gpCappedL1

Implements cappedL1 regularization for general purpose optimization problems. The penalty function is given by: $$p( x_j) = \lambda \min(| x_j|, \theta)$$ where \(\theta > 0\). The cappedL1 penalty is identical to the lasso for parameters which are below \(\theta\) and identical to a constant for parameters above \(\theta\). As adding a constant to the fitting function will not change its minimum, larger parameters can stay unregularized while smaller ones are set to zero.

Usage

gpCappedL1(
  par,
  fn,
  gr = NULL,
  ...,
  regularized,
  lambdas,
  thetas,
  method = "glmnet",
  control = lessSEM::controlGlmnet()
)

Arguments

par

labeled vector with starting values

fn

R function which takes the parameters AND their labels as input and returns the fit value (a single value)

gr

R function which takes the parameters AND their labels as input and returns the gradients of the objective function. If set to NULL, numDeriv will be used to approximate the gradients

...

additional arguments passed to fn and gr

regularized

vector with names of parameters which are to be regularized.

lambdas

numeric vector: values for the tuning parameter lambda

thetas

parameters whose absolute value is above this threshold will be penalized with a constant (theta)

method

which optimizer should be used? Currently implemented are ista and glmnet.

control

used to control the optimizer. This element is generated with the controlIsta and controlGlmnet functions. See ?controlIsta and ?controlGlmnet for more details.

Value

Object of class gpRegularized

Details

The interface is similar to that of optim. Users have to supply a vector with starting values (important: This vector must have labels) and a fitting function. This fitting functions must take a labeled vector with parameter values as first argument. The remaining arguments are passed with the ... argument. This is similar to optim.

The gradient function gr is optional. If set to NULL, the numDeriv package will be used to approximate the gradients. Supplying a gradient function can result in considerable speed improvements.

CappedL1 regularization:

• Zhang, T. (2010). Analysis of Multi-stage Convex Relaxation for Sparse Regularization. Journal of Machine Learning Research, 11, 1081–1107.

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Examples

# This example shows how to use the optimizers
# for other objective functions. We will use
# a linear regression as an example. Note that
# this is not a useful application of the optimizers
# as there are specialized packages for linear regression
# (e.g., glmnet)

# This example shows how to use the optimizers
# for other objective functions. We will use
# a linear regression as an example. Note that
# this is not a useful application of the optimizers
# as there are specialized packages for linear regression
# (e.g., glmnet)

library(lessSEM)
set.seed(123)

# first, we simulate data for our
# linear regression.
N <- 100 # number of persons
p <- 10 # number of predictors
X <- matrix(rnorm(N*p),  nrow = N, ncol = p) # design matrix
b <- c(rep(1,4),
       rep(0,6)) # true regression weights
y <- X%*%matrix(b,ncol = 1) + rnorm(N,0,.2)

# First, we must construct a fiting function
# which returns a single value. We will use
# the residual sum squared as fitting function.

# Let's start setting up the fitting function:
fittingFunction <- function(par, y, X, N){
  # par is the parameter vector
  # y is the observed dependent variable
  # X is the design matrix
  # N is the sample size
  pred <- X %*% matrix(par, ncol = 1) #be explicit here:
  # we need par to be a column vector
  sse <- sum((y - pred)^2)
  # we scale with .5/N to get the same results as glmnet
  return((.5/N)*sse)
}

# let's define the starting values:
b <- c(solve(t(X)%*%X)%*%t(X)%*%y) # we will use the lm estimates
names(b) <- paste0("b", 1:length(b))
# names of regularized parameters
regularized <- paste0("b",1:p)

# optimize
cL1 <- gpCappedL1(
  par = b,
  regularized = regularized,
  fn = fittingFunction,
  lambdas = seq(0,1,.1),
  thetas = c(0.001, .5, 1),
  X = X,
  y = y,
  N = N
)

# optional: plot requires plotly package
# plot(cL1)

# for comparison

fittingFunction <- function(par, y, X, N, lambda, theta){
  pred <- X %*% matrix(par, ncol = 1)
  sse <- sum((y - pred)^2)
  smoothAbs <- sqrt(par^2 + 1e-8)
  pen <- lambda * ifelse(smoothAbs < theta, smoothAbs, theta)
  return((.5/N)*sse + sum(pen))
}

round(
  optim(par = b,
      fn = fittingFunction,
      y = y,
      X = X,
      N = N,
      lambda =  cL1@fits$lambda[15],
      theta =  cL1@fits$theta[15],
      method = "BFGS")$par,
  4)
cL1@parameters[15,]