A Short Note of Comparison between Convex and Non-convex Penalized Likelihood

Kasy Du

REVIEW 4 major objections 5 minor 29 references

A Short Note of Comparison between Convex and Non-convex Penalized Likelihood

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper argues that non-convex penalties such as SCAD and MCP can reach ground-truth sparsity without sacrificing estimation accuracy, while Lasso trades one against the other.

desk verdict A correct but unoriginal teaching note whose own experiments don't back the headline claim. read the letter →

arxiv 2502.07655 v2 pith:I3YJARQZ submitted 2025-02-11 stat.ME

Kasy Du This is my paper · ORCID

classification stat.ME MSC 62J07

keywords penalizedlikelihoodLassoSCADMCPnon-convexregularizationcoordinatedescenthigh-dimensionalregressionvariableselection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This short note tries to establish a practical trade-off between convex and non-convex penalized likelihood methods in high-dimensional regression. Lasso is computationally efficient and convex but biases large coefficients, while SCAD and MCP shrink mainly small coefficients and can match the true sparsity level without losing estimation accuracy. A toy simulation with n=200, p=1000 and ten nonzero coefficients shows Lasso cannot simultaneously reach ground-truth sparsity and low MSE, whereas SCAD and MCP can. The note also reports that coordinate descent for SCAD and MCP converges more slowly and with less stable loss variation, and that on a rat gene-expression data set SCAD gives a sparser model with lower residual sum of squares than Lasso. If the claims hold, practitioners can choose Lasso when fast convex optimization matters and SCAD or MCP when bias on large coefficients matters.

What carries the argument

The machinery is the first derivative of the penalty function $p_\lambda(t)$: Lasso has $p_\lambda'(t)=\lambda$ constant, SCAD has $p_\lambda'(t)=\lambda[I(t\le\lambda)+((a\lambda-t)_+/((a-1)\lambda))I(t>\lambda)]$, and MCP has $p_\lambda'(t)=(\lambda-t/a)_+$. This derivative determines the shrinkage pattern: a constant derivative means equal shrinkage of all coefficients, while a derivative that declines to zero means only small coefficients are penalized. The paper reads the bias, sparsity, and accuracy comparison off this derivative, then uses coordinate descent with closed-form soft-thresholding-type updates for each penalty to test the behavior computationally.

What would settle it

Run the n=200, p=1000 simulation with SCAD and MCP over a grid of shape parameters (e.g., $a=2.1,3.7,10$) and several random initializations; if some runs stop at visibly higher loss or sparsity levels far from 10, the claimed sparsity-without-accuracy-loss is an optimizer artifact rather than a penalty property. A cleaner check is to solve small instances to certified global optimality and compare the coordinate-descent solution against the global solution.

Watch

Extended reading notes

Core claim

The central claim is that non-convex penalties are not merely a theoretical alternative to Lasso: in a high-dimensional linear model with n=200, p=1000 and 10 nonzero coefficients, SCAD and MCP produce estimators with ground-truth sparsity level without sacrificing estimation accuracy, while Lasso requires a much larger penalty to reach that sparsity and then suffers high MSE. The mechanism is the derivative of the penalty: Lasso applies a constant penalty to every coefficient regardless of size, shrinking large coefficients and creating bias, whereas SCAD and MCP keep a positive penalty only for small values and let it drop to zero for large values, so large coefficients are left nearly unpenalized. The note derives closed-form coordinate-descent updates for all three penalties and reports that SCAD and MCP converge about three times slower than Lasso with unstable loss variation across iterations. On the rat eye gene-expression data, under cross-validated tuning, SCAD yields a sparser model with lower residual sum of squares than Lasso.

Load-bearing premise

The load-bearing premise is that the coordinate-descent solutions for SCAD and MCP show the penalties' true behavior rather than a quirk of the optimizer: Section 3 shows unstable loss variation and does not report the shape parameter, the convergence tolerance, or the starting values, so if the optimizer settles in a poor local optimum the claimed sparsity without accuracy loss could be an algorithmic accident.

Editorial extensions

If this is right

In high-dimensional sparse linear models, Lasso forces a trade-off: a penalty level that recovers true sparsity over-shrinks the nonzero coefficients and inflates MSE.
With SCAD or MCP, a single penalty level can deliver both the ground-truth sparsity level and low estimation error in the simulated setting.
Non-convex penalties carry a computational cost: coordinate descent takes roughly three times as long on the toy problem and shows unstable loss variation.
On the rat gene-expression data, SCAD gives a sparser model with lower residual sum of squares than Lasso under cross-validated tuning.
The practical guide is to prefer Lasso when fast convex optimization matters and SCAD or MCP when bias on large coefficients matters.
The choice between convex and non-convex penalties is not about one being universally better; it depends on whether accuracy or computational feasibility is the binding constraint.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension of the derivative-based explanation is that any penalty whose derivative reaches zero for large coefficients should reproduce the sparsity-without-bias pattern, while any constant-derivative penalty should inherit Lasso's bias.
Because Section 3 omits the shape parameter $a$, convergence tolerance, and initialization, rerunning the simulation across $a$ values and random starts would separate penalty behavior from optimizer behavior.
The real-data conclusion rests on a single data set and a single cross-validation run; repeating the comparison with repeated CV splits or multiple data sets would show whether the sparser SCAD fit is stable.
A practical extension the paper does not test is initializing SCAD or MCP from a Lasso solution, which could reduce the reported convergence time while keeping the favorable sparsity-accuracy profile.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a competent teaching note, not a research contribution. It repeats known results about Lasso vs SCAD/MCP, and its own simulation doesn't actually demonstrate the headline claim.

What's good: the coordinate descent updates in Section 3 are correctly written, and the figures do show the expected bias-sparsity tension. The paper is easy to read, and for someone who hasn't seen these penalties before, it would be a serviceable introduction. The real-data example uses the standard Scheetz eye data, which is a nice illustration.

The soft spots are real. The central assertion in Section 2—that SCAD and MCP can achieve ground-truth sparsity without losing accuracy—is read off Figure 3, where MSE and sparsity are plotted against λ as separate curves. For Lasso the text is careful to name specific λ values; for SCAD and MCP it never identifies a single λ where both criteria are met. That's the load-bearing gap. Also, the non-convex optimization details are missing: the shape parameter a, the convergence tolerance, and the initialization are not reported, so the plotted behavior could reflect local minima rather than the typical estimator. The real-data comparison reports training RSS, which tells us nothing about generalization. And the reference list contains several works on 3D pose estimation and other topics that are irrelevant to the content; that looks like citation padding.

None of this is fatal to the qualitative message, which is standard and correct. But the paper doesn't give the reader a reproducible or rigorous demonstration of its own claim. It reads like an undergraduate project note, not a peer-reviewed research paper.

Who is this for? A student or practitioner looking for a quick overview. Not for a researcher. I would not send it to peer review as a research article; I would suggest the authors either reposition it as an explicit review note or add the missing experimental details and out-of-sample evaluation. If they do that, it could become a decent teaching resource.

For now: desk reject if it lands on a research journal.

Referee Report

4 major / 5 minor

Summary. This short note compares Lasso with the non-convex SCAD and MCP penalties in high-dimensional penalized likelihood estimation. The paper reviews the penalties, states their known bias/oracle properties, presents a small simulation study (Section 2), derives closed-form coordinate descent updates (Section 3), and reports a real-data comparison on a rat eye eQTL dataset (Section 4). The central empirical assertion is that SCAD and MCP can achieve the ground-truth sparsity level without sacrificing estimation accuracy, while Lasso cannot simultaneously satisfy both goals.

Significance. If the claims were fully supported, the paper would offer a useful, concise comparison with practical guidance on choosing between convex and non-convex penalties. The qualitative statements accord with the existing literature, and the coordinate descent formulas are standard. However, the paper provides no new theory, and its own empirical evidence is too incomplete to establish the central claim: the simulation lacks paired sparsity/MSE statistics, the non-convex optimization settings are unreported, and the real-data comparison relies on in-sample RSS. The manuscript is a reasonable pedagogical note, but the load-bearing quantitative assertions need substantial strengthening before the paper can be accepted as a standalone research contribution.

major comments (4)

[Section 2, after Figure 3] The claim that 'the SCAD and the MCP penalties can give us an estimator with ground truth sparsity level without sacrificing estimation accuracy' is not established by the displayed results. Figure 3 plots MSE and sparsity level separately against lambda; for Lasso the text explicitly identifies lambda=0.12 (minimum MSE) and lambda>0.35 (sparsity 10), but no lambda is identified for SCAD or MCP at which the average sparsity equals 10 and the MSE is simultaneously low. Because the two criteria are not paired at a single lambda, the curves only suggest that some intermediate lambda may look favorable. Please report a paired summary: for each method and each lambda, the average sparsity and MSE together, or identify a concrete lambda where both objectives are met, with standard errors across the 100 replications.
[Section 3, coordinate descent formulas] The closed-form updates for Lasso, SCAD, and MCP are not correctly normalized with respect to the objective in Eq. (1.1), which includes the factor 1/(2n). With L(beta)=||Y-Xbeta||^2, the coordinate-wise solution for Lasso is beta_j = S(X_j^T r, n lambda)/X_j^T X_j (or, if columns are standardized to unit L2 norm, S(X_j^T r, lambda)). The manuscript writes beta_j = S(X_j^T r, lambda) without stating the column normalization. For the real data, variables are standardized to zero mean and unit standard deviation, so X_j^T X_j is approximately n, making the displayed threshold inconsistent with Eq. (1.1). Please clarify the normalization used in the simulations and correct the formulas, since the lambda grid in Figure 3 depends on this scaling.
[Section 3, Figure 4 and Table 1] The empirical comparison of convergence is not reproducible because the optimization settings are unreported. The paper never states the SCAD shape parameter a, the MCP shape parameter a, the initialization, whether warm starts were used, the convergence tolerance, the maximum number of iterations, or the lambda grid. For non-convex objectives, coordinate descent can converge to different local minima depending on these choices; the unstable loss variation in Figure 4 may therefore reflect the algorithm's terminal points rather than a property of the penalty family. Please report all of these settings and consider a sensitivity check with multiple starting values or a comparison against a well-tested implementation such as ncvreg.
[Section 4, Figure 5] The real-data comparison uses the residual sum of squares (RSS) to conclude that SCAD 'performs better' than Lasso. Since RSS is an in-sample measure and the two methods have different sparsity levels under the selected lambdas, a lower RSS does not establish better predictive performance or better model selection. Please use out-of-sample prediction error, cross-validated deviance, or another criterion that accounts for model complexity. In addition, the real-data section omits MCP entirely, so it does not support the paper's stated three-way comparison.

minor comments (5)

[Section 2] The sentence 'the Lasso penalty always introduces a bias on its large components without sacrificing model selection consistency' is ambiguous and potentially misleading: model selection consistency of Lasso is known to require additional conditions such as the irrepresentable condition. Please rephrase to state the precise theoretical result.
[Section 1, Eq. (1.1)] The scaling 1/(2n) is introduced for the least-squares case, but L(beta) is defined generally as the negative log-likelihood. Please clarify whether Eq. (1.1) is meant to be generic or specific to the linear model.
[Section 3, Figure 4] The quantity 'Loss Variation' plotted in Figure 4 is not defined. Specify whether it is |Q(beta^(k)) - Q(beta^(k-1))|, a relative change, or something else.
[Throughout] There are several typographical issues: 'coordinate decent' should be 'coordinate descent', 'right penal' should be 'right panel', 'BardetBiedl' should be 'Bardet-Biedl', and 'Not surprised' should be 'It is not surprising'.
[References] The paper should cite relevant algorithmic references for coordinate descent with SCAD and MCP, such as Breheny and Huang (2011) and Mazumder et al. (2011), and the oracle property results of Fan and Li (2001) and Zhang (2010).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the comparison is expository, the penalty formulas are standard definitions, and no fitted parameter, self-citation chain, or definitional reduction forces the paper's conclusion.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The penalized likelihood objective (1.1) and the Lasso/SCAD/MCP penalty functions are standard definitions taken from the cited literature; the coordinate-descent update formulas in Section 3 are derived algebraically from those definitions, not fitted to the conclusion. The qualitative claims in Section 2 (Lasso shrinks large coefficients, SCAD/MCP flatten the penalty for large |t|) follow directly from the plotted derivative functions and are consistent with the externally cited results of Fan and Li (2001) and Zhang et al. (2010). The toy simulation in Model 1 is an illustration: no parameter is fitted to make SCAD/MCP look better, and no 'prediction' is produced from a fit that was calibrated on the same target. The central empirical assertion—that SCAD/MCP can give an estimator with ground-truth sparsity without sacrificing estimation accuracy—rests on visual comparison of the MSE and sparsity curves in Figure 3 rather than a paired statistic, and Section 3 omits the shape parameter a, initialization, and convergence tolerance for the non-convex coordinate-descent runs. These are transparency and evidentiary weaknesses, not circularity, because the curves are generated from the model and penalties rather than being defined to equal the conclusion. There are no load-bearing self-citations, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. The comparison is therefore not circular; the relevant risk is correctness and rigor of the empirical support, which is outside the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities and no fitted constants in the theoretical sense, but the experiments depend on unreported tuning constants, namely the shape parameter a for SCAD and MCP and the convergence tolerance, as well as standard sparse linear model assumptions. The oracle-property claims are imported from Fan and Li (2001) and Zhang et al. (2010) without verifying their conditions.

free parameters (4)

SCAD shape parameter a = not reported
SCAD penalty defined with a>2; the value used in simulation and real data analysis is never stated, so all SCAD results are conditional on an unknown choice.
MCP shape parameter a = not reported
MCP penalty defined with a>1; the chosen value affects thresholding and is not reported.
Regularization parameter lambda = simulation path 0.05 to 0.4; real data Lasso 0.146, SCAD 0.158
lambda is selected by cross-validation or scanned in Figure 3; the reported MSE/sparsity trade-off is conditional on this tuning parameter.
Convergence tolerance and maximum iterations = not reported
Table 1 convergence times (31s, 86s, 92s) depend on stopping criteria that are not described.

assumptions (4)

domain assumption The true coefficient vector is exactly sparse (10 non-zero coefficients in simulation)
Section 2 Model 1 sets beta1..beta10=1 and betaj=0 for j>10; the comparison and oracle-property discussion rely on exact sparsity.
domain assumption The data follow a linear model with i.i.d. normal errors and standardized predictors
Section 1 defines L(beta)=||Y-X beta||2 as the negative log-likelihood for normal data; the simulation and real-data analyses assume this model.
domain assumption Coordinate descent on the non-convex objectives reaches a relevant optimum
Section 3 applies coordinate descent to SCAD/MCP and reports unstable loss variation (Figure 4); no convergence theorem or initialization strategy is provided.
domain assumption The oracle properties of SCAD/MCP cited from Fan and Li (2001) and Zhang et al. (2010) apply to the simulation and real-data settings
Section 2 asserts reduced bias and oracle properties based on citations without verifying the required regularity conditions.

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Cite this review

Pith. "Pith review of A Short Note of Comparison between Convex and Non-convex Penalized Likelihood." pith.science (2026). https://pith.science/paper/I3YJARQZ

@misc{pith2026250207655,
  author       = {Pith},
  title        = {Pith review of: A Short Note of Comparison between Convex and Non-convex Penalized Likelihood},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3YJARQZ}},
  note         = {Machine review of arXiv:2502.07655}
}

read the original abstract

This paper compares convex and non-convex penalized likelihood methods in high-dimensional statistical modeling, focusing on their strengths and limitations. Convex penalties, like LASSO, offer computational efficiency and strong theoretical guarantees but often introduce bias in parameter estimation. Non-convex penalties, such as SCAD and MCP, reduce bias and achieve oracle properties but pose optimization challenges due to non-convexity. The paper highlights key differences in bias-variance trade-offs, computational complexity, and robustness, offering practical guidance for method selection. It concludes that the choice depends on the problem context, balancing accuracy

Figures

Figures reproduced from arXiv: 2502.07655 by the authors.

**Figure 2.** Derivative of some penalty functions. 2 Properties of Different Penalties The Lasso owns its popularity to its convexity and computational properties in statistical learning and modeling. However several authors such as Zou and Hastie (2005) and Zhao and Yu (2006) have pointed out in theory that the Lasso penalty always introduces a bias on its large components without sacrificing model selection consistency. The … view at source ↗

**Figure 3.** Left: MSE of different penalties under different λ Right: Sparsity level of different penalties under different λ. 3 Algorithms and Experimental Results For the SCAD and the MCP penalties, most of the authors do not talk much about algorithms they used for optimization, especially the performance of convergence, such as escape time and frequency of convergence. In this section I derive the formula of coordinate dece… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Loss Variation. • SCAD: S(z, λ) =    sgn(z)(|z| − λ)+, when |z| ≤ 2λ sgn(z)[(a − 1)|z| − aλ]/(a − 2), when 2λ < |z| ≤ aλ z, when |z| ≥ aλ • MCP: S(z, λ) =    sgn(z)(|z| − λ)+/(1 − 1/a), when |z| < aλ; z, when |z| ≥ aλ We still consider Model 1 in sectio…

**Figure 5.** Figure 5: Left: Sparsity level Comparison. Right: Residual sum of squares comparison. References Candes, E., Tao, T., et al. (2007). The dantzig selector: Statistical estimation when p is much larger than n. The annals of Statistics, 35(6):2313–2351. Chen, J., de Hoogh, K., Gull…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Candes, E., Tao, T., et al. (2007). The dantzig selector: Statistical estimation when p is much larger than n. The annals of Statistics , 35(6):2313--2351

work page 2007

[2] [2]

A., Katsouyanni, K., et al

Chen, J., de Hoogh, K., Gulliver, J., Hoffmann, B., Hertel, O., Ketzel, M., Bauwelinck, M., Van Donkelaar, A., Hvidtfeldt, U. A., Katsouyanni, K., et al. (2019). A comparison of linear regression, regularization, and machine learning algorithms to develop europe-wide spatial models of fine particles and nitrogen dioxide. Environment international , 130:104934

work page 2019

[3] [3]

Ci, H., Liu, M., Pan, X., Zhong, F., and Wang, Y. (2023a). Proactive multi-camera collaboration for 3d human pose estimation. arXiv preprint arXiv:2303.03767

work page Pith review arXiv 2023

[4] [4]

Ci, H., Wang, C., Ma, X., and Wang, Y. (2019). Optimizing network structure for 3d human pose estimation. In Proceedings of the IEEE/CVF international conference on computer vision , pages 2262--2271

work page 2019

[5] [5]

Ci, H., Wu, M., Zhu, W., Ma, X., Dong, H., Zhong, F., and Wang, Y. (2023b). Gfpose: Learning 3d human pose prior with gradient fields. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 4800--4810

work page 2023

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Cui, S., Guo, X., Li, R., Yang, S., and Zhang, Z. (2024a). Double-estimation-friendly inference for high dimensional misspecified measurement error models. arXiv preprint arXiv:2409.16463

work page Pith review arXiv 2024

[7] [7]

Cui, S., Guo, X., and Zhang, Z. (2024b). Estimation and inference in ultrahigh dimensional partially linear single-index models. arXiv preprint arXiv:2404.04471

work page Pith review arXiv 2024

[8] [8]

Cui, S., Li, D., Li, R., and Xue, L. (2024c). Hypothesis testing for high-dimensional matrix-valued data. arXiv preprint arXiv:2412.07987

work page Pith review arXiv 2024