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REVIEW 1 major objections 2 minor 59 references

Relax and Follow: L0-Path Computation with L0-Bregman Relaxations

T0 review · 1 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read L0PathBrex computes L0 solution paths by relaxing the objective with Bregman functions that preserve global minimizers and support warm-start tracking.

desk verdict L0PathBrex turns B-rex relaxations into a warm-start path algorithm that outperforms baselines on sparse regression tasks. read the letter →

arxiv 2606.02176 v1 pith:LNA2TU75 submitted 2026-06-01 math.OC

Mhamed Essafri , Luca Calatroni , Emmanuel Soubies This is my paper · ORCID
classification math.OC
keywords L0regularizationsolutionpathBregmanrelaxationsparseregressionwarm-startleastsquareslogistic
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

The paper presents L0PathBrex as a method to estimate the solution path of L0-regularized problems. It relies on L0-Bregman relaxations that reformulate the discrete L0 penalty as a continuous objective while keeping the original global minimizers. These relaxations remove some unwanted local minima, so standard numerical solvers can be applied. The method uses warm starts to generate candidate minimizers and determines the intervals of the regularization parameter where each candidate stays valid. Experiments on sparse least-squares and logistic regression show better performance than existing approaches on both synthetic and real data.

What carries the argument

L0 Bregman relaxations (B-rex): continuous reformulations of the L0 objective that preserve global minimizers while eliminating part of the undesirable local minima.

What would settle it

A sparse least-squares instance where the true global L0 minimizer for some regularization value is never recovered as a local minimizer of any B-rex relaxation would show the method misses solutions.

Watch

Extended reading notes

Core claim

L0PathBrex dynamically constructs a collection of local minimizers that are candidates for the L0-solution path. It does so by solving L0 Bregman relaxations with any numerical solver, exploiting warm-start strategies, and identifying ranges of the regularization parameter for which each minimizer remains valid under the corresponding relaxation.

Load-bearing premise

L0 Bregman relaxations preserve the global minimizers of the original L0 problem and allow warm-start strategies to trace valid candidates across regularization values.

Editorial extensions

If this is right

  • The method applies directly to non-quadratic data-fidelity terms such as logistic or Poisson regression.
  • Any off-the-shelf numerical solver for the relaxed problem can generate the path.
  • The approach identifies explicit intervals of the regularization parameter on which each local minimizer remains valid.
  • Systematic gains over state-of-the-art baselines appear on both synthetic and real-world sparse regression tasks.

Reading between the lines

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

  • The same relaxation-plus-warm-start pattern could be tested on other combinatorial penalties such as group sparsity or rank constraints.
  • Because the method re-uses existing solvers, it may enable path computation on problem sizes where branch-and-bound or combinatorial search becomes prohibitive.
  • Different choices of the B-rex family could be compared systematically on Poisson regression to see which family yields the most accurate paths.
  • The identified validity intervals for each minimizer might support adaptive selection of the regularization parameter without exhaustive grid search.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper introduces L0PathBrex, a method for estimating the solution path of L0-regularized problems via L0-Bregman relaxations (B-rex). These provide continuous reformulations applicable to non-quadratic fidelity terms that preserve global minimizers while removing some undesirable local minima. Given any numerical solver, the approach dynamically constructs candidate local minimizers for the L0 path using warm-starts and identifies valid regularization-parameter ranges. Experiments on sparse least-squares, logistic regression, and Poisson regression tasks show systematic outperformance over state-of-the-art baselines on synthetic and real-world datasets, with additional analysis of B-rex choice in the Poisson setting.

Significance. If the empirical outperformance holds under rigorous validation, the work offers a practical extension of prior B-rex literature to L0 path computation for non-quadratic losses, potentially improving efficiency and solution quality in sparse regression. The dynamic warm-start construction and sensitivity analysis to B-rex functions add concrete value for applied optimization.

major comments (1)
  1. [Experiments] Experiments section: the central claim of systematic outperformance (abstract and §4) asserts superiority across metrics and datasets but provides no details on error bars, statistical significance tests, data exclusion rules, or the precise implementation of the dynamic construction and warm-start strategy. This is load-bearing for the empirical contribution and prevents assessment of robustness.
minor comments (2)
  1. Abstract: the phrase 'various evaluation metrics' is vague; explicitly naming the primary metrics (e.g., support recovery, prediction error) would improve clarity.
  2. Notation: ensure consistent use of the B-rex family parameters across sections describing the relaxation and the path-construction algorithm.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment regarding the Experiments section. We will revise the manuscript to address the lack of implementation and statistical details.

read point-by-point responses

  1. Referee: [Experiments] Experiments section: the central claim of systematic outperformance (abstract and §4) asserts superiority across metrics and datasets but provides no details on error bars, statistical significance tests, data exclusion rules, or the precise implementation of the dynamic construction and warm-start strategy. This is load-bearing for the empirical contribution and prevents assessment of robustness.

    Authors: We agree that these details are necessary for assessing robustness. In the revised manuscript we will: (i) report error bars computed from 10–20 independent runs with different random seeds for all synthetic and real-world experiments; (ii) add statistical significance tests (paired t-tests or Wilcoxon signed-rank tests with p-values) comparing L0PathBrex against each baseline on the primary metrics; (iii) explicitly state that no data points were excluded beyond standard preprocessing steps described in §4; and (iv) include a new subsection (or appendix) with pseudocode and parameter settings that fully specify the dynamic local-minimizer construction, warm-start procedure, and range-identification logic. These additions will be placed in §4 and will not alter any numerical results. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's central contribution is the L0PathBrex algorithm for constructing candidate L0-solution paths via warm-started numerical solves on B-rex relaxations. The key properties (preservation of global minimizers, removal of some local minima) are explicitly attributed to 'recently introduced and analyzed' prior literature rather than derived or fitted inside this manuscript. No equations reduce a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from the same authors' prior work to force the method, and the experimental outperformance claims rest on new comparisons rather than self-referential renaming or ansatz smuggling. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on properties of L0-Bregman relaxations from prior work and the assumption that warm-start strategies can track validity ranges; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • domain assumption L0 Bregman relaxations preserve global minimizers while eliminating undesirable local minima
    Invoked in the abstract as the key property enabling the continuous reformulation and dynamic construction approach.

how reviews work

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Cite this review

Pith. "Pith review of Relax and Follow: L0-Path Computation with L0-Bregman Relaxations." pith.science (2026). https://pith.science/paper/LNA2TU75

@misc{pith2026260602176,
  author       = {Pith},
  title        = {Pith review of: Relax and Follow: L0-Path Computation with L0-Bregman Relaxations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LNA2TU75}},
  note         = {Machine review of arXiv:2606.02176}
}
read the original abstract

This work introduces L0PathBrex, a novel method for estimating the solution path of L0-regularized problems through the use of L0 Bregman relaxations (B-rex). Recently introduced and analyzed in the literature, these relaxations provide continuous reformulations of the original objective, are applicable to possibly non-quadratic data fidelity terms, and depend on a family of functions designed to preserve the global minimizers while eliminating part of the undesirable local minima. Given any numerical solver for the relaxation, the proposed approach dynamically constructs a collection of local minimizers that are candidates for the L0-solution path. It exploits warm-start strategies and identifies ranges of the regularization parameter for which each minimizer remains valid under the corresponding relaxation. Experiments on sparse least-squares and logistic regression problems demonstrate that L0PathBrex systematically outperforms state-of-the-art baselines across both synthetic and real-world datasets in terms of various evaluation metrics; additionally, the study investigates how the choice of the B-rex affects the quality of the estimated path in the sparse Poisson regression setting.

Figures

Figures reproduced from arXiv: 2606.02176 by the authors.

Figure 1
Figure 1. The ℓ0-curve. Plot of the affine functions λ0 7→ (Fy(Axˆ) + λ2 2 ∥xˆ∥ 2 ) + λ0∥xˆ∥0 for all fifteen (strict local) minimizers xˆ of a ℓ0-regularized least-squares problem (Fy(A·) = 1 2 ∥A·−y∥ 2 , λ2 = 0) of size (M, N) = (3, 4). The ℓ0-curve corresponds to the lower concave envelope of these affine functions, and is represented by the solid line. It is piecewise linear, with breakpoints (λˆ1 0 , λˆ2 0 , λˆ3 0 ) that… view at source ↗
Figure 2
Figure 2. The ℓ0-curve computed using B-rex. Plots of the segments corresponding to strict local minimizers of J0 preserved by the exact relaxation JΨ. Each segment is supported on an interval [λ0 (xˆ), λ¯ 0(xˆ)] given by Theorem 2. This graph is the counterpart of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. L0PathBrex-A applied to the ℓ0-regularized least-squares problem illustrated in Fig￾ures 1 and 2. Here, a single forward and backward pass is performed (Npass = 1), with k max = 3 and ρ = 0.98. Each subplot corresponds to a step of the algorithm. In the four top plots, solid colored segments represent the current points in X , while the purple cross marks the initially selected point x0 at λ0 = ρλ0 (x0) for the forw… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: LS. Average F1-score (top) and RMSE (bottom) as functions of the number of sam [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: LS. Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Estimated Pareto fronts for the LS problem on the four real datasets from Table [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: LR. Average F1-score (top) and RMSE (bottom) as functions of the number of samples, [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: LR. Same as in Figure [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Estimated Pareto fronts for the LR problem on the two real datasets from Table [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]
Figure 13
Figure 13. Figure 13: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.