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arxiv: 2604.26446 · v1 · submitted 2026-04-29 · 💻 cs.LG

Near-Optimal Cryptographic Hardness of Learning With Homogeneous Halfspaces Under Gaussian Marginals

Jizhou Huang , Brendan Juba This is my paper

Pith reviewed 2026-05-07 13:36 UTC · model grok-4.3

classification 💻 cs.LG
keywords homogeneous halfspacesagnostic learningGaussian marginalsLearning With Errorscomputational hardnessfairness auditingreliable learning
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The pith

LWE hardness implies no efficient algorithms exist for learning homogeneous halfspaces under Gaussian marginals, closing gaps with known upper bounds.

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

The paper shows that three tasks—agnostically learning a homogeneous halfspace, one-sided reliable learning, and auditing for fairness violations—cannot be solved efficiently when input vectors follow a standard Gaussian distribution but labels are arbitrary. It establishes this by giving polynomial-time reductions from the Learning With Errors problem, so that any algorithm succeeding on the learning tasks would also break LWE. A reader should care because earlier hardness proofs applied only to general (non-homogeneous) halfspaces; these results extend the hardness directly to the homogeneous case and tighten the separation between what is known to be possible and what is provably hard.

Core claim

Under the standard hardness assumption for Learning With Errors, there is no polynomial-time algorithm that, given labeled Gaussian examples, outputs a homogeneous halfspace whose loss is within a small additive factor of the best possible homogeneous halfspace, for the agnostic, one-sided reliable, and fairness-auditing loss measures. The same lower bounds apply even when the algorithm is allowed to output any halfspace (not necessarily homogeneous) in the agnostic case.

What carries the argument

Polynomial-time reductions from LWE instances to distributions over labeled Gaussian vectors that preserve the optimal halfspace loss up to small additive error.

If this is right

  • Agnostic learning of homogeneous halfspaces under Gaussians requires super-polynomial time or suffers an approximation gap unless LWE is easy.
  • One-sided reliable learning and fairness auditing of homogeneous halfspaces inherit the same computational barrier.
  • The hardness applies even when the learner may return a non-homogeneous halfspace for the agnostic task.
  • Prior lower bounds for general halfspaces now extend to the homogeneous restriction without loss of tightness.

Where Pith is reading between the lines

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

  • If these hardness results hold, practical systems that rely on Gaussian data for linear classification may need to accept either slow runtimes or suboptimal accuracy.
  • The reductions could be adapted to show hardness for other linear concepts or loss functions that admit similar embedding of LWE noise.
  • Closing the remaining small gap to the best known upper bounds would require either new algorithms or a stronger hardness assumption than LWE.

Load-bearing premise

The Learning With Errors problem is computationally hard and the feature vectors are drawn exactly from the standard Gaussian distribution.

What would settle it

An explicit polynomial-time algorithm that, for any constant epsilon greater than zero, outputs a homogeneous halfspace whose 0-1 loss is at most the optimal loss plus epsilon, on any distribution whose marginal on x is standard Gaussian.

Figures

Figures reproduced from arXiv: 2604.26446 by Brendan Juba, Jizhou Huang.

Figure 1
Figure 1. Figure 1: A demonstration of the closed contour. Proof. Notice that, by Euler’s formula, we have that E z∼N (µ,σ2) [sin(ωz)] = E z∼N (µ,σ2) [Im(e iωz )] =Im  E z∼N (µ,σ2) [e iωz ]  (i) =Im(e iµωe −σ 2ω 2/2 ) = sin(µω)e −σ 2ω 2/2 where Equation (i) holds because of Fact 28. The last equation is obtained by another application of Euler’s formula. Lemma 31 (Centering of Partial Sub-Gaussian FT). For any a > 0, α ≥ 0,… view at source ↗
read the original abstract

We study three problems that involve identifying homogeneous halfspaces under Gaussian distributions: agnostic learning, one-sided reliable learning, and fairness auditing. In each of these problems, we are given labeled examples $(\mathbf{x}, \mathrm{y})$ drawn from an unknown distribution on $\mathbb{R}^d\times\{-1, +1\}$, whose marginal distribution on $\mathbf{x}$ is standard Gaussian and on $\mathrm{y}$ is arbitrary. The goal of each problem is to output a homogeneous halfspace that approaches the best-fitting homogeneous halfspace in terms of its corresponding loss measure. We prove near-optimal computational hardness results for these problems under the widely believed hardness assumption of the Learning With Errors (LWE) problem. Prior hardness results for these problems were mostly established for general halfspaces; our findings extend some of these hardness results to homogeneous halfspaces. Remarkably, our lower bound strictly generalizes over prior works and narrows the gap between the upper and lower bounds for agnostically learning homogeneous halfspaces under Gaussian marginals.

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 manuscript establishes near-optimal computational hardness results for agnostic learning, one-sided reliable learning, and fairness auditing of homogeneous halfspaces under standard Gaussian marginals. Hardness is shown via reductions from the Learning With Errors (LWE) problem, extending prior results that applied to general halfspaces and narrowing the gap to known upper bounds specifically for agnostic learning of homogeneous halfspaces.

Significance. If the LWE reductions and error analyses hold, the results are significant: they provide tight, assumption-based lower bounds for a practically relevant subclass of halfspaces (homogeneous) under Gaussian marginals, which are standard in learning theory. The work strengthens prior hardness results by specializing to homogeneous halfspaces while preserving near-optimality, and the use of a standard cryptographic assumption (LWE) with explicit Gaussian marginals is a methodological strength that enables falsifiable predictions.

major comments (2)
  1. [§4 (reduction construction)] The central reduction from LWE to the three problems must preserve both homogeneity of the target halfspace and the exact Gaussian marginal; any slack in the error analysis or distribution shift could weaken the near-optimality claim. This needs explicit verification in the main reduction theorem (likely §4 or §5).
  2. [Introduction and §6 (comparison to upper bounds)] The claim that the lower bound 'narrows the gap' to upper bounds for agnostic learning requires a precise parameter comparison (e.g., the dependence on dimension d and error rate); without it, the 'near-optimal' qualifier is not fully substantiated.
minor comments (2)
  1. [§2 (preliminaries)] Notation for the loss measures (e.g., 0-1 loss vs. one-sided reliable loss) should be unified across definitions and theorems to avoid reader confusion.
  2. [Table 1 or new comparison table] The paper should include a short table summarizing the new hardness parameters versus prior work for each of the three problems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the recommendation for minor revision. We appreciate the positive assessment of the significance of our results. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [§4 (reduction construction)] The central reduction from LWE to the three problems must preserve both homogeneity of the target halfspace and the exact Gaussian marginal; any slack in the error analysis or distribution shift could weaken the near-optimality claim. This needs explicit verification in the main reduction theorem (likely §4 or §5).

    Authors: The reduction presented in Section 4 is explicitly designed to maintain the homogeneity of the target halfspace and the standard Gaussian marginal distribution on the features. The LWE secret is used to define the labels in a way that the optimal homogeneous halfspace corresponds to the LWE solution, while the input vectors remain i.i.d. standard Gaussians. We have verified that there is no distribution shift or slack that compromises the near-optimality, as the error terms are controlled to match the LWE hardness parameters. To make this verification more prominent, we will add an explicit statement or corollary in the main theorem statement in the revised version. revision: partial

  2. Referee: [Introduction and §6 (comparison to upper bounds)] The claim that the lower bound 'narrows the gap' to upper bounds for agnostic learning requires a precise parameter comparison (e.g., the dependence on dimension d and error rate); without it, the 'near-optimal' qualifier is not fully substantiated.

    Authors: We do provide a comparison in the introduction and Section 6, noting that our lower bound for agnostic learning matches the upper bound up to logarithmic factors in d, with the same dependence on the error rate. However, to strengthen the substantiation of the 'near-optimal' claim, we will include a detailed table or explicit parameter comparison of the dependencies on d and the error parameter in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

Hardness via external LWE reduction; no internal circularity

full rationale

The paper derives its near-optimal hardness results for agnostic learning, one-sided reliable learning, and fairness auditing of homogeneous halfspaces under Gaussian marginals by reduction from the standard external LWE assumption. The abstract and description explicitly state this as the basis for the lower bounds, which generalize prior results without introducing self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to the paper's own inputs. The Gaussian marginal is a stated assumption, not derived internally. No load-bearing steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard LWE hardness assumption from cryptography and the modeling choice of Gaussian marginals; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The Learning With Errors (LWE) problem is computationally hard
    This external cryptographic assumption is invoked as the basis for all hardness reductions.

pith-pipeline@v0.9.0 · 5476 in / 1291 out tokens · 56241 ms · 2026-05-07T13:36:01.926147+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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    Running the algorithm mentioned above may return us a s′ such that uD(s′) ≥C/ √ηlogd−ϵ

    For the alternative hypothesis as described in Theorem 24, we have thatuD(s) ≥C/ √ηlogd , where C > 0is a universal constant. Running the algorithm mentioned above may return us a s′ such that uD(s′) ≥C/ √ηlogd−ϵ . Again, by theHoeffding Bound, we can estimate uD(s′) up to C/√ηlogd−ϵ−ε 1 by drawing O(1/ε2 1)examples from the distribution constructed in th...

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