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arxiv: 2604.12694 · v2 · submitted 2026-04-14 · 📊 stat.CO

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Adaptive Sparse Group Lasso Penalized Quantile Regression via Dual ADMM

Huayan Kou , Yuwen Gu , Yi Lian , Rui Zhang , Jun Fan
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:43 UTC · model grok-4.3

classification 📊 stat.CO
keywords quantile regressionsparse group lassoadaptive lassoADMMvariable selectionhigh-dimensional datapenalized regression
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The pith

Adaptive sparse group lasso penalized quantile regression achieves simultaneous within- and between-group sparsity via dual ADMM.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a penalized quantile regression model for high-dimensional data with grouped predictors. It combines adaptive lasso penalties on individual variables with adaptive group lasso penalties on entire groups to enforce sparsity at both levels. The resulting optimization problem is solved by applying the alternating direction method of multipliers to the dual formulation, which also yields a proof of global convergence. Simulations and real-data examples indicate that the approach recovers the correct sparse pattern while running faster than comparable penalized quantile methods.

Core claim

We introduce the adaptive sparse group lasso penalized quantile regression, which integrates adaptive lasso and adaptive group lasso penalties. We optimize the model parameters via the alternating direction method of multipliers (ADMM) applied to the dual problem, and establish global convergence. Through extensive simulation studies and real data analyses, we demonstrate the efficacy of the proposed method in achieving simultaneous within- and between-group sparsity, and the computational efficiency of our algorithm relative to existing alternatives.

What carries the argument

The adaptive sparse group lasso penalty within a quantile regression objective, solved by the dual ADMM algorithm that simultaneously enforces individual and group-level sparsity.

If this is right

  • The method achieves simultaneous sparsity within groups and between groups in high-dimensional quantile regression.
  • The dual ADMM solver guarantees global convergence of the iterates.
  • Computational efficiency exceeds that of existing penalized quantile regression alternatives.
  • Simulations and real data analyses confirm accurate recovery of sparse structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dual formulation may allow straightforward addition of further constraints or different loss functions without redesigning the solver.
  • The method's performance will degrade if the supplied group structure is misspecified or if the initial weight estimator is inconsistent.
  • Similar adaptive dual-penalty constructions could be applied to other regression models with grouped predictors, such as mean or logistic regression.

Load-bearing premise

The adaptive weights can be reliably estimated from an initial consistent estimator and the grouped structure of the variables is known in advance.

What would settle it

A dataset with known group structure in which the method fails to recover the true sparse support or in which the ADMM iterations diverge would contradict the claimed sparsity recovery and convergence.

read the original abstract

Sparse penalized quantile regression provides an effective framework for variable selection and robust estimation in high-dimensional data analysis. When ex planatory variables are organized into groups, achieving sparsity both within and between groups is essential. However, existing quantile regression methods often fail to meet this dual objective. To address this gap, we introduce the adaptive sparse group lasso penalized quantile regression, which integrates adaptive lasso and adaptive group lasso penalties. We optimize the model parameters via the alternating direction method of multipliers (ADMM) applied to the dual problem, and establish global convergence. Through extensive simulation studies and real data analyses, we demonstrate (i) the efficacy of the proposed method in achieving simultaneous within- and between-group sparsity, and (ii) the computational efficiency of our algorithm relative to existing alternatives.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes an adaptive sparse group lasso penalized quantile regression that combines adaptive lasso and adaptive group lasso penalties to induce simultaneous within-group and between-group sparsity. Model parameters are estimated by applying the alternating direction method of multipliers (ADMM) to the dual problem, for which global convergence is established. The approach is assessed via simulation studies and real-data examples that compare its sparsity recovery and computational efficiency against existing penalized quantile regression methods.

Significance. If the global convergence result holds under the stated conditions and the empirical advantages prove robust, the method supplies a practical extension of adaptive penalties to grouped high-dimensional quantile regression. The dual-ADMM formulation is a constructive choice that can improve scalability relative to primal solvers; the combination of adaptive weights with group structure is a natural and previously underexplored direction for this loss.

major comments (2)
  1. [§3, Theorem 1] §3 (Convergence Analysis), Theorem 1: the global convergence claim for dual ADMM is asserted on the basis of standard ADMM theory, yet the non-smooth quantile check loss together with the non-separable adaptive group penalty requires explicit verification that the dual objective satisfies the requisite saddle-point existence and that the chosen step-size sequence guarantees boundedness of the iterates; without these details the theorem cannot be confirmed as load-bearing for the central algorithmic contribution.
  2. [§2.2] §2.2 (Adaptive Weights): the construction of the adaptive weights from an initial consistent estimator is presented without a quantitative statement of the required convergence rate of that estimator or of the resulting oracle-type properties (if any) for the final estimator; this assumption is load-bearing for the claimed superiority in sparsity recovery.
minor comments (3)
  1. [Abstract] The abstract contains a typographical error (“ex planatory”).
  2. [§2] Notation for the group index set and the adaptive weight matrices should be introduced once and used consistently across the model definition, the dual formulation, and the ADMM updates.
  3. [§5] Simulation tables would benefit from explicit reporting of the tuning-parameter selection procedure (e.g., cross-validation folds, grid size) and of any data-exclusion rules applied to the real-data examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our paper. We address the major comments point by point below, indicating the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3 (Convergence Analysis), Theorem 1: the global convergence claim for dual ADMM is asserted on the basis of standard ADMM theory, yet the non-smooth quantile check loss together with the non-separable adaptive group penalty requires explicit verification that the dual objective satisfies the requisite saddle-point existence and that the chosen step-size sequence guarantees boundedness of the iterates; without these details the theorem cannot be confirmed as load-bearing for the central algorithmic contribution.

    Authors: We agree that a more explicit verification is necessary to make the convergence result fully rigorous. In the revised version, we will expand the proof of Theorem 1 to include a direct verification of the saddle-point existence for the dual problem, leveraging the convexity of the quantile check loss and the structure of the adaptive penalties. Additionally, we will specify the conditions on the step-size sequence that ensure boundedness of the iterates, referencing relevant extensions of ADMM theory to non-smooth and non-separable cases. This will be added to §3. revision: yes

  2. Referee: [§2.2] §2.2 (Adaptive Weights): the construction of the adaptive weights from an initial consistent estimator is presented without a quantitative statement of the required convergence rate of that estimator or of the resulting oracle-type properties (if any) for the final estimator; this assumption is load-bearing for the claimed superiority in sparsity recovery.

    Authors: We acknowledge this point. The manuscript assumes an initial consistent estimator for constructing the adaptive weights, as is standard in adaptive lasso literature. In the revision, we will add a quantitative statement specifying that the initial estimator needs to be consistent at rate o_p(n^{-1/2}) or better for the adaptive weights to achieve the desired sparsity properties. We will also discuss the oracle-type properties under these conditions, noting that while full oracle results are established in related works for non-grouped cases, for the grouped quantile regression setting they follow similarly but require additional technical conditions on the group structure. We will clarify that the superiority in sparsity recovery is supported by the simulation studies, which use consistent initial estimators. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new penalty (adaptive sparse group lasso for quantile regression) by combining standard adaptive lasso and group lasso terms, then solves the resulting convex program via dual ADMM whose global convergence follows from established ADMM theory under standard convexity and constraint qualifications. The adaptive weights are constructed from a preliminary consistent estimator, which is the conventional two-step procedure for adaptive penalties and does not make the main existence or convergence claim tautological. No load-bearing step reduces by definition or by self-citation to the target result; the derivation remains self-contained against external ADMM convergence theorems and does not rename empirical patterns or import uniqueness via author-overlapping citations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects typical assumptions in penalized quantile regression literature. The central claim depends on the validity of adaptive weight construction and the convergence theory for dual ADMM under the stated penalties.

free parameters (1)
  • penalty tuning parameters
    The regularization parameters controlling the adaptive lasso and group lasso terms are not specified in the abstract and are typically chosen by cross-validation or information criteria.
axioms (2)
  • domain assumption Global convergence holds for the dual ADMM algorithm under the proposed penalties
    Stated in the abstract but without the technical conditions or proof sketch provided.
  • domain assumption Initial estimators yield consistent adaptive weights
    Standard assumption for adaptive lasso methods; required for the penalty to achieve the desired oracle properties.

pith-pipeline@v0.9.0 · 5429 in / 1430 out tokens · 34759 ms · 2026-05-10T13:43:48.201066+00:00 · methodology

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Reference graph

Works this paper leans on

39 extracted references · 2 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Introductory lectures on convex programming volume i: Basic course

    Nesterov, Y. Introductory lectures on convex programming volume i: Basic course. Lecture notes. volume I. Kluwer Academic Publishers, 2004

    2004

  4. [4]

    and Toh, K.C

    Zhang, Y., Zhang, N., Sun, D. and Toh, K.C. (2017). An efficient Hessian based algorithm for solving large-scale sparse group lasso problems. arXiv preprint arXiv:1712.05910

    work page arXiv 2017

  5. [5]

    Regularization in statistics[J]

    Bickel P J, Li B, Tsybakov A B, et al. Regularization in statistics[J]. Test, 2006, 15: 271-344

    2006

  6. [6]

    Discussion: The Dantzig selector: Statistical estimation when p is much larger than n[J]

    Efron B, Hastie T, Tibshirani R. Discussion: The Dantzig selector: Statistical estimation when p is much larger than n[J]. The Annals of Statistics, 2007, 35(6): 2358-2364

    2007

  7. [7]

    Simultaneous analysis of Lasso and Dantzig selector[J]

    Bickel P J, Ritov Y, Tsybakov A B. Simultaneous analysis of Lasso and Dantzig selector[J]. The Annals of Statistics,2009, 37(4): 1705-1732

    2009

  8. [8]

    Robust low-rank multiple kernel learning with compound regularization[J]

    Jiang H, Tao C, Dong Y, et al. Robust low-rank multiple kernel learning with compound regularization[J]. European Journal of Operational Research, 2021, 295(2): 634-647

    2021

  9. [9]

    A statistical view of some chemometrics regression tools[J]

    Frank L L E, Friedman J H. A statistical view of some chemometrics regression tools[J]. Technometrics, 1993, 35(2): 109-135

    1993

  10. [10]

    Regression shrinkage and selection via the lasso[J]

    Tibshirani R. Regression shrinkage and selection via the lasso[J]. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1996, 58(1): 267-288

    1996

  11. [11]

    Variable selection via nonconcave penalized likelihood and its oracle properties[J]

    Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties[J]. Journal of the American statistical Association, 2001, 96(456): 1348-1360

    2001

  12. [12]

    Nearly unbiased variable selection under minimax concave penalty[J].The Annals of Statistics, 2010,38(2): 894-942

    Zhang C H. Nearly unbiased variable selection under minimax concave penalty[J].The Annals of Statistics, 2010,38(2): 894-942

    2010

  13. [13]

    The adaptive lasso and its oracle properties[J]

    Zou H. The adaptive lasso and its oracle properties[J]. Journal of the American statistical association, 2006, 101(476): 1418-1429

    2006

  14. [14]

    Regression quantiles[J]

    Koenker R, Bassett Jr G. Regression quantiles[J]. Econometrica: journal of the Econometric Society, 1978, 46(1): 33-50

    1978

  15. [15]

    _1 -penalized quantile regression in high-dimensional sparse models[J].Annals of Statistics, 2011,39(1):82-130

    Belloni A, Chernozhukov V. _1 -penalized quantile regression in high-dimensional sparse models[J].Annals of Statistics, 2011,39(1):82-130

    2011

  16. [16]

    Quantile regression for analyzing heterogeneity in ultra-high dimension[J]

    Wang L, Wu Y, Li R. Quantile regression for analyzing heterogeneity in ultra-high dimension[J]. Journal of the American Statistical Association, 2012, 107(497): 214-222

    2012

  17. [17]

    Adaptive robust variable selection[J]

    Fan J, Fan Y, Barut E. Adaptive robust variable selection[J]. Annals of statistics, 2014, 42(1): 324-351

    2014

  18. [18]

    Quantile regression via an MM algorithm[J]

    Hunter D R, Lange K. Quantile regression via an MM algorithm[J]. Journal of Computational and Graphical Statistics, 2000, 9(1): 60-77

    2000

  19. [19]

    Coordinate descent algorithms for lasso penalized regression[J].The Annals of Applied Statistics,2008, 2(1):224-244

    Wu T T, Lange K. Coordinate descent algorithms for lasso penalized regression[J].The Annals of Applied Statistics,2008, 2(1):224-244

    2008

  20. [20]

    An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression[J]

    Peng B, Wang L. An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression[J]. Journal of Computational and Graphical Statistics, 2015, 24(3): 676-694

    2015

  21. [21]

    Semismooth Newton coordinate descent algorithm for elastic-net penalized Huber loss regression and quantile regression[J]

    Yi C, Huang J. Semismooth Newton coordinate descent algorithm for elastic-net penalized Huber loss regression and quantile regression[J]. Journal of Computational and Graphical Statistics, 2017, 26(3): 547-557

    2017

  22. [22]

    ADMM for high-dimensional sparse penalized quantile regression[J]

    Gu Y, Fan J, Kong L, et al. ADMM for high-dimensional sparse penalized quantile regression[J]. Technometrics, 2018, 60(3): 319-331

    2018

  23. [23]

    Model selection and estimation in regression with grouped variables[J]

    Yuan M, Lin Y. Model selection and estimation in regression with grouped variables[J]. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2006, 68(1): 49-67

    2006

  24. [24]

    Quantile regression with group lasso for classification[J]

    Hashem H, Vinciotti V, Alhamzawi R, et al. Quantile regression with group lasso for classification[J]. Advances in Data Analysis and Classification, 2016, 10: 375-390

    2016

  25. [25]

    Group Lasso for high dimensional sparse quantile regression Models[EB/OL].(2011-05-25)

    Kato K. Group Lasso for high dimensional sparse quantile regression Models[EB/OL].(2011-05-25)

    2011

  26. [26]

    Group penalized quantile regression[J]

    Ouhourane M, Yang Y, Benedet A L, et al. Group penalized quantile regression[J]. Statistical Methods & Applications, 2022,31: 495-529

    2022

  27. [27]

    Quantile regression feature selection and estimation with grouped variables using Huber approximation[J]

    Sherwood B, Li S. Quantile regression feature selection and estimation with grouped variables using Huber approximation[J]. Statistics and Computing, 2022, 32(5): 75

    2022

  28. [28]

    A note on adaptive group lasso[J]

    Wang H, Leng C. A note on adaptive group lasso[J]. Computational statistics & data analysis, 2008, 52(12): 5277-5286

    2008

  29. [29]

    Adaptive group LASSO selection in quantile models[J]

    Ciuperca G. Adaptive group LASSO selection in quantile models[J]. Statistical Papers, 2019, 60: 173-197

    2019

  30. [30]

    Regularization Paths for Generalized Linear Models via Coordinate Descent[J]

    Friedman J, Hastie T, Tibshirani R. Regularization Paths for Generalized Linear Models via Coordinate Descent[J]. Journal of Statistical Software, 2010, 33(1): 1-22

    2010

  31. [31]

    Asymptotic theory of the adaptive Sparse Group Lasso[J]

    Poignard B. Asymptotic theory of the adaptive Sparse Group Lasso[J]. Annals of the Institute of Statistical Mathematics, 2020, 72: 297-328

    2020

  32. [32]

    Adaptive sparse group LASSO in quantile regression[J]

    Mendez-Civieta A, Aguilera-Morillo M C, Lillo R E. Adaptive sparse group LASSO in quantile regression[J]. Advances in Data Analysis and Classification, 2021, 15(3): 547-573

    2021

  33. [33]

    A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions[J]

    Li X, Sun D, Toh K C. A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions[J]. Mathematical Programming, 2016, 155(1-2): 333-373

    2016

  34. [34]

    An efficient Hessian based algorithm for solving large-scale sparse group lasso problems[J]

    Zhang Y, Zhang N, Sun D, et al. An efficient Hessian based algorithm for solving large-scale sparse group lasso problems[J]. Mathematical Programming, 2020, 179: 223-263

    2020

  35. [35]

    A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems[J]

    Li X, Sun D, Toh K C. A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems[J]. SIAM Journal on Optimization, 2018, 28(1): 433-458

    2018

  36. [36]

    An efficient inexact symmetric Gauss--CSeidel based majorized ADMM for high-dimensional convex composite conic programming[J]

    Chen L, Sun D, Toh K C. An efficient inexact symmetric Gauss--CSeidel based majorized ADMM for high-dimensional convex composite conic programming[J]. Mathematical Programming, 2017, 161: 237-270

    2017

  37. [37]

    A fast unified algorithm for solving group-lasso penalize learning problems[J]

    Yang Y, Zou H. A fast unified algorithm for solving group-lasso penalize learning problems[J]. Statistics and Computing, 2015, 25(6): 1129-1141

    2015

  38. [38]

    A coordinate descent algorithm for computing penalized smooth quantile regression.[J] Statistics and Computing, 2017,27: 865--C883

    Mkhadri A, Ouhourane M, Oualkacha K. A coordinate descent algorithm for computing penalized smooth quantile regression.[J] Statistics and Computing, 2017,27: 865--C883

    2017

  39. [39]

    sparsegl: An R Package for Estimating Sparse Group Lasso[EB/OL].(2022-08-05)[2023-06-28].https://doi.org/10.48550/arXiv.2208.02942

    Liang X, Cohen A, Heinsfeld A S, et al. sparsegl: An R Package for Estimating Sparse Group Lasso[EB/OL].(2022-08-05)[2023-06-28].https://doi.org/10.48550/arXiv.2208.02942

    work page doi:10.48550/arxiv.2208.02942 2022