Sparse Convex Quantile Regression (SCQR)

• SCQR is a method that fits conditional quantiles under convexity constraints while enforcing sparsity for variable selection.
• It combines quantile regression with nonparametric convex fitting, ℓ2 regularization, and an ε-insensitive loss to stabilize estimation.
• The approach employs a Generalized Benders Decomposition with local-search acceleration to efficiently solve the mixed-integer optimization problem.

Searching arXiv for the cited SCQR paper and closely related quantile-regression papers to support the article. I’m checking whether arXiv search is available in this environment. Sparse Convex Quantile Regression (SCQR) denotes a class of sparse quantile-regression formulations in which the fitted conditional quantile is constrained to be convex, while variable selection is imposed for interpretability and generalization. In the formulation introduced in “Sparse Convex Quantile Regression: A Generalized Benders Decomposition Approach,” SCQR combines quantile regression, nonparametric convex fitting, binary support selection, an $\ell_2$ regularizer on subgradients, and an $\varepsilon$-insensitive zone for small residuals, together with a decomposition algorithm based on Generalized Benders Decomposition (GBD) and a local-search Benders matheuristic (Luo et al., 2 Sep 2025).

1. Conceptual position within quantile regression

SCQR marries the robustness and distributional insight of quantile regression with the flexibility of nonparametric convex fitting, while enforcing sparsity for interpretability and generalization (Luo et al., 2 Sep 2025). The quantile component targets a specified $\tau \in (0,1)$ rather than the conditional mean; the convexity component treats the fitted function as a convex surface represented through sample-point values and subgradients; the sparsity component limits the number of active covariates through explicit support variables.

In the 2025 convex-fitting formulation, the object being estimated is not a linear coefficient vector alone. Instead, each observation $x_i$ carries a function value $\theta_i$ and a subgradient $\beta_i$, so the regression surface is reconstructed through Afriat-type convexity inequalities. This distinguishes SCQR from sparse linear quantile regression, even though both use asymmetric quantile loss.

The broader literature in the supplied sources shows that the acronym “SCQR” is not fully standardized. Earlier and adjacent uses include structured sparse quantile regression with a mixed $\ell_{1,\infty}$ norm (Nassiri et al., 2013), convolution-smoothed high-dimensional quantile regression with concave regularization (Tan et al., 2021, Moon et al., 2022), sparse quantile Huber regression (Aravkin et al., 2014), and sparse-smooth spatially varying coefficient quantile regression on graphs (Jian et al., 22 Nov 2025). This suggests that, in contemporary usage, SCQR is best understood as a family of sparse quantile-regression constructions rather than a single universally fixed model.

2. Mathematical formulation of the convex sparse model

Given observations $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$, let $\beta_i \in \mathbb{R}^d$ be the subgradient of the estimated convex quantile function at $x_i$, and $\varepsilon$0 its value. Nonnegative slack variables $\varepsilon$1 represent the asymmetric quantile loss outside an $\varepsilon$2-insensitive zone, and $\varepsilon$3 is a regularization coefficient. With fixed support, the regularized convex quantile regression problem is (Luo et al., 2 Sep 2025)

$\varepsilon$4

subject to

$\varepsilon$5

$\varepsilon$6

The convexity constraints

$\varepsilon$7

encode global convexity of the fitted quantile function. The loss is the asymmetric quantile, or “pinball,” loss, but only outside the $\varepsilon$8-insensitive zone. The penalty $\varepsilon$9 shrinks subgradients and renders strong convexity.

To enforce sparsity, binary variables $\tau \in (0,1)$0 are introduced with $\tau \in (0,1)$1 and Big–$\tau \in (0,1)$2 constraints $\tau \in (0,1)$3. Writing $\tau \in (0,1)$4, the full SCQR model becomes (Luo et al., 2 Sep 2025)

$\tau \in (0,1)$5

This formulation makes the source of combinatorial difficulty explicit: support selection occurs through $\tau \in (0,1)$6, while convex quantile fitting occurs through $\tau \in (0,1)$7.

3. Regularization, quantile targeting, and statistical interpretation

The 2025 SCQR paper emphasizes two modifications to classical convex quantile regression: the $\tau \in (0,1)$8-norm penalty and the $\tau \in (0,1)$9-insensitive zone. Both are presented as theoretically justified and empirically validated (Luo et al., 2 Sep 2025).

The $x_i$0 penalty yields strong convexity and controls the growth of subgradients near extreme data points, thereby improving out-of-sample generalization. The $x_i$1-insensitive region prevents penalizing small residuals and further stabilizes fits against noise. In this framework, neither term is a purely algorithmic device; each is part of the statistical design of the estimator.

Under mild conditions, the model satisfies a quantile-coverage property. If

$x_i$2

then

$x_i$3

This ensures that the fitted function genuinely targets the $x_i$4-th conditional quantile (Luo et al., 2 Sep 2025).

A common misunderstanding is to view the $x_i$5-insensitive zone as merely a heuristic tolerance band. In the SCQR formulation it is instead tied to the loss definition and is used to prevent penalizing small residuals. A plausible analogy appears in sparse quantile Huber regression, where a quadratic region around zero residuals likewise avoids over-penalizing small errors and smooths the local behavior of the loss (Aravkin et al., 2014). The two constructions are not identical, but the comparison clarifies why central residual downweighting can improve stability.

4. Decomposition by Generalized Benders and local-search acceleration

The central algorithmic contribution of the 2025 SCQR work is the first Generalized Benders Decomposition algorithm tailored to sparse convex quantile regression (Luo et al., 2 Sep 2025). The decomposition separates support selection from convex quantile fitting. Defining

$x_i$6

the full problem reduces to

$x_i$7

By dualizing the inner convex quantile regression subproblem, Theorem 1 yields

$x_i$8

subject to flow-conservation and box constraints

$x_i$9

From this representation one extracts a subgradient $\theta_i$0 and constructs linear under-estimators, namely Benders cuts (Luo et al., 2 Sep 2025).

The Benders master problem is

$\theta_i$1

Algorithm 1 iterates two principal steps (Luo et al., 2 Sep 2025):

• Subproblem step: solve the convex quantile subproblem at $\theta_i$2 by cutting-plane to obtain $\theta_i$3 and $\theta_i$4, then update the upper bound.
• Master step: add the Benders cut, resolve the master problem for $\theta_i$5, and update the lower bound and incumbent.

Finite convergence to optimality is guaranteed, though worst-case exponential, and in practice only a few iterations suffice (Luo et al., 2 Sep 2025).

To further accelerate solution improvement, the paper embeds Benders within a local-search neighborhood. Given the current best support $\theta_i$6, define the Hamming neighborhood

$\theta_i$7

A restricted master problem is then solved with $\theta_i$8 while incorporating all cuts generated so far; if an improved support is found, the neighborhood is re-centered and the process repeats. By focusing on a small Hamming ball, the method avoids oscillation and tightens the incumbent rapidly (Luo et al., 2 Sep 2025).

5. Empirical performance and the Sustainable Development Goals application

The empirical program in the 2025 SCQR study comprises both simulation experiments and a real-world application, and the reported findings are specific about scaling, support recovery, and comparative accuracy (Luo et al., 2 Sep 2025).

Across two data-generating processes involving nonlinear convex functions of a few covariates plus noise, adding $\theta_i$9 regularization reduces out-of-sample MAE dramatically, and the $\beta_i$0-insensitive zone further stabilizes performance, matching or outperforming Lipschitz-constrained baselines. For the convex quantile subproblem, the primal cutting-plane algorithm scales to $\beta_i$1, $\beta_i$2 within minutes, achieving primal-infeasibility below $\beta_i$3.

For the sparse model itself, across $\beta_i$4, dimension $\beta_i$5 up to $\beta_i$6, multiple SNRs, several $\beta_i$7 levels, and varying feature correlations $\beta_i$8, the GBD+LSB combination recovers $\beta_i$9–$\ell_{1,\infty}$0 of true supports within a few hundred seconds and $\ell_{1,\infty}$1–$\ell_{1,\infty}$2 iterations. Comparisons against the CNLS-A primal cutting-plane method show that GBD reaches higher selection accuracy in a fraction of the time. When the true active features vary with $\ell_{1,\infty}$3, using data from Lee et al. 2014, GBD correctly identifies median- versus upper-quantile-specific variables, whereas mean-regression cuts cannot (Luo et al., 2 Sep 2025).

The real-data benchmark concerns Sustainable Development Goals performance for $\ell_{1,\infty}$4 OECD countries over three years, giving $\ell_{1,\infty}$5 points, with $\ell_{1,\infty}$6 SDG indicators as inputs and GDP growth as output. At $\ell_{1,\infty}$7, the GBD algorithm, with cross-validated $\ell_{1,\infty}$8, yields MAE(in/out) lower than CNLS-A at all $\ell_{1,\infty}$9, sparser models with $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$0 between $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$1 and $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$2, and better out-of-sample quantile loss. Feature selections adapt to different quantiles, which offers nuanced policy insights (Luo et al., 2 Sep 2025).

These results anchor the practical claim of the framework: it is intended not only to recover convex conditional quantiles accurately, but also to do so under sparsity and at scales that were computationally difficult in earlier convex-quantile formulations.

6. Related variants, extensions, and recurrent themes

The supplied literature places the 2025 SCQR model within a broader sparse quantile-regression ecosystem. Earlier work on structured sparse quantile regression introduced a mixed $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$3 norm,

$$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$4

to impose group sparsity in linear quantile regression, and derived a piecewise-linear solution path algorithm (Nassiri et al., 2013). This line of work focuses on group structure in a linear model rather than convex nonparametric fitting.

A separate high-dimensional line studies convolution-smoothed quantile losses and iteratively reweighted $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$5 regularization. In “High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization,” the smoothed empirical loss is twice continuously differentiable and locally strongly convex with high probability, and repeated reweighted solves achieve the oracle rate and, under additional conditions, the strong oracle property (Tan et al., 2021). “High-Dimensional Composite Quantile Regression: Optimal Statistical Guarantees and Fast Algorithms” similarly uses a convolution-smoothed composite loss, iteratively reweighted $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$6 regularization, and a majorize-minimize first-order solver, with near-minimax and near-oracle rates under heavy-tailed errors (Moon et al., 2022). These papers concern sparse linear or composite quantile regression rather than convex shape-constrained quantile fitting, but they share with SCQR a reliance on smoothing or strong convexity to improve both theory and computation.

Sparse quantile Huber regression provides another related construction. It replaces the piecewise-linear check loss near zero with a smooth quadratic cap, combines this with $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$7 or greedy sparsity mechanisms, and exploits a piecewise-linear-quadratic representation for efficient interior-point optimization (Aravkin et al., 2014). This parallels, at a conceptual level, the role of the $$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$8-insensitive region in the 2025 SCQR formulation.

An extension toward structured heterogeneity appears in “Sparse-Smooth Spatially Varying Coefficient Quantile Regression,” where each coefficient is decomposed into a global term and a spatial deviation,

$$\{(x_i,y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$$9

with adaptive group penalization for selecting whether a predictor varies over space and graph-Laplacian smoothing for continuity over irregular networks. That framework provides ADMM and smoothed proximal-gradient solvers, selection consistency for deviation groups, mean-squared error bounds, and root-$\beta_i \in \mathbb{R}^d$0 asymptotic normality for global coefficients with an oracle property (Jian et al., 22 Nov 2025). This suggests that SCQR ideas naturally extend beyond Euclidean variable selection to graph-structured and geographically varying quantile effects.

Within this broader landscape, the defining feature of the 2025 SCQR model remains its combination of convex shape constraints, sparse support selection, and exact mixed-integer decomposition. Its stated future directions include adaptive cut-strengthening or dynamic $\beta_i \in \mathbb{R}^d$1-selection, theoretical analysis of convergence rates under problem structure, piecewise-model complexity control by limiting the number of supporting hyperplanes, and extensions to multivariate outputs or expectiles (Luo et al., 2 Sep 2025).