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arxiv: 2507.05084 · v2 · submitted 2025-07-07 · 💻 cs.LG · stat.ML

Distribution-dependent Generalization Bounds for Tuning Linear Regression Across Tasks

Maria-Florina Balcan , Saumya Goyal , Dravyansh Sharma This is my paper

Pith reviewed 2026-05-19 06:18 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords generalization boundsmulti-task learninglinear regressionregularization tuningsub-Gaussian distributionsdistribution-dependent boundshigh-dimensional datavalidation loss
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The pith

Under common distributional assumptions, tuning regularization hyperparameters in multi-task linear regression yields generalization bounds that do not worsen with feature dimension.

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

The paper develops generalization bounds for the validation loss when tuning L1 and L2 regularization coefficients across multiple related tasks in linear regression. These bounds depend on the data distribution and improve as the distribution becomes nicer, such as when instances are i.i.d. sub-Gaussian. This contrasts with prior uniform bounds that necessarily degrade as the feature dimension d increases. The results apply to ridge, lasso, and elastic net, and extend to a generalized ridge regression that incorporates an estimate of the mean. A sympathetic reader would care because this allows for effective hyperparameter tuning in high-dimensional settings without the bounds blowing up.

Core claim

The central claim is that distribution-dependent generalization bounds for tuning the L1 and L2 coefficients in multi-task linear regression do not get worse with increasing feature dimension d when instances within each task are i.i.d. draws from sub-Gaussian or similar distributions, and these bounds are much sharper than those from prior work for very large d. The bounds are obtained for the validation loss and also extended to a generalized form of ridge regression using mean estimates of the ground truth.

What carries the argument

Distribution-dependent generalization bounds on the validation loss for hyperparameter tuning, derived using concentration properties of sub-Gaussian random variables across tasks.

If this is right

  • The generalization error bounds stay controlled even for arbitrarily large feature dimensions.
  • Performance guarantees are tighter than uniform bounds when data is well-behaved.
  • Results cover standard regularization methods including ridge, lasso, and elastic net.
  • Tighter bounds are possible for generalized ridge regression by estimating the mean of the target distribution.

Where Pith is reading between the lines

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

  • This approach could enable scaling multi-task learning to very high-dimensional problems in practice without bound degradation.
  • Similar distribution-dependent analyses might apply to hyperparameter tuning in other models beyond linear regression.
  • Practitioners could leverage these bounds for more reliable regularization choices in high-d regimes with structured data.

Load-bearing premise

The instances within each task must be i.i.d. draws from sub-Gaussian or analogous well-studied distribution classes.

What would settle it

Empirical observation that the generalization error of the tuned parameters increases with feature dimension d on data satisfying the i.i.d. sub-Gaussian assumption would disprove the bounds.

read the original abstract

Modern regression problems often involve high-dimensional data and a careful tuning of the regularization hyperparameters is crucial to avoid overly complex models that may overfit the training data while guaranteeing desirable properties like effective variable selection. We study the recently introduced direction of tuning regularization hyperparameters in linear regression across multiple related tasks. We obtain distribution-dependent bounds on the generalization error for the validation loss when tuning the L1 and L2 coefficients, including ridge, lasso and the elastic net. In contrast, prior work develops bounds that apply uniformly to all distributions, but such bounds necessarily degrade with feature dimension, d. While these bounds are shown to be tight for worst-case distributions, our bounds improve with the "niceness" of the data distribution. Concretely, we show that under additional assumptions that instances within each task are i.i.d. draws from broad well-studied classes of distributions including sub-Gaussians, our generalization bounds do not get worse with increasing d, and are much sharper than prior work for very large d. We also extend our results to a generalization of ridge regression, where we achieve tighter bounds that take into account an estimate of the mean of the ground truth distribution.

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

0 major / 2 minor

Summary. The manuscript derives distribution-dependent generalization bounds for the validation loss when tuning L1 and L2 regularization hyperparameters (including ridge, lasso, and elastic net) in a multi-task linear regression setting. Under the explicit assumption that instances within each task are i.i.d. draws from sub-Gaussian or similar well-studied distribution classes, the bounds remain independent of feature dimension d and are sharper than prior uniform bounds for large d. The results are extended to a generalized ridge regression that incorporates an estimate of the mean of the ground truth distribution.

Significance. If the central derivations hold, the work offers a meaningful advance by replacing worst-case uniform bounds (which necessarily degrade with d) with sharper, distribution-dependent guarantees that exploit common niceness properties of real data. This is particularly relevant for practical high-dimensional multi-task regression, where hyperparameter tuning must balance overfitting and variable selection without overly pessimistic dimension dependence.

minor comments (2)
  1. [Introduction / Problem Setup] The abstract and introduction refer to 'broad well-studied classes of distributions including sub-Gaussians' without listing the precise classes for which the theorems are proved; this should be stated explicitly in the problem setup section.
  2. [Preliminaries] In the multi-task notation, the symbols for the number of tasks T and samples per task n are introduced without a dedicated table or clear cross-reference to the single-task case; this creates minor ambiguity when comparing to prior single-task bounds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation for minor revision. The referee's summary accurately reflects the paper's focus on deriving distribution-dependent generalization bounds for tuning L1/L2 regularization in multi-task linear regression, which remain independent of dimension d under sub-Gaussian i.i.d. assumptions and improve upon uniform bounds for large d.

Circularity Check

0 steps flagged

No significant circularity; bounds derived from explicit distributional assumptions

full rationale

The paper derives distribution-dependent generalization bounds for tuning L1/L2 regularization in multi-task linear regression. The central argument invokes the explicit assumption that instances within each task are i.i.d. draws from sub-Gaussian or similar well-studied distribution classes; this directly yields bounds independent of dimension d that improve over uniform worst-case bounds for large d. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input. The derivation remains self-contained with independent content supplied by the stated distributional niceness and standard concentration tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard statistical assumptions about data distributions rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Instances within each task are i.i.d. draws from sub-Gaussian or similar well-studied distribution classes
    Invoked in the abstract as the additional assumption enabling bounds that do not worsen with increasing d.

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Reference graph

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