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arxiv: 2501.09189 · v3 · submitted 2025-01-15 · 💻 cs.LG · cs.DS

Testing Noise Assumptions of Learning Algorithms

Surbhi Goel , Adam R. Klivans , Konstantinos Stavropoulos , Arsen Vasilyan This is my paper

Pith reviewed 2026-05-23 04:58 UTC · model grok-4.3

classification 💻 cs.LG cs.DS
keywords testable learningMassart noisehalfspacesGaussian marginalsnoise model testinglearning theorypolynomial time algorithms
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The pith

There exists a fully polynomial-time testable learning algorithm for halfspaces over Gaussian marginals with Massart noise.

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

The paper asks whether we can efficiently test if a training set satisfies the assumptions of a given noise model. It extends the testable learning framework so that the learner must run a test that accepts only on data drawn from the assumed marginal distribution and noise model, and if the test passes must output a classifier together with a certificate of optimality. For the specific case of halfspaces under Massart noise, where each label is flipped with probability less than 1/2 depending on the features, the authors give an algorithm that runs in fully polynomial time. They also prove a separation showing that testable learning requires super-polynomial time for the simpler random classification noise model at rate 1/2, even though ordinary learning is trivial. The result matters because it supplies a concrete way to verify the modeling assumptions that many noise-tolerant learning algorithms implicitly rely on.

Core claim

We pose a fundamental question in computational learning theory: can we efficiently test whether a training set satisfies the assumptions of a given noise model? We extend the testable learning framework of Rubinfeld and Vasilyan (2023) and give a fully-polynomial time testable learning algorithm for learning halfspaces over Gaussian marginals with Massart noise. We also show a separation: for random classification noise with flip probability 1/2, testable learning requires super-polynomial time while classical learning is trivial.

What carries the argument

The testable learner in the extended framework: it runs an associated test on the input dataset that must accept whenever the data is drawn from the modeled marginal and noise, and upon acceptance outputs a classifier plus a certificate of optimality.

If this is right

  • Testable learning is possible in polynomial time for halfspaces under Massart noise.
  • Testable learning is computationally harder than classical learning for random classification noise at rate 1/2.
  • The extended framework supplies both a hypothesis and a certificate of optimality when the noise assumption holds.
  • Noise-model testing can be separated from ordinary learning in complexity.

Where Pith is reading between the lines

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

  • Similar testable algorithms might exist for other structured noise models or other marginal distributions.
  • In practice one could run the test on real data and only trust the learned classifier when the test accepts.
  • The separation result suggests that the computational cost of verification depends strongly on the precise noise model.

Load-bearing premise

The test must pass for any dataset drawn according to the specified modeling assumption on both the marginal distribution and the noise model.

What would settle it

Generate a dataset from a Gaussian marginal with Massart noise and run the proposed testable algorithm; if the test rejects or the output classifier fails to be optimal on that data, the claim is false.

Figures

Figures reproduced from arXiv: 2501.09189 by Adam R. Klivans, Arsen Vasilyan, Konstantinos Stavropoulos, Surbhi Goel.

Figure 1
Figure 1. Figure 1: The shaded region is {x ∈ R d : sign(v · x) ̸= sign(v ∗ · x)}. Left: red square points have label +1, blue round points have label −1. Right: green square points are in S¯ g and purple round points are in S¯ b. Towards a testable bound. We have assumed that if the noise was indeed RCN, then v = v ∗ . Therefore, in this case, |S¯ b| = |S¯ g| = 0. However, if the noise assumption is not guaranteed, given S¯,… view at source ↗
Figure 2
Figure 2. Figure 2: For vectors v, v ′ ∈ S d−1 , the region {x ∈ R d : sign(v · x) ̸= sign(v ′ · x)} is contained in the union of green regions and it contains the union of blue regions. In the diagram we highlight one of the green regions (top left) and one of the blue regions (bottom right). We choose µ = c and run the tester above on the datapoints in S¯ and S¯ False = {(x, y) ∈ S¯ : y ̸= sign(v · x)}. If the tester accept… view at source ↗
read the original abstract

We pose a fundamental question in computational learning theory: can we efficiently test whether a training set satisfies the assumptions of a given noise model? This question has remained unaddressed despite decades of research on learning in the presence of noise. In this work, we show that this task is tractable and present the first efficient algorithm to test various noise assumptions on the training data. To model this question, we extend the recently proposed testable learning framework of Rubinfeld and Vasilyan (2023) and require a learner to run an associated test that satisfies the following two conditions: (1) whenever the test accepts, the learner outputs a classifier along with a certificate of optimality, and (2) the test must pass for any dataset drawn according to a specified modeling assumption on both the marginal distribution and the noise model. We then consider the problem of learning halfspaces over Gaussian marginals with Massart noise (where each label can be flipped with probability less than $1/2$ depending on the input features), and give a fully-polynomial time testable learning algorithm. We also show a separation between the classical setting of learning in the presence of structured noise and testable learning. In fact, for the simple case of random classification noise (where each label is flipped with fixed probability $\eta = 1/2$), we show that testable learning requires super-polynomial time while classical learning is trivial.

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 / 1 minor

Summary. The paper extends the testable learning framework of Rubinfeld and Vasilyan (2023) so that a learner must execute a test that accepts with high probability on data drawn from a specified marginal-plus-noise model and, on acceptance, outputs a classifier together with an optimality certificate. It claims a fully polynomial-time testable learner for halfspaces over Gaussian marginals under Massart noise and proves a super-polynomial separation: testable learning under random classification noise at η=1/2 requires super-polynomial time while classical learning is trivial.

Significance. If the algorithmic construction and lower-bound argument hold, the work is significant: it supplies the first efficient testable learner for a standard noise model in a canonical distribution setting and cleanly separates testable from classical learning. The framework extension itself is a reusable contribution, and the paper supplies concrete algorithmic descriptions together with the required analysis.

minor comments (1)
  1. [Abstract] Abstract: the claims of a fully polynomial-time algorithm and a super-polynomial lower bound are stated without even a one-sentence proof sketch or pointer to the relevant section; adding such pointers would improve readability without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the work, the recognition of its significance, and the recommendation for minor revision. The provided report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the testable learning framework of Rubinfeld and Vasilyan (2023) to define a testable learner that must accept on data from the stated marginal-plus-noise model and output a certified classifier. It then presents new algorithmic results (a fully polynomial-time testable learner for halfspaces under Massart noise) and a new lower-bound separation (superpolynomial time for testable learning under random classification noise at η=1/2). These contributions are constructive and hardness results that stand independently of the cited framework; no derivation step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities; the work rests on standard domain assumptions from learning theory.

axioms (3)
  • domain assumption Gaussian marginal distribution on the input features
    Invoked for the halfspace learning result with Massart noise.
  • domain assumption Massart noise model (label flip probability < 1/2 depending on features)
    Core modeling assumption for which the testable learner is claimed.
  • domain assumption Random classification noise with fixed flip probability eta = 1/2
    Used for the separation result showing super-polynomial testable learning time.

pith-pipeline@v0.9.0 · 5789 in / 1440 out tokens · 47474 ms · 2026-05-23T04:58:39.359599+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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    (For how to compute this approximation, see Claim 3)

    For allaandbin{−∞,−k 1ϵ,−(k 1 −1)ϵ,· · ·,−ϵ,0,+ϵ,· · ·,(k 1 −1)ϵ, k 1ϵ,+∞} (a) For all monomialsmof degree at mostk 2 overR d: i.A a,b m ←E x∼N(0,I d)[m(x)·1 a≤x·v<b]± 60(2k2(d+1))k2+2 δ logN N 1/4 . (For how to compute this approximation, see Claim 3). ii. If Ex∼S[m(x)·1 a≤x·v<b]−A a,b m > 200(2k2(d+1))k2+2 δ logN N 1/4 , then outputReject

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    X A∈C pA down(x)1 x∈A , X A∈C pA up(x)1 x∈A # ,(B.2) as well as E x∼N(0,I d)

    If did not reject in any previous step, outputAccept. It is immediate that the algorithm indeed runs in timepoly dN ϵδ . B.1 Completeness Suppose the set datasetSconsists of i.i.d. samples fromN(0, I d). We observe that the collectionHof sets of the form1 a≤v·x<b has VC dimension at most(d+ 1) 2. This allows us to use Lemma A.1, to conclude that with prob...

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    For allzinR d, the derivativeρ ′(z) =α 1e−z2/2zα1−11 α1≥1 −e −z2/2zα1+1 we have|ρ ′(z)| ≤ (k2 + 1)k2+1

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    For allz 0 inR d satisfyingz 0 >4k 2 + 2the value R |z|>z0 e−z2/2zα1 dzis at most R |z|>z0 e−z2/4 dz which in turn is at moste −z2 0 /4. The three properties above imply that one can approximate the value of R b a ρ(z)dzup to an additive error of β′ via discretization, i.e., by splitting the interval[a, b]∩[− p 2 ln(β′), p 2 ln (β′)]into intervals of size...

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    (b) If the following does not hold: 1 U X x∈S x⊗k2 x⊗k2 ⊤ 1 a≤x·v<b ⪯W a,b + 3∆I(d+1 k2 )×(d+1 k2 ),(C.2) then outputReject

    For allaandbin{−∞,−k 1ϵ,−(k 1 −1)ϵ,· · ·,−ϵ,0,+ϵ,· · ·,(k 1 −1)ϵ, k 1ϵ,+∞} (a) ComputeW a,b such that W a,b −∆I (d+1 k2 )×(d+1 k2 ) ⪯E x∼N x⊗k2 x⊗k2 ⊤ ·1 a≤x·v<b ⪯W a,b + ∆I(d+1 k2 )×(d+1 k2 ) (C.1) (For how to compute this approximation, see Claim 6). (b) If the following does not hold: 1 U X x∈S x⊗k2 x⊗k2 ⊤ 1 a≤x·v<b ⪯W a,b + 3∆I(d+1 k2 )×(d+1 k2 ),(C.2...

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    It is immediate that the algorithm indeed runs in timepoly dU ϵδ , because step (2b) can be performed by computing the largest eigenvalue of a d+1 k2 × d+1 k2 -sized matrix

    If did not reject in any previous step, outputAccept. It is immediate that the algorithm indeed runs in timepoly dU ϵδ , because step (2b) can be performed by computing the largest eigenvalue of a d+1 k2 × d+1 k2 -sized matrix. Monotonicity over datapoint removal also follows immediately since ifS ′ ⊂Sthen 1 U X x∈S′ x⊗k2 x⊗k2 ⊤ ⪯ 1 U X x∈S x⊗k2 x⊗k2 ⊤ , ...

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    Without loss of generality we can assume that the algorithm is deterministic, because we can use some of the points in ¯Sfor random seeds

    Letvbe the output of the algorithm of Theorem 2.8 run on the dataset ¯Swith accuracy parameter ϵ′ = ϵ3/2 100 √ d−1 and failure probabilityδ. Without loss of generality we can assume that the algorithm is deterministic, because we can use some of the points in ¯Sfor random seeds. 2.S← x: (x, y)∈ ¯S andN← ¯S

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  17. [17]

    Run the testerT disagreement from Theorem B.1, on input(S,v, ϵ, δ,0.1)

    work page

  18. [18]

    IfT disagreement rejects in the previous step, output outputReject

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  19. [19]

    6.S far False ←S False ∩ n x∈R d : ∡(x,v)− π 2 > ϵ3/2 √ d−1 o ;S near False ←S False \S far False

    If|S False|> 2 5 N, then outputReject. 6.S far False ←S False ∩ n x∈R d : ∡(x,v)− π 2 > ϵ3/2 √ d−1 o ;S near False ←S False \S far False

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  20. [20]

    IfS near False >4ϵN, then outputReject

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  21. [21]

    TakeU= 2 5 Nand then run the spectral testerT spectral from Theorem C.1 with the input param- eters(U, S far False,v, ϵ, δ,0.1)

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  22. [22]

    IfT spectral rejects in the previous step, output outputReject

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  23. [23]

    From the run-time guarantees given in Theorem B.1 and Theorem C.1, we see immideately that the run-time of the algorithmA Massart ispoly d ϵ log 1 δ

    Otherwise, output(Accept,v). From the run-time guarantees given in Theorem B.1 and Theorem C.1, we see immideately that the run-time of the algorithmA Massart ispoly d ϵ log 1 δ . D.1 Soundness We first show the soundness condition. For any dataset ¯Sof sizeN≥ Cd ϵδ C , we need to show that if AMassart accepts then the vectorvgiven byA Massart satisfies E...

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  24. [24]

    The setSis such that for all unit vectorsv ′ the algorithmT disagreement accepts when given the input (S,v ′, ϵ, δ,0.1)

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  25. [25]

    For all vectorsuinR d, we have P (x,y)∼ ¯S [sign(x·u)̸=y]−P (x,y)∼EXMassart N,f,η 0 [sign(x·v)̸=y] ≤2d r logN N log 1 δ

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  26. [26]

    For all vectorsuinR d and scalarsθ, we have P x∼S [∡(x,u)≤θ]−P x∼N(0,I d) [∡(x,u)≤θ] ≤2d r logN N log 1 δ

    work page

  27. [27]

    It is the case that∡(v,v ∗)≤ ϵ3/2 10 √ d−1

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  28. [28]

    ∡(x,v)− π 2 ≤ ϵ3/2 √ d−1 # −P x∼N(0,I d)

    It is the case that|S False| ≤ 2 5 Nand Saugmented ≤ 2 5 N. 6.S augmented is such that for all unit vectorsv ′ the algorithmT spectral accepts when given as input on the input U, Saugmented,v ′, ϵ, δ,0.1 (we remind the reader thatU= 2 5 N). Proof.Event 1 holds with probability at least1−O(δ)by Theorem B.1. The Event (2) holds with probability at least1−O(...

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