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Proximal Iteration for Nonlinear Adaptive Lasso

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Treating adaptive Lasso penalty strengths as decision variables yields one proximal-gradient algorithm for debiased variable selection with any smooth likelihood and sparsity structure.

desk verdict A genuinely useful framework for debiased structured sparsity with a new proximal operator, but Theorem 2 is numerically wrong as stated (swapped inequality) and needs correction before the method can be trusted. read the letter →

arxiv 2412.05726 v1 pith:JAHUTQEX submitted 2024-12-07 stat.ML cs.LG

Nathan Wycoff , Lisa O. Singh , Ali Arab , Katharine M. Donato This is my paper · ORCID
classification stat.MLcs.LG MSC 62J0765K0562F12
keywords variableselectionproximalalgorithmsadaptiveLassostructuredsparsitypenalizedregressionoraclepropertyMAPestimationnon-Gaussian
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

Adaptive Lasso shrinks each coefficient with its own penalty weight, but choosing those weights is a bottleneck in complex models. This paper treats the weights themselves as decision variables, placing a prior on them and estimating them jointly with the coefficients in a maximum-a-posteriori sense. The key move is a closed-form proximal operator for the two-dimensional function $\lambda|\beta|$, which makes joint optimization by proximal gradient tractable and lets any sparsity structure be encoded through a smooth prior on $\lambda$ rather than through a bespoke nonsmooth penalty. The authors prove an oracle property for the resulting estimator and show empirically that it matches or beats specialized packages on Gaussian and non-Gaussian regressions while scaling to large nonlinear applications such as vaccine-behavior and migration models.

What carries the argument

The load-bearing object is the proximal operator — the map sending a point to the minimizer of the function plus a quadratic proximity term — applied to the variable-penalty norm $g(\beta,\lambda)=\lambda|\beta|$ and its log-augmented variant $\lambda|\beta|+a\log\lambda$. Because the MAP objective separates across coordinates, the proximal problem reduces to a two-dimensional scalar problem, and the paper gives its closed-form solution: for $s_\beta s_\lambda<1$, the optimal $\lambda^*$ is $\lambda_0$ when $\lambda_0\ge|\beta_0|/s_\beta$, and otherwise $(\lambda_0-s_\lambda|\beta_0|)_+/(1-s_\lambda s_\beta)$, with $\beta^*=(|\beta_0|-s_\beta\lambda^*)_+\operatorname{sgn}(\beta_0)$; the log term replaces this with a square-root formula that keeps $\lambda$ strictly positive. This operator lets a single proximal-gradient loop optimize $\beta$ and $\lambda$ together, and because sparsity structure lives in the smooth prior $p_\lambda$, the same operator handles arbitrary structure.

What would settle it

Take a small negative-binomial regression with correlated predictors, run the paper's proximal-gradient algorithm from many random starting points with its default step-size schedule, and check whether all runs settle on the same coefficients at a point where no small perturbation improves the joint objective; disagreement or non-stationary endpoints would falsify the claim that one algorithm reliably handles arbitrary smooth likelihoods.

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Extended reading notes

Core claim

The central claim is that adaptive Lasso can be learned end to end: instead of preselecting penalty coefficients $\lambda_p$ with a pilot estimator or iteratively reweighting them, one minimizes $$\min_{\$\beta$,\$\lambda$} L(\$\beta$)+\sum_p(\tau\lambda_p|\beta_p|-\log\lambda_p)-\log p_\$\lambda$(\$\lambda$)$$ over coefficients and penalty strengths jointly. The authors establish a closed-form proximal operator for the nonconvex term $\lambda|\beta|$, including a log-barrier variant, so proximal gradient iterations can update $\beta$ and $\lambda$ together. They prove that with an independent sufficiently diffuse prior the estimator has the oracle property: it recovers the zero pattern with probability tending to one and is asymptotically normal with the same covariance as if the zero coefficients were known. The same machinery, they argue, imposes group, overlapping-group, or hierarchical sparsity simply by choosing a smooth hyperprior on $\lambda$, shifting structure from the nonsmooth to the smooth part of the objective. Numerical studies on synthetic and real data and two social-science applications are offered as evidence that one algorithm is competitive with structure-specific implementations.

Load-bearing premise

The whole method depends on the iterative optimization landing on a good solution rather than stalling or drifting, and the paper does not verify that its chosen step sizes satisfy the conditions its convergence guarantee requires.

Editorial extensions

If this is right

  • A single proximal operator replaces the bespoke prox computations for group Lasso, overlapping group Lasso, and hierarchical sparsity; users can impose new structures by writing a smooth prior.
  • For large-data nonlinear models the method scales to objects such as the hurdle gravity model with about two hundred thousand candidate interaction terms, by building the interactions on the fly inside minibatches.
  • The oracle property means that, asymptotically, the estimator selects the zero pattern correctly and has the same limiting distribution for nonzero coefficients as an oracle that knew the pattern.
  • On synthetic non-Gaussian regressions and eight benchmark datasets the method is competitive in accuracy and usually faster than specialized packages, and it also handles Cauchy and negative-binomial likelihoods that those packages do not.
  • Active coefficients are less biased than under a fixed-penalty Lasso because each $\lambda_p$ can shrink to a small value for strong signals.

Reading between the lines

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

  • If the convergence gap is closed, this framework could become a default black-box sparse-estimation routine, since the same proximal operator would serve any likelihood and any sparsity structure; a testable extension is to compare it against bespoke solvers on graph- and tree-structured sparsity benchmarks.
  • The same variable-coefficient prox idea may extend to other penalties such as SCAD, MCP, or bridge penalties by letting their tuning parameters vary; closed forms for those proximal operators would generalize the approach beyond Lasso-type terms.
  • Numerical evidence that structured priors can improve prediction even when the effect-hierarchy principle is weak suggests that group assumptions in complicated likelihoods are best chosen by validation rather than by prior belief.
  • With the log term removed, the bare $\lambda|\beta|$ proximal operator exhibits the possibility of driving $\lambda$ to zero, hinting at a formulation that induces sparsity directly in penalty strengths as well as coefficients.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a unified proximal-gradient framework for adaptive Lasso problems in which the per-coordinate penalty coefficients λ are treated as decision variables and optimized jointly with the regression coefficients β. The central objects are closed-form proximal operators for the variable-penalty L1 function λ|β| (Theorem 2) and for its log-penalized variant λ|β| + a log λ (Theorem 5). The authors state oracle properties for the resulting estimator, introduce priors that encode group and overlapping-group sparsity structure, and evaluate the method on synthetic data, UCI benchmarks, and two social-science case studies. The main claim is that this one-algorithm-fits-all approach is competitive with, and often faster than, specialized solvers such as glmnet and MLGL across Gaussian, Bernoulli, Negative Binomial, and Cauchy likelihoods.

Significance. If the central construction were correct, the paper would make a useful contribution: moving sparsity structure from the nonsmooth penalty into a smooth prior while keeping a single proximal operator is an attractive idea, and the empirical scope is broad, including non-Gaussian likelihoods, structured sparsity, real data, and large-scale case studies. The authors provide code and appear to have run a substantial battery of comparisons. However, the load-bearing proximal operator is stated incorrectly in Theorem 2, and several theoretical statements either are unverified or appear false as written. These issues affect the correctness of the algorithm actually deployed, not merely the presentation, so the current version cannot be accepted. With careful correction of the proximal operators and a honest restatement of the convergence and oracle-property results, the framework could be a valuable addition to the sparse-regression toolbox.

major comments (4)
  1. [Section 3.1, Theorem 2] The sβsλ ≥ 1 branch is incorrect as stated. The theorem gives λ* = λ0 when λ0√sλ > |β0|√sβ and λ* = 0 otherwise, but the proof in Appendix A compares the costs λ0²/(2sλ) and β0²/(2sβ) and therefore yields the opposite placement of sβ and sλ: λ* = λ0 when |β0|√sλ < λ0√sβ. A concrete counterexample: with sβ = 4, sλ = 1, β0 = 1, λ0 = 1.5, the theorem's condition is false (1.5 > 2), so it predicts λ* = 0 with proximal cost 1.125, whereas (β*,λ*) = (0,1.5) has proximal cost 0.125 and is the true minimizer. Since the algorithm in Section 3.3 applies exactly this proximal operator, the central construction is not correctly specified as written. Remark 3 reproduces the same incorrect branch and should be corrected together with the theorem.
  2. [Section 3.2, Theorem 5] The log-term proximal operator has several apparent errors. In the zero branch, Eq. (12) and the surrounding condition use λ0² where the stationarity equation gives λ0; the correct λ* at β = 0 is (λ0 + sqrt(λ0² + 4 sλ a))/2, not (λ0² + sqrt(λ0² + 4 sλ a))/2. The condition should also involve |β0|, not β0, since the theorem is stated for arbitrary real β0. In addition, the theorem is written for the objective λ|β| + a log λ, while the algorithm in Section 3.3 applies the prox of λ|β| − (1/τ) log λ; with the implemented sign, a is negative and the square root in the formulas can fail to be real. The relation between a and τ and the sign convention must be clarified and the formulas re-derived for the objective actually used in the experiments.
  3. [Section 3.3.1, Convergence] The convergence discussion invokes Li and Pong (2015), Theorem 5, but none of the hypotheses are verified for the objectives in this paper. The theorem requires a twice continuously differentiable convex q such that ∇²h + ∇²q has spectrum bounded by l, and a step size below 1/l; the paper does not establish such a bound for any of the negative log-likelihoods or priors used in Section 6. Moreover, the actual algorithm uses Adam-based step sizes and stochastic minibatches, for which the deterministic proximal-gradient result does not directly apply. The text itself concedes that the theorem does not imply the existence of cluster points. This leaves the main algorithmic claim—that the proximal iteration reliably finds a useful stationary point—without theoretical support, and the gap is load-bearing for the 'one-algorithm-fits-all' statement in Section 7.1.
  4. [Section 4, Theorem 8] Part 2 of Theorem 8 is not established by the given proof and appears to be false under the paper's own assumptions. The proof states that both |β| and τλ* are decreasing in |β|, but the first term is increasing. For the half-Cauchy prior used in Section 5.1, one can compute the derivative of |β| + g'_{τn}(|β|) at |β| = 0 as 1 − τn²/(ρ'(1)² + ρ''(1)) = 1 − τn² (with the paper's normalization), which is negative for the linearly growing τn = nτ0 used in Theorem 12. Hence the minimum is not at β = 0 for large n. The proof of the oracle property in Appendix B is also a sketch: the zero-coordinate result is asserted after a sign-of-derivative argument, and the expansion mixes terms of different orders in √N without a complete justification. Since the oracle property is one of the paper's stated contributions, this needs a correct statement and a complete proof.
minor comments (5)
  1. [Throughout] There are numerous typos, including 'Deonte' for 'Denote' in Section 1, 'wieghts' in Section 2.1, and 'sythetic' in the abstract; the paper would benefit from a careful proofreading pass.
  2. [Figure 2 and Section 3.1] The notation is inconsistent: the main text uses sβ and sλ, while the figure caption and the reduced form in Eq. (11) use sx and sλ (or sxsλ); unify the notation.
  3. [Section 5.2] The notation pγ = C(0,1)+ and λi|γg(i) ∼ N(γg(i), 1/√N)+ is nonstandard; please define the half-Cauchy and truncated-normal conventions explicitly, including which side is truncated.
  4. [Lemma 7 and Theorem 8] Lemma 7, part 4, contains an n in the denominator that is absent from the corresponding statement in Theorem 8; the subscripted τn and τ notation should be made consistent throughout Section 4.
  5. [Appendix B and theorem numbering] The oracle property is stated as Theorem 12 in the main text but is labeled Theorem 9 in Appendix B; align the numbering and the assumptions between the two statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proximal-operator and oracle-property derivations are self-contained, and the only self-citations are data-source references that do not carry the load.

full rationale

The paper's derivation chain is not circular. The central construction, the closed-form proximal operators in Section 3.1 and Section 3.2, is derived from the definition of the proximal cost (P1 and P2) using nested optimization and the known soft-thresholding operator; it does not assume the target proximal mapping as an input. The statistical claims in Section 4 likewise start from an explicit profiling identity (Eq. 14-15) and then prove properties of the resulting penalty via implicit differentiation and a Fan-Li-style argument, rather than importing the oracle property as an assumption. The reweighted-L1 connection in Section 2.4 is presented as an a posteriori interpretation, not as a premise for the main results. Numerical comparisons are run against external baselines (glmnet, MLGL, unpenalized regression), so the empirical claims are not forced by construction. The only self-citations are to Singh et al. (2024) for the vaccination survey data, which is a data provenance reference and is not load-bearing for the method's derivation. The reviewer-flagged algebraic discrepancy in Theorem 2's stated condition is a correctness concern about the closed form, not a circularity: a miscalculated expression can still be derived from the stated cost rather than assumed from the conclusion. The paper also explicitly limits its convergence claims in Section 3.3.1, which further supports a non-circular reading. Overall, no step reduces by definition to its own inputs, so no circularity is found.

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

No new physical or model entities are introduced; the free parameters are hyperparameters of the priors and optimizer. The main unverified premises are the convergence conditions for the nonconvex algorithm and the regularity conditions for the oracle property.

free parameters (4)
  • Global penalty strength tau = 0.025N (independent), 0.015N (structured)
    Chosen by hand per experiment (Section 6.1.2), not selected by a data-driven criterion. Reported results depend on this choice.
  • Group prior variance scale = 1/sqrt(N)
    Variance of lambda around group center in hierarchical group prior. The paper states 'We found empirically that a factor of 1/sqrt(N) provided the right balance' (Section 5.2).
  • Softmax temperature for overlapping groups = sqrt(P)
    Temperature in softmax approximation to min over groups. The paper states 'We found empirically that using a temperature of sqrt(P) in the softmax lead to better performance' (Section 5.3).
  • Adam learning rate = 1e-2
    Optimizer step size used in all experiments (Section 6.1.1). Affects convergence but is not part of the statistical model.
assumptions (4)
  • standard math The proximal operator concept extends to nonconvex functions.
    Implicitly used in defining the proximal operator for lambda|beta|, following Hare and Sagastizabal (2009) and related work cited in Section 3.1.
  • domain assumption The conditions of Li and Pong (2015) Theorem 5 hold for the proposed proximal gradient iteration.
    Invoked in Section 3.3.1 without verifying that the smooth part of the objective has Hessian with bounded spectrum or that the Adam step sizes satisfy the required bound. The paper admits 'We suspect that more precise statements could be made'.
  • domain assumption The likelihood satisfies standard regularity conditions (identifiability, bounded third derivatives, positive definite Fisher information).
    Assumed in Theorem 12 and Appendix B. Not verified for the Cauchy, Negative Binomial, or hurdle likelihoods used in the experiments.
  • domain assumption The prior density on lambda is decreasing with bounded logarithmic derivative.
    Required by Theorem 8 and Theorem 12. Half-Cauchy satisfies this, but the structured group priors used in experiments (truncated Normal with group centers) are not shown to satisfy these assumptions.

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Cite this review

Pith. "Pith review of Proximal Iteration for Nonlinear Adaptive Lasso." pith.science (2026). https://pith.science/paper/JAHUTQEX

@misc{pith2026241205726,
  author       = {Pith},
  title        = {Pith review of: Proximal Iteration for Nonlinear Adaptive Lasso},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAHUTQEX}},
  note         = {Machine review of arXiv:2412.05726}
}
abstract

Augmenting a smooth cost function with an $\ell_1$ penalty allows analysts to efficiently conduct estimation and variable selection simultaneously in sophisticated models and can be efficiently implemented using proximal gradient methods. However, one drawback of the $\ell_1$ penalty is bias: nonzero parameters are underestimated in magnitude, motivating techniques such as the Adaptive Lasso which endow each parameter with its own penalty coefficient. But it's not clear how these parameter-specific penalties should be set in complex models. In this article, we study the approach of treating the penalty coefficients as additional decision variables to be learned in a \textit{Maximum a Posteriori} manner, developing a proximal gradient approach to joint optimization of these together with the parameters of any differentiable cost function. Beyond reducing bias in estimates, this procedure can also encourage arbitrary sparsity structure via a prior on the penalty coefficients. We compare our method to implementations of specific sparsity structures for non-Gaussian regression on synthetic and real datasets, finding our more general method to be competitive in terms of both speed and accuracy. We then consider nonlinear models for two case studies: COVID-19 vaccination behavior and international refugee movement, highlighting the applicability of this approach to complex problems and intricate sparsity structures.

Figures

Figures reproduced from arXiv: 2412.05726 by the authors.

Figure 1
Figure 1. Left: The function g(β, λ) = λ|β|. Subsequently: The proximal cost of as a function of β and λ (center) and marginal for λ (right) with λ0 = β0 = 1; sλ = sβ = 2. 3 Joint Nonsmooth Optimization This section studies the joint optimization of Problem 2 via proximal gradient methods. We give only brief outlines of proofs in this section; see Appendix A for details. 3.1 The Variable-Penalty ℓ1 Proximal Problem Consider t… view at source ↗
Figure 2
Figure 2. The Action of the Proximal Operator: Plots of the reduced proximal operator (Eq 11) for various fixed b := sxsλ < 1 and with λ0, |x0| sx ∈ (0, 2). Values b = sxsλ ∈ {0.1, 0.35, 0.65, 0.99} are shown left to right. Remark 4. Since when sxsλ > 1 the λ ∗ is either 0 or λ0, we will focus on the case where sxsλ < 1. Then, let a := |x0| sx and b := sxsλ yielding a function of just three variables: λ(λ0, a, b) = ( λ0 λ0 ≥ … view at source ↗
Figure 3
Figure 3. Comparison on synthetic data with independent sparsity. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Comparison on synthetic data with group sparsity. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Comparison on synthetic data with hierarchical sparsity. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Comparison on real datasets in terms of prediction accuracy. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Regularized Deep Active Subspace reveals two vaccination hesitant groups. Plots show the active subspace projection of the training data Left: without penalty Mid: with a penalty strength of τ = 10. Right: 10 fold Cross Validation reveals that the penalty improves pred…
Figure 8
Figure 8. Figure 8: International migration case study. Left: Regression coefficient trajectory versus regularization strength τ . Right: Predictive NLL versus τ ; lower is better. this may seem a priori reasonable. But our study takes place amidst a contemporary legal-political context i…
Figure 9
Figure 9. Figure 9: Inverse Regressions for the high penalty (left) and low penalty (right) settings. [PITH_FULL_IMAGE:figures/full_fig_p035_9.png]

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Pith tools

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