must be numeric; for family = "binomial" or family = "multinomial", it can be a factor.
family	model family (objective); see Families for details.
intercept	whether to fit an intercept
center	whether to center predictors or not by their mean. Defaults to TRUE if x is dense and FALSE otherwise.
scale	type of scaling to apply to predictors. "l1" scales predictors to have L1 norms of one. "l2" scales predictors to have L2 norms of one.#' "sd" scales predictors to have a population standard deviation one. "none" applies no scaling.
alpha	scale for regularization path: either a decreasing numeric vector (possibly of length 1) or a character vector; in the latter case, the choices are: "path", which computes a regularization sequence where the first value corresponds to the intercept-only (null) model and the last to the almost-saturated model, and "estimate", which estimates a single alpha using Algorithm 5 in Bogdan et al. (2015). When a value is manually entered for alpha, it will be scaled based on the type of standardization that is applied to x. For scale = "l2", alpha will be scaled by \sqrt n. For scale = "sd" or "none", alpha will be scaled by n, and for scale = "l1" no scaling is applied. Note, however, that the alpha that is returned in the resulting value is the unstandardized alpha.
lambda	either a character vector indicating the method used to construct the lambda path or a numeric non-decreasing vector with length equal to the number of coefficients in the model; see section Regularization sequences for details.
alpha_min_ratio	smallest value for lambda as a fraction of lambda_max; used in the selection of alpha when alpha = "path".
path_length	length of regularization path; note that the path returned may still be shorter due to the early termination criteria given by tol_dev_change, tol_dev_ratio, and max_variables.
q	parameter controlling the shape of the lambda sequence, with usage varying depending on the type of path used and has no effect is a custom lambda sequence is used. Must be greater than 1e-6 and smaller than 1.
theta1	parameter controlling the shape of the lambda sequence when lambda == "OSCAR". This parameter basically sets the intercept for the lambda sequence and is equivalent to \lambda_1 in the original OSCAR formulation.
theta2	parameter controlling the shape of the lambda sequence when lambda == "OSCAR". This parameter basically sets the slope for the lambda sequence and is equivalent to \lambda_2 in the original OSCAR formulation.
tol_dev_change	the regularization path is stopped if the fractional change in deviance falls below this value; note that this is automatically set to 0 if a alpha is manually entered
tol_dev_ratio	the regularization path is stopped if the deviance ratio 1 - \mathrm{deviance}/\mathrm{(null-deviance)} is above this threshold
max_variables	criterion for stopping the path in terms of the maximum number of unique, nonzero coefficients in absolute value in model. For the multinomial family, this value will be multiplied internally with the number of levels of the response minus one.
solver	type of solver use, either "auto", "hybrid", "pgd", or "fista"; "auto" means that the solver is automatically selected, which currently means that "hybrid" is used for all objectives except multinomial ones, in which case FISTA ("fista") is used.
max_passes	maximum number of passes (outer iterations) for solver
tol	stopping criterion for the solvers in terms of the relative duality gap
threads	number of threads to use in the solver; if NULL, half of the available (logical) threads will be used
diagnostics	whether to save diagnostics from the solver (timings and other values depending on type of solver)
patterns	whether to return the SLOPE pattern (cluster, ordering, and sign information) as a list of sparse matrices, one for each step on the path.
gamma	relaxation mixing parameter, between 0 and 1. Has no effect if set to 0. If larger than 0, the solver will mix the coefficients from the ordinary SLOPE solutions with the coefficients from the relaxed solutions (fitting OLS on the SLOPE pattern).
tol_abs	DEPRECATED
tol_rel	relative DEPRECATED
tol_rel_gap	DEPRECATED
tol_infeas	DEPRECATED
tol_rel_coef_change	DEPRECATED
prox_method	DEPRECATED
screen	DEPRECATED
verbosity	DEPRECATED
screen_alg	DEPRECATED

Details

SLOPE() solves the convex minimization problem

f(\beta) + \alpha \sum_{i=j}^p \lambda_j |\beta|_{(j)},

where f(\beta) is a smooth and convex function and the second part is the sorted L1-norm. In ordinary least-squares regression, f(\beta) is simply the squared norm of the least-squares residuals. See section Families for specifics regarding the various types of f(\beta) (model families) that are allowed in SLOPE().

By default, SLOPE() fits a path of models, each corresponding to a separate regularization sequence, starting from the null (intercept-only) model to an almost completely unregularized model. These regularization sequences are parameterized using \lambda and \alpha, with only \alpha varying along the path. The length of the path can be manually, but will terminate prematurely depending on arguments tol_dev_change, tol_dev_ratio, and max_variables. This means that unless these arguments are modified, the path is not guaranteed to be of length path_length.

Value

An object of class "SLOPE" with the following slots:

coefficients	a list of the coefficients from the model fit, not including the intercept. The coefficients are stored as sparse matrices.
nonzeros	a three-dimensional logical array indicating whether a coefficient was zero or not
lambda	the lambda vector that when multiplied by a value in alpha gives the penalty vector at that point along the regularization path
alpha	vector giving the (unstandardized) scaling of the lambda sequence
class_names	a character vector giving the names of the classes for binomial and multinomial families
passes	the number of passes the solver took at each step on the path
deviance_ratio	the deviance ratio (as a fraction of 1)
null_deviance	the deviance of the null (intercept-only) model
family	the name of the family used in the model fit
diagnostics	a data.frame of objective values for the primal and dual problems, as well as a measure of the infeasibility, time, and iteration; only available if diagnostics = TRUE in the call to SLOPE().
call	the call used for fitting the model

Families

Gaussian

The Gaussian model (Ordinary Least Squares) minimizes the following objective:

\frac{1}{2} \Vert y - X\beta\Vert_2^2

Binomial

The binomial model (logistic regression) has the following objective:

\sum_{i=1}^n \log\left(1+ \exp\left( - y_i \left(x_i^T\beta + \beta_0 \right) \right) \right)

with y \in \{-1, 1\}.

Poisson

In poisson regression, we use the following objective:

-\sum_{i=1}^n \left(y_i\left( x_i^T\beta + \beta_0\right) - \exp\left(x_i^T\beta + \beta_0 \right)\right)

Multinomial

In multinomial regression, we minimize the full-rank objective

-\sum_{i=1}^n\left( \sum_{k=1}^{m-1} y_{ik}(x_i^T\beta_k + \beta_{0,k}) - \log\sum_{k=1}^{m-1} \exp\big(x_i^T\beta_k + \beta_{0,k}\big) \right)

with y_{ik} being the element in a n by (m-1) matrix, where m is the number of classes in the response.

Regularization Sequences

There are multiple ways of specifying the lambda sequence in SLOPE(). It is, first of all, possible to select the sequence manually by using a non-increasing numeric vector, possibly of length one, as argument instead of a character. The greater the differences are between consecutive values along the sequence, the more clustering behavior will the model exhibit. Note, also, that the scale of the \lambda vector makes no difference if alpha = NULL, since alpha will be selected automatically to ensure that the model is completely sparse at the beginning and almost unregularized at the end. If, however, both alpha and lambda are manually specified, then the scales of both do matter, so make sure to choose them wisely.

Instead of choosing the sequence manually, one of the following automatically generated sequences may be chosen.

BH (Benjamini–Hochberg)

If lambda = "bh", the sequence used is that referred to as \lambda^{(\mathrm{BH})} by Bogdan et al, which sets \lambda according to

\lambda_i = \Phi^{-1}(1 - iq/(2p)),

for i=1,\dots,p, where \Phi^{-1} is the quantile function for the standard normal distribution and q is a parameter that can be set by the user in the call to SLOPE().

Gaussian

This penalty sequence is related to BH, such that

\lambda_i = \lambda^{(\mathrm{BH})}_i \sqrt{1 + w(i-1)\cdot \mathrm{cumsum}(\lambda^2)_i},

for i=1,\dots,p, where w(k) = 1/(n-k-1). We let \lambda_1 = \lambda^{(\mathrm{BH})}_1 and adjust the sequence to make sure that it's non-increasing. Note that if p is large relative to n, this option will result in a constant sequence, which is usually not what you would want.

OSCAR

This sequence comes from Bondell and Reich and is a linear non-increasing sequence, such that

\lambda_i = \theta_1 + (p - i)\theta_2.

for i = 1,\dots,p. We use the parametrization from Zhong and Kwok (2021) but use \theta_1 and \theta_2 instead of \lambda_1 and \lambda_2 to avoid confusion and abuse of notation.

lasso

SLOPE is exactly equivalent to the lasso when the sequence of regularization weights is constant, i.e.

\lambda_i = 1

for i = 1,\dots,p. Here, again, we stress that the fact that all \lambda are equal to one does not matter as long as alpha == NULL since we scale the vector automatically. Note that this option is only here for academic interest and to highlight the fact that SLOPE is a generalization of the lasso. There are more efficient packages, such as glmnet and biglasso, for fitting the lasso.

Solvers

There are currently three solvers available for SLOPE: Hybrid (Beck and Teboulle 2009), proximal gradient descent (PGD), and FISTA (Beck and Teboulle, 2009). The hybrid method is the preferred and generally fastest method and is therefore the default for the Gaussian and binomial families, but not currently available for multinomial and disabled for Poisson due to convergence issues.

References

Bogdan, M., van den Berg, E., Sabatti, C., Su, W., & Candès, E. J. (2015). SLOPE – adaptive variable selection via convex optimization. The Annals of Applied Statistics, 9(3), 1103–1140.

Larsson, J., Klopfenstein, Q., Massias, M., & Wallin, J. (2023). Coordinate descent for SLOPE. In F. Ruiz, J. Dy, & J.-W. van de Meent (Eds.), Proceedings of the 26th international conference on artificial intelligence and statistics (Vol. 206, pp. 4802–4821). PMLR. https://proceedings.mlr.press/v206/larsson23a.html

Bondell, H. D., & Reich, B. J. (2008). Simultaneous Regression Shrinkage, Variable Selection, and Supervised Clustering of Predictors with OSCAR. Biometrics, 64(1), 115–123. JSTOR.

Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2010). Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends® in Machine Learning, 3(1), 1–122.

Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202.

See Also

plot.SLOPE(), plotDiagnostics(), score(), predict.SLOPE(), trainSLOPE(), coef.SLOPE(), print.SLOPE(), print.SLOPE(), deviance.SLOPE(), sortedL1Prox()

Examples

# Gaussian response, default lambda sequence
fit <- SLOPE(bodyfat$x, bodyfat$y)

# Multinomial response, custom alpha and lambda
m <- length(unique(wine$y)) - 1
p <- ncol(wine$x)

alpha <- 0.005
lambda <- exp(seq(log(2), log(1.8), length.out = p * m))

fit <- SLOPE(
  wine$x,
  wine$y,
  family = "multinomial",
  lambda = lambda,
  alpha = alpha
)

jolars/SLOPE documentation built on June 15, 2025, 1:45 p.m.