Restricted Bridge Estimation

Implements bridge regression with linear coefficient restrictions using the local quadratic approximation described by Yuzbasi, Arashi and Akdeniz (2021) . Provides coefficient paths, cross-validation, prediction and coefficient extraction for restricted and unrestricted linear models.

rbridge fits bridge-penalized linear models with optional linear coefficient restrictions. The R interface provides coefficient paths, cross-validation, predictions and coefficient extraction through compiled RcppArmadillo solvers.

The methodology is described in Yuzbasi, Arashi and Akdeniz (2021), Penalized regression via the restricted bridge estimator. The banner is decorative artwork.

Read the online documentation, getting started guide and function reference.

Install from source

The package is currently archived on CRAN. The 1.0.3 maintenance update is available from GitHub; CRAN reinstatement is being prepared. It requires R, a working C++ toolchain, and the Rcpp, RcppArmadillo and Matrix packages.

Install the GitHub version from R with the optional remotes package:

remotes::install_github("byuzbasi/rbridge")

Run this command in the package source directory:

R CMD INSTALL .

Alternatively, install the built source archive from R:

install.packages("rbridge_1.0.3.tar.gz", repos = NULL, type = "source")

Penalty geometry and estimator

The bridge exponent controls the shape of a coefficient region. For two coefficient coordinates and a unit penalty budget, the region contains points whose absolute coordinates raised to q sum to at most one. This gives a nonconvex region for q < 1, the L1 diamond for q = 1 and the L2 disk for q = 2. The following figure is generated from the mathematical boundary, using the exponents in Figure 1 of the cited article.

The fitting functions support 0 < q <= 2; the last two panels show geometry only. Restricted bridge combines the bridge penalty with linear restrictions on standardized slopes. At each local quadratic approximation (LQA) step, it corrects the base update by an adjustment weighted by the LQA system matrix to satisfy the supplied restrictions. Iteration uses the historical thresholding and stopping rules. The getting started guide defines the penalized target, LQA weights and restriction correction, with the matrix assumptions and complete plotting code.

A small working example

This is a demonstration with synthetic data, not a reproduction of the paper's reported numerical results. It compares penalty exponents q = 0.5, 1, 2 using seed 2019, the same four folds and 100 logarithmically spaced penalties from 0.001 to 100 for every exponent. The 80 observations have 30 correlated predictors, three nonzero population slopes and Gaussian noise with standard deviation 0.75. This example was selected to illustrate an interior CV minimum; the pre-specified selection rule is documented in the getting started guide.

library(rbridge)
set.seed(2019)
n <- 80
p <- 30
# Adjacent predictor correlation is 0.5; covariance decays with index distance.
Sigma <- outer(seq_len(p), seq_len(p), function(i, j) 0.5^abs(i - j))
X <- matrix(rnorm(n * p), n, p) %*% chol(Sigma)
colnames(X) <- paste0("x", seq_len(p))
beta <- c(1, 0.5, -0.2, rep(0, p - 3))
sigma <- 0.75
y <- as.numeric(1 + X %*% beta + sigma * rnorm(n))

# Restrict the fourth standardized predictor coefficient to zero.
R <- matrix(0, nrow = 1, ncol = p)
R[1, 4] <- 1
r <- matrix(0, nrow = 1)
lambda_cv <- exp(seq(log(100), log(0.001), length.out = 100))
q_values <- c(0.5, 1, 2)
cv_fits <- lapply(q_values, function(q) {
  cv.rbridge(X, y, q = q, R = R, r = r, lambda = lambda_cv,
             nfolds = 4, num_threads = 1)
})
names(cv_fits) <- paste("q =", q_values)
fit <- cv_fits[["q = 2"]]
# Display the intercept and first five slopes; coef() returns the full matrix.
coef(fit, s = "lambda.min")[1:6, , drop = FALSE]
predict(fit, newx = X[1:3, , drop = FALSE])

The left panel marks each curve's lambda.min; the right panel compares the minimum CV errors on a closer error scale. These are minima over the fitted grid for this demonstration. They do not establish a preferred exponent for other data. The getting started guide contains the comparison table and the complete plotting code. The existing plot(fit) method remains available for an individual CV object.

Functions

The supported penalty exponent is positive and at most two. The exponent controls the bridge penalty; one and two give the L1 and L2 penalty forms. The solver uses local quadratic approximation and the historical elimination and convergence thresholds.

Coefficient scales and selectors

Restrictions apply to standardized predictor coefficients. Predictors are centered and divided by their root-mean-square centered deviations before fitting. Returned coefficients include a reconstructed intercept and are on the original predictor scale. Consequently, applying the same restriction matrix directly to returned slopes generally tests a different restriction. The intercept is not part of the restriction matrix.

For CV objects, prediction defaults to lambda.min; coefficient extraction defaults to lambda.1se. For path objects, omitting s returns the entire path. A numeric s selects exact values already present in the fitted lambda grid; there is no interpolation. A path object's s = "lambda.min" selects the smallest fitted penalty, while lambda.1se requires a CV object. Nonzero indices are reported over the whole fitted path, preserving the historical behavior.

The historical contiguous fold allocation and CV-error aggregation are retained. Use num_threads = 1 for a sequential demonstration; multithreading depends on the build's OpenMP support. Inspect it with rbridge_info(). Supported builds use at most one worker per fold; unsupported builds run sequentially. Set BLAS threads to one before starting R when requesting parallel CV.

The maintenance update caches fixed and active-set cross-products, reuses CV data slices and solves the restricted coefficient and constraint systems together. These are implementation changes to the historical estimator. Speed ratios depend on design size, penalty, toolchain and hardware; no universal speedup is claimed. Singular or numerically unresolved systems fail explicitly without substituting an approximate fit.

Measured performance

The completed benchmark compares the optimized engine with the first repaired 1.0.3 candidate on four synthetic designs, three penalty exponents and all four fitting interfaces. Each engine uses two warmups and 15 timed fits per case. The speed ratio divides the baseline median time by the optimized median time.

All 96 timing shards and 48 numerical comparisons pass validation. Selected penalties, variable support and variable counts match exactly; the maximum relative coefficient/CV difference is 5.51e-14. These timings use one CV worker and one BLAS thread on an Apple M3 Pro with R 4.6.0. The fixed engine order and single machine limit generalization. This measures serial optimization against the repaired candidate, rather than a direct speed comparison with CRAN 1.0.2 or a reproduction of the article's numerical study.

See the benchmark report, case comparisons, individual timings and recorded protocol and environment.

Guides and citation

• Reproducibility and validation scope
• Getting started guide
• Vignette source
• Version notes
• Contribution guidelines: CONTRIBUTING.md in the source repository
• Source repository
• Report a reproducible issue

citation("rbridge")

Yuzbasi, B., Arashi, M. and Akdeniz, F. (2021). Penalized regression via the restricted bridge estimator. Soft Computing, 25, 8401-8416. doi:10.1007/s00500-021-05763-9.

The package is licensed under GPL-3. Utility-function attribution inherited from the original source is retained.

From a source-repository checkout with rbridge installed, build the site with the optional pkgdown tool and the repository's documentation script:

system2("Rscript", c("--vanilla", "tools/build-site.R", ".", "docs"))

The site includes the working guide, curated function reference and validation scope. The script also renders reference equations as native MathML using Pandoc, so they can be read offline. Building it does not publish a website.