Sorted L-One Penalized Estimation

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 = !inherits(x, "sparseMatrix"),
  scale = c("l2", "l1", "sd", "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 = if (alpha[1] == "estimate") 1 else 20,
  q = 0.1 * min(1, NROW(x)/NCOL(x)),
  theta1 = 1,
  theta2 = 0.5,
  prox_method = c("stack", "pava"),
  screen = TRUE,
  screen_alg = c("strong", "previous"),
  tol_dev_change = 1e-05,
  tol_dev_ratio = 0.995,
  max_variables = NROW(x),
  solver = c("fista", "admm"),
  max_passes = 1e+06,
  tol_abs = 1e-05,
  tol_rel = 1e-04,
  tol_rel_gap = 1e-05,
  tol_infeas = 0.001,
  tol_rel_coef_change = 0.001,
  diagnostics = FALSE,
  verbosity = 0,
  sigma,
  n_sigma,
  lambda_min_ratio
)

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.

prox_method

method for calculating the proximal operator for the Sorted L1 Norm (the SLOPE penalty). Please see sortedL1Prox() for more information.

screen

whether to use predictor screening rules (rules that allow some predictors to be discarded prior to fitting), which improve speed greatly when the number of predictors is larger than the number of observations.

screen_alg

what type of screening algorithm to use.

"strong" uses the set from the strong screening rule and check against the full set
"previous" first fits with the previous active set, then checks against the strong set, and finally against the full set if there are no violations in the strong set

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 "fista" or "admm"; all families currently support FISTA but only family = "gaussian" supports ADMM.

max_passes

maximum number of passes (outer iterations) for solver

tol_abs

absolute tolerance criterion for ADMM solver

tol_rel

relative tolerance criterion for ADMM solver

tol_rel_gap

stopping criterion for the duality gap; used only with FISTA solver.

tol_infeas

stopping criterion for the level of infeasibility; used with FISTA solver and KKT checks in screening algorithm.

tol_rel_coef_change

relative tolerance criterion for change in coefficients between iterations, which is reached when the maximum absolute change in any coefficient divided by the maximum absolute coefficient size is less than this value.

diagnostics

whether to save diagnostics from the solver (timings and other values depending on type of solver)

verbosity

level of verbosity for displaying output from the program. Setting this to 1 displays basic information on the path level, 2 a little bit more information on the path level, and 3 displays information from the solver.

sigma

deprecated; please use alpha instead

n_sigma

deprecated; please use path_length instead

lambda_min_ratio

deprecated; please use alpha_min_ratio instead

Value

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

coefficients: a three-dimensional array of the coefficients from the model fit, including the intercept if it was fit. There is one row for each coefficient, one column for each target (dependent variable), and one slice for each penalty.
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
violations: the number of violations of the screening rule at each step on the path; only available if diagnostics = TRUE in the call to SLOPE().
active_sets: a list where each element indicates the indices of the coefficients that were active at that point in the regularization path
unique: the number of unique predictors (in absolute value)
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

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.

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 two solvers available for SLOPE: FISTA (Beck and Teboulle 2009) and ADMM (Boyd et al. 2008). FISTA is available for families but ADMM is currently only available for family = "gaussian".

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.

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.

Examples


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

# Poisson response, OSCAR-type lambda sequence
fit <- SLOPE(
  abalone$x,
  abalone$y,
  family = "poisson",
  lambda = "oscar",
  theta1 = 1,
  theta2 = 0.9
)

# 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
)