**Con**volution-type smoothed
**qu**antil**e**
**r**egression

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
*ℓ _{1}*-penalization and iteratively reweighted

**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(n ^{2})* to

**2023-02-05 (Version 1.3.2)**:

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

**2022-09-12 (Version 1.3.1)**:

Add flexibility into the `conquer`

function:

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

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)**:

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

Add more flexible penalties (elastic-net, group Lasso and sparse group Lasso) into

`conquer.reg`

and`conquer.cv.reg`

functions.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:

Add a function

`conquer.process`

for conquer process over a quantile range.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:

Add logistic kernel for all the functions.

Modify initialization using asymmetric Huber regression.

Default number of tightening iterations is now 3.

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} = β_{0} + <x_{i},
β> + ε_{i}*, for

`MASS`

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

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*.

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

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

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

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

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

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.

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

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 xmhe@umich.edu, Xiaoou Pan xip024@ucsd.edu, Kean Ming Tan keanming@umich.edu and Wen-Xin Zhou wez243@ucsd.edu

Xiaoou Pan xip024@ucsd.edu

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)
*ℓ _{1}*-penalized quantile regression in high-dimensional
sparse models.

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