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On Coreset for LASSO Regression Problem with Sensitivity Sampling

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
LASSO regressionsampling algorithm
Abstract:

In this paper, we study coreset construction for LASSO regression, where a coreset is a small, weighted subset of the data that approximates the original problem with provable guarantees. For unregularized regression problems, sensitivity sampling is a successful and widely applied technique for constructing coresets. However, extending these methods to LASSO typically requires coreset size to scale with O(\mathcal{G}d), where d is the VC dimension and \mathcal{G} is the total sensitivity, following existing generalization bounds. A key challenge in improving upon this general bound lies in the difficulty of capturing the sparse and localized structure of the function space induced by the \ell_1 penalty in LASSO objective. To address this, we first provide an empirical process-based method of sensitivity sampling for LASSO, localizing the procedure by decomposing the functional space into independent spaces, which leads to tighter estimation error. By carefully leveraging the geometric properties of these localized spaces, we establish tight empirical process bounds on the required coreset size. These techniques enable us to achieve a coreset of size \tilde{O}(\epsilon^{-2}d\cdot((\log d)^3\cdot\min(1,\log d/\lambda^2)+\log(1/\delta))), which ensures a (1\pm\epsilon)-approximation for any \epsilon,\delta\in(0,1) and \lambda > 0. Furthermore, we give a lower bound showing that any algorithm achieving a (1+\epsilon)-approximation must select at least \Omega(\frac{d\log{d}}{\epsilon^2}) rows in the regime where \lambda=O(d^{-1/2}). Empirical experiments show that our proposed algorithm is at least 4 times faster than the existing LASSO solver and more than 9 times faster on half of the datasets, while ensuring high solution quality and sparsity.

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Overview

Overall Novelty Assessment

The paper develops a localized empirical process method for constructing coresets in LASSO regression, aiming to improve upon the standard O(Gd) coreset size bound by decomposing the functional space into independent localized regions. It resides in the 'LASSO and L1-Regularized Regression' leaf of the taxonomy, which contains only four papers total, indicating a relatively sparse research direction within the broader coreset literature. This leaf sits under 'Regularized Regression Coresets,' distinguishing L1-penalized methods from ridge regression and general norm-based approaches, suggesting the paper targets a specialized but well-defined niche.

The taxonomy reveals neighboring leaves focused on ridge regression coresets and general norm-based regularization, alongside broader branches covering sensitivity sampling, determinantal point processes, and distributed learning applications. The paper's localization strategy appears to bridge theoretical foundations—such as empirical process bounds—with algorithmic sampling techniques, positioning it at the intersection of 'Sampling and Sketching Techniques' and 'Theoretical Foundations and Complexity.' The scope notes clarify that LASSO-specific methods must exploit L1 geometry rather than generic optimization frameworks, which the paper explicitly addresses through its decomposition approach.

Across three contributions, the analysis examined 27 candidate papers with zero refutable pairs identified. The localized empirical process method (8 candidates examined, 0 refutable), improved size bound (9 candidates, 0 refutable), and matching lower bound (10 candidates, 0 refutable) all appear free of direct prior overlap within this limited search scope. The absence of refutations suggests either genuine novelty or that the top-30 semantic matches did not capture closely related theoretical work. The improved bound and lower bound contributions show slightly broader examination (9–10 candidates) yet remain unrefuted, hinting at potential originality in the theoretical analysis.

Based on the limited search of 27 candidates drawn from top-K semantic similarity, the work appears to introduce fresh theoretical machinery for LASSO coresets, particularly through localized space decomposition. However, the sparse taxonomy leaf and modest search scope mean the analysis cannot rule out relevant prior work outside the examined set. The combination of algorithmic innovation and matching lower bounds suggests substantive contributions, though a more exhaustive literature review would strengthen confidence in the novelty claims.

Taxonomy

Core-task Taxonomy Papers
16
3
Claimed Contributions
27
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: coreset construction for LASSO regression. The field centers on building compact, weighted subsets of data that approximate the solution to L1-regularized regression problems while preserving statistical and computational guarantees. The taxonomy reveals several main branches: Regularized Regression Coresets focuses on methods tailored to sparsity-inducing penalties such as LASSO and related L1 formulations, often leveraging sensitivity sampling or leverage-score techniques (e.g., Regularized Regression Coresets[1], Sensitivity Sampling[16]). Sampling and Sketching Techniques encompasses randomized dimension-reduction and subsampling strategies—ranging from classical leverage-score sampling (L1 Regression Sampling[6]) to more recent sketching constructions (Householder Sketch[7])—that trade off approximation quality and runtime. Application Domains and Extensions addresses specialized settings such as streaming data (Streaming Regression Coresets[4]), fairness constraints (Fair Regression Coresets[5]), and distributed or sparse classification scenarios (Distributed Sparse Classification[3]). Theoretical Foundations and Complexity investigates lower bounds, unconditional coreset existence (Unconditional Coresets[10]), and complexity analyses for constrained Lp regression (Constrained Lp Regression[11]). Finally, Survey and Cross-Domain Applications collects broader perspectives on interpretability (Interpretable ML Survey[13]) and emerging paradigms like quantum-classical hybrids (Quantum Classical ML[15]). A particularly active line of work explores how negative dependence and advanced sampling schemes can tighten coreset size bounds (Negative Dependence Coresets[2]) or accelerate solver convergence (Fast LMS Solvers[8], Weighted SGD Regression[9]). Trade-offs between sample complexity, approximation error, and computational overhead remain central: some methods prioritize provable worst-case guarantees, while others optimize empirical performance in streaming or distributed regimes. LASSO Sensitivity Sampling[0] sits squarely within the Regularized Regression Coresets branch, emphasizing sensitivity-based importance sampling tailored to L1 penalties. Its approach aligns closely with classical sensitivity frameworks (Sensitivity Sampling[16]) yet contrasts with more general constrained-regression techniques (Constrained Lp Regression[11]) by exploiting the specific geometry of LASSO. Compared to neighboring works like Weighted SGD Regression[9], which blends stochastic optimization with coreset ideas, LASSO Sensitivity Sampling[0] focuses more directly on constructing a static, reweighted subset that faithfully approximates the regularized objective.

Claimed Contributions

Localized empirical process method for LASSO coreset construction

The authors introduce a localized empirical process framework that decomposes the LASSO function space into independent residual and l1 penalty components. This decomposition enables tighter bounds on Gaussian diameter and metric entropy, improving upon standard union-bound-based epsilon-net methods for coreset construction.

8 retrieved papers
Improved coreset size bound for LASSO regression

The authors establish a coreset of size Õ(ε^−2 d · ((log d)^3 · min{1, log d/λ^2} + log(1/δ))) that achieves (1 ± ε)-approximation for LASSO regression. This bound improves upon the standard Õ(Gd/ε^2) bound from existing generalization theory by carefully leveraging geometric properties of localized spaces.

9 retrieved papers
Matching lower bound for LASSO coreset size

The authors prove an information-theoretic lower bound showing that any algorithm achieving (1+ε)-approximation for LASSO must use at least Ω(d log d / ε^2) rows when λ = O(d^−1/2). This demonstrates that their upper bound is nearly tight up to polylogarithmic factors in this regime.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Localized empirical process method for LASSO coreset construction

The authors introduce a localized empirical process framework that decomposes the LASSO function space into independent residual and l1 penalty components. This decomposition enables tighter bounds on Gaussian diameter and metric entropy, improving upon standard union-bound-based epsilon-net methods for coreset construction.

Contribution

Improved coreset size bound for LASSO regression

The authors establish a coreset of size Õ(ε^−2 d · ((log d)^3 · min{1, log d/λ^2} + log(1/δ))) that achieves (1 ± ε)-approximation for LASSO regression. This bound improves upon the standard Õ(Gd/ε^2) bound from existing generalization theory by carefully leveraging geometric properties of localized spaces.

Contribution

Matching lower bound for LASSO coreset size

The authors prove an information-theoretic lower bound showing that any algorithm achieving (1+ε)-approximation for LASSO must use at least Ω(d log d / ε^2) rows when λ = O(d^−1/2). This demonstrates that their upper bound is nearly tight up to polylogarithmic factors in this regime.

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