R: Sorted L-One Penalized Estimation

SLOPE {SLOPE}

R Documentation

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 = !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

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 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

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
)

[Package SLOPE version 0.5.1 Index]