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arxiv: 2603.26227 · v2 · submitted 2026-03-27 · 📊 stat.ML · cs.LG

Recognition: no theorem link

Privacy-Accuracy Trade-offs in High-Dimensional LASSO under Perturbation Mechanisms

Ayaka Sakata , Haruka Tanzawa
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:54 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords differential privacyLASSOhigh-dimensional regressionsparse estimationapproximate message passingprivacy-accuracy trade-offperturbation mechanismsstatistical estimation
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The pith

Stronger regularization improves privacy in high-dimensional sparse LASSO by stabilizing estimators against single data changes.

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

The paper analyzes privacy-preserving sparse linear regression in high dimensions using the LASSO estimator under two differential privacy mechanisms: output perturbation and objective perturbation. It employs approximate message passing to describe the typical performance of these estimators when the design is random and privacy noise is added. The central finding is that sparsity shapes the privacy-accuracy trade-off, with stronger regularization reducing the estimator's sensitivity to individual data points and thereby improving privacy as measured by on-average KL divergence. The two mechanisms behave differently, and objective perturbation can exhibit non-monotonic effects where added noise eventually destabilizes the estimator.

Core claim

Using approximate message passing under random design, the typical behavior of LASSO estimators with output or objective perturbation is characterized; stronger regularization improves privacy by making the estimator less sensitive to single-point data changes, while objective perturbation shows non-monotonic dependence on noise level that can increase sensitivity when noise becomes excessive.

What carries the argument

Approximate message passing characterization of perturbed LASSO estimators that tracks both estimation error and on-average KL divergence as the privacy measure.

If this is right

  • Increasing the regularization parameter can simultaneously reduce estimation error in sparse regimes and tighten the privacy guarantee.
  • Output perturbation yields monotonic improvement in privacy with added noise, whereas objective perturbation can reverse this trend at high noise levels.
  • The on-average KL divergence admits a direct hypothesis-testing interpretation for distinguishability of neighboring datasets.
  • AMP supplies explicit formulas for the privacy-accuracy frontier that depend on sparsity level and noise variance.

Where Pith is reading between the lines

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

  • Practitioners may tune regularization to meet a target privacy budget rather than solely minimizing prediction error.
  • The non-monotonic behavior for objective perturbation suggests an optimal noise level that balances stability and privacy.
  • The framework could be extended to other convex sparse estimators or to non-Gaussian designs by modifying the state evolution equations.
  • Design of future perturbation mechanisms might exploit the stabilizing effect of sparsity to achieve better trade-offs.

Load-bearing premise

The design matrix is random and the approximate message passing equations accurately describe the typical behavior of the perturbed estimators.

What would settle it

An empirical simulation on random-design data in which the observed mean squared error or the empirical distinguishability between neighboring datasets deviates substantially from the AMP state evolution predictions.

read the original abstract

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

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

2 major / 2 minor

Summary. The paper claims to characterize the typical privacy-accuracy trade-offs for the high-dimensional LASSO under output perturbation and objective perturbation mechanisms for differential privacy. Using approximate message passing (AMP) under random design assumptions, it derives the behavior of the estimators and adopts the on-average KL divergence (with a hypothesis-testing interpretation) as the privacy measure. Central results are that sparsity plays a key role, stronger regularization can improve privacy by stabilizing the estimator against single-point changes, and the mechanisms differ qualitatively—with objective perturbation exhibiting non-monotonic noise effects that can destabilize the estimator at high noise levels.

Significance. If the AMP characterizations hold, the work supplies a precise typical-case analysis of privacy-accuracy trade-offs in sparse high-dimensional regression that goes beyond worst-case bounds and highlights the beneficial role of regularization and sparsity; this could inform mechanism selection and parameter tuning in practical DP applications. The adoption of on-average KL divergence as a privacy metric is a constructive choice that admits a clear statistical interpretation.

major comments (2)
  1. [AMP state evolution for objective perturbation] The state-evolution equations for the on-average KL divergence under objective perturbation (the section deriving the AMP fixed-point equations after adding the random linear term) assume that the privacy-induced perturbation preserves asymptotic Gaussianity and message independence. The random linear term perturbs the effective noise distribution and residual correlations in a manner that may violate these standard AMP assumptions when the privacy scale interacts with λ; this is load-bearing for the non-monotonicity and destabilization claims, yet no error bounds or alternative derivations are supplied.
  2. [sensitivity analysis via AMP fixed points] The claim that stronger regularization stabilizes the estimator against single-point data changes (and thereby improves privacy) is obtained directly from the AMP fixed-point equations for sensitivity; without reported finite-dimensional simulations, exact computations on small instances, or explicit verification that the perturbed noise distribution remains compatible with the AMP assumptions, the qualitative distinction between the two mechanisms rests on an unvalidated extension of the framework.
minor comments (2)
  1. [privacy measure definition] Clarify the precise definition of the on-average KL divergence (including how it is computed from the AMP state variables) and ensure it is distinguished from worst-case DP notions throughout the text.
  2. [numerical results] The simulation figures comparing theoretical curves to empirical behavior should report the number of trials and include variability measures to allow assessment of agreement with the AMP predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our work. We address each major comment below and outline the revisions we plan to make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [AMP state evolution for objective perturbation] The state-evolution equations for the on-average KL divergence under objective perturbation (the section deriving the AMP fixed-point equations after adding the random linear term) assume that the privacy-induced perturbation preserves asymptotic Gaussianity and message independence. The random linear term perturbs the effective noise distribution and residual correlations in a manner that may violate these standard AMP assumptions when the privacy scale interacts with λ; this is load-bearing for the non-monotonicity and destabilization claims, yet no error bounds or alternative derivations are supplied.

    Authors: We appreciate this observation. Our derivation treats the objective perturbation as an additive random term that, under the random design and high-dimensional limit, integrates into the effective noise variance in the state evolution equations, preserving the Gaussianity and message independence properties of standard AMP. The non-monotonicity arises from the fixed-point interaction between the regularization λ and the privacy noise scale. We acknowledge the absence of rigorous error bounds for this extension; this is a limitation of the current analysis. In the revision, we will include a more detailed justification of the assumptions and a remark on potential future work for concentration bounds. revision: partial

  2. Referee: [sensitivity analysis via AMP fixed points] The claim that stronger regularization stabilizes the estimator against single-point data changes (and thereby improves privacy) is obtained directly from the AMP fixed-point equations for sensitivity; without reported finite-dimensional simulations, exact computations on small instances, or explicit verification that the perturbed noise distribution remains compatible with the AMP assumptions, the qualitative distinction between the two mechanisms rests on an unvalidated extension of the framework.

    Authors: The sensitivity claims follow from solving the AMP fixed-point equations, which quantify the estimator's response to data perturbations in the typical case. While the manuscript focuses on the asymptotic analysis, we agree that empirical validation would strengthen the presentation. We will add finite-dimensional numerical experiments in the revision to verify the AMP predictions for both perturbation mechanisms and illustrate the qualitative differences, including the non-monotonic behavior. revision: yes

Circularity Check

0 steps flagged

Derivations rely on external AMP theory without reducing to self-defined inputs

full rationale

The paper applies established approximate message passing (AMP) state evolution from the high-dimensional statistics literature to the perturbed LASSO under random design. These external characterizations are not derived from or fitted to the privacy measures in the present work. No equations reduce the target privacy-accuracy trade-offs to parameters defined by those same trade-offs, and any self-citations are peripheral rather than load-bearing for the central claims about regularization effects and non-monotonic noise behavior.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of AMP state evolution for perturbed estimators and the interpretation of on-average KL divergence as a typical-case privacy measure; no new free parameters or invented entities are introduced beyond standard regularization and noise variance.

axioms (2)
  • domain assumption Random design assumption for the measurement matrix in high-dimensional regime
    Invoked to enable AMP characterization of typical behavior
  • domain assumption Approximate message passing accurately tracks the fixed-point behavior of perturbed LASSO
    Core technical assumption enabling the privacy-accuracy analysis

pith-pipeline@v0.9.0 · 5487 in / 1224 out tokens · 43712 ms · 2026-05-14T22:54:52.709791+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation

    stat.ML 2026-05 unverdicted novelty 7.0

    A new Gram-based anisotropic objective perturbation stabilizes private LASSO under heterogeneous covariates and improves efficiency via AMP state evolution analysis.