Convolution-type smoothed quantile regression

The conquer library performs fast and accurate convolution-type smoothed quantile regression (Fernandes, Guerre and Horta, 2021, He et al., 2022, Tan, Wang and Zhou, 2022 for low/high-dimensional estimation and bootstrap inference.

In the low-dimensional setting, efficient gradient-based methods are employed for fitting both a single model and a regression process over a quantile range. Normal-based and (multiplier) bootstrap confidence intervals for all slope coefficients are constructed. In high dimensions, the conquer methods complemented with ℓ₁-penalization and iteratively reweighted ℓ₁-penalization are used to fit sparse models.

2023-03-05 (Version 1.3.3):

When calling conquer function with ci = "asymptotic", an n by n diagonal matrix was involved for estimating asymptotic covariance matrix. This space allocation was expensive and unnecessary. In practice, on data with large n, computing the asymptotic confidence interval was infeasible.

This issue is mitigated via a more computationally efficient matrix multiplication. The space complexity is released from O(n²) to O(np).

2023-02-05 (Version 1.3.2):

1. Fix an issue in the conquer.reg function: when the penalties were group lasso, sparse group lasso or elastic-net, and the input λ was a sequence, the estimated coefficients were not reasonable. This didn't affect cross-validation (conquer.cv.reg), or conquer.reg with other penalties or when input λ was a scalar.

2. When the input λ of conquer.reg function was a sequence, the output estimation was a vector instead of a matrix, which was not consistent with the description of the function.

3. Update the default version of C++ as required by CRAN.

2022-09-12 (Version 1.3.1):

Add flexibility into the conquer function:

1. The step size of Barzilai-Borweincan gradient descent can be unbounded, or the upper bound can be user-specified.

2. The smoothing bandwidth can be specified as any positive value. In previous versions, it has to be bounded away from zero.

2022-03-24 (Version 1.3.0):

1. Add inference methods based on estimated asymptotic covariance matrix for low-dimensional conquer.

2. Add more flexible penalties (elastic-net, group Lasso and sparse group Lasso) into conquer.reg and conquer.cv.reg functions.

3. Speed up cross-validation using warm start along a sequence of λ's.

2022-02-12 (Version 1.2.2):

Remove the unnecessary dependent packge caret for a cleaner installation.

2021-10-24 (Version 1.2.1):

Major updates:

1. Add a function conquer.process for conquer process over a quantile range.

2. Add functions conquer.reg, conquer.cv.reg for high-dimensional conquer with Lasso, SCAD and MCP penalties. The first function is called with a prescribed λ, and the second function calibrate λ via cross-validation. The candidates of λ can be user-specified, or automatically generated by simulating the pivotal quantity proposed in Belloni and Chernozhukov, 2011.

Minor updates:

1. Add logistic kernel for all the functions.

2. Modify initialization using asymmetric Huber regression.

3. Default number of tightening iterations is now 3.

4. Parameters for SCAD (default = 3.7) and MCP (default = 3) are added as arguments into the functions.

conquer is available on CRAN, and it can be installed into R environment:

install.packages("conquer")

Compilation errors by install.packages("conquer") in R:

It usually takes several days to build a binary package after we submit a source packge to CRAN. During that time period, only a source package for the new version is available. However, installing source packges (especially Rcpp-based ones) may cause various compilation errors. Hence, when users see the prompt "There is a binary version available but the source version is later. Do you want to install from sources the package which needs compilation?", we strongly recommend selecting no.

Below are a collection of error / warning messages and their solutions:

• Error: smqr.cpp: 'quantile' is not a member of 'arma’. Solution: 'quantile' function was added into RcppArmadillo version 0.9.850.1.0 (2020-02-09), so reinstalling / updating the library RcppArmadillo will fix this issue.

• Error: unable to load shared object.. Symbol not found: _EXTPTR_PTR. Solution: This issue is common in some specific versions of R when we load Rcpp-based libraries. It is an error in R caused by a minor change about EXTPTR_PTR. Upgrading R to 4.0.2 will solve the problem.

• Error: function 'Rcpp_precious_remove' not provided by package 'Rcpp'. Solution: This happens when a package is compiled against a recent Rcpp release, but users load it using an older version of Rcpp. Reinstalling the package Rcpp will solve the problem.

There are 4 functions in this library:

• conquer: convolution-type smoothed quantile regression
• conquer.process: convolution-type smoothed quantile regression process
• conquer.reg: convolution-type smoothed quantile regression with regularization
• conquer.cv.reg: cross-validated convolution-type smoothed quantile regression with regularization

Let us illustrate conquer by a simple example. For sample size n = 5000 and dimension p = 500, we generate data from a linear model y_i = β₀ + <x_i, β> + ε_i, for i = 1, 2, ... n. Here we set β₀ = 1, β is a p-dimensional vector with every entry being 1, x_i follows p-dimensional standard multivariate normal distribution (available in the library MASS), and ε_i is from t₂ distribution.

library(MASS)
library(quantreg)
library(conquer)
n = 5000
p = 500
beta = rep(1, p + 1)
set.seed(2021)
X = mvrnorm(n, rep(0, p), diag(p))
err = rt(n, 2)
Y = cbind(1, X) %*% beta + err

Then we run both quantile regression using package quantreg, with a Frisch-Newton approach after preprocessing (Portnoy and Koenker, 1997), and conquer (with Gaussian kernel) on the generated data. The quantile level τ is fixed to be 0.5.

tau = 0.5
start = Sys.time()
fit.qr = rq(Y ~ X, tau = tau, method = "pfn")
end = Sys.time()
time.qr = as.numeric(difftime(end, start, units = "secs"))
est.qr = norm(as.numeric(fit.qr$coefficients) - beta, "2")

start = Sys.time()
fit.conquer = conquer(X, Y, tau = tau)
end = Sys.time()
time.conquer = as.numeric(difftime(end, start, units = "secs"))
est.conquer = norm(fit.conquer$coeff - beta, "2")

It takes 7.4 seconds to run the standard quantile regression but only 0.2 seconds to run conquer. In the meanwhile, the estimation error is 0.5186 for quantile regression and 0.4864 for conquer. For readers’ reference, these runtimes are recorded on a Macbook Pro with 2.3 GHz 8-Core Intel Core i9 processor, and 16 GB 2667 MHz DDR4 memory. We refer to He et al., 2022 for a more extensive numerical study.

We can also run conquer over a quantile range

fit.conquer.process = conquer.process(X, Y, tauSeq = seq(0.2, 0.8, by = 0.05))
beta.conquer.process = fit.conquer.process$coeff

Let us switch to the setting of high-dimensional sparse regression with (n, p, s) = (200, 500, 5), and generate data accordingly.

n = 200
p = 500
s = 5
beta = c(runif(s + 1, 1, 1.5), rep(0, p - s))
X = mvrnorm(n, rep(0, p), diag(p))
err = rt(n, 2)
Y = cbind(1, X) %*% beta + err

Regularized conquer can be executed with flexible penalitis, including Lasso, elastic-net, SCAD and MCP. For all the penalties, the bandwidth parameter h is self-tuned, and the regularization parameter λ is selected via cross-validation.

fit.lasso = conquer.cv.reg(X, Y, tau = 0.5, penalty = "lasso")
beta.lasso = fit.lasso$coeff

fit.elastic = conquer.cv.reg(X, Y, tau = 0.5, penalty = "elastic", para.elastic = 0.7)
beta.elastic = fit.elastic$coeff

fit.scad = conquer.cv.reg(X, Y, tau = 0.5, penalty = "scad")
beta.scad = fit.scad$coeff

fit.mcp = conquer.cv.reg(X, Y, tau = 0.5, penalty = "mcp")
beta.mcp = fit.mcp$coeff

Finally, group Lasso is also incorporated in to account for more complicated sparse structure. The group argument stands for group indices, and it has to be specified for group Lasso.

n = 200
p = 500
s = 5
beta = c(1, rep(1.3, 2), rep(1.5, 3), rep(0, p - s))
X = matrix(rnorm(n * p), n, p)
err = rt(n, 2)
Y = cbind(1, X) %*% beta + err

group = c(rep(1, 2), rep(2, 3), rep(3, p - s))
fit.group = conquer.cv.reg(X, Y,tau = 0.5, penalty = "group", group = group)
beta.group = fit.group$coeff

Help on the functions can be accessed by typing ?, followed by function name at the R command prompt.

For example, ?conquer will present a detailed documentation with inputs, outputs and examples of the function conquer.

GPL-3.0

C++17

Xuming He [email protected], Xiaoou Pan [email protected], Kean Ming Tan [email protected] and Wen-Xin Zhou [email protected]

Xiaoou Pan [email protected]

Barzilai, J. and Borwein, J. M. (1988). Two-point step size gradient methods. IMA J. Numer. Anal. 8 141-148. Paper

Belloni, A. and Chernozhukov, V. (2011) ℓ₁-penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82-130. Paper

Fan, J., Liu, H., Sun, Q. and Zhang, T. (2018). I-LAMM for sparse learning: Simultaneous control of algorithmic complexity and statistical error. Ann. Statist. 46 814-841. Paper

Fernandes, M., Guerre, E. and Horta, E. (2021). Smoothing quantile regressions. J. Bus. Econ. Statist. 39 338-357, Paper

He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2023). Smoothed quantile regression with large-scale inference. J. Econometrics, 232(2) 367-388, Paper

Koenker, R. (2005). Quantile Regression. Cambridge Univ. Press, Cambridge. Book

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. Paper

Portnoy, S. and Koenker, R. (1997). The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators. Statist. Sci. 12 279–300. Paper

Tan, K. M., Wang, L. and Zhou, W.-X. (2022). High-dimensional quantile regression: convolution smoothing and concave regularization. J. Roy. Statist. Soc. Ser. B 84(1) 205-233. Paper