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arxiv: 2604.05063 · v2 · submitted 2026-04-06 · 🧮 math.ST · stat.TH

Robust mean estimation under star-shaped constraints with heavy-tailed noise

Tuorui Peng , Akshay Prasadan , Matey Neykov This is my paper

Pith reviewed 2026-05-10 18:46 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords robust estimationmean estimationstar-shaped setsheavy-tailed distributionsminimax rateslocal entropyadversarial contaminationstatistical estimation
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The pith

Robust mean estimation under star-shaped constraints with heavy-tailed noise achieves minimax rate max(δ*², ε σ²) ∧ d² for small contamination ε.

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

The paper establishes a minimax rate for estimating the mean of a distribution under adversarial contamination and heavy-tailed noise, when the mean is constrained to lie in a star-shaped set. Specifically, when the contamination fraction ε is below a constant, the rate for squared Euclidean error is the maximum of a term δ* squared derived from the local covering entropy of the set and ε times the noise variance, and this is further bounded by the square of the set's diameter. This extends previous results from convex sets to star-shaped ones while requiring only finite second moments on the noise. A reader cares because it shows that robust estimation is possible in more general geometric settings with minimal assumptions on the data distribution, as long as the sample size exceeds the entropy of the constraint set.

Core claim

We show that for contamination level ε below some constant, the minimax rate of the squared ℓ₂ loss is max(δ*², ε σ²) ∧ d² for a star-shaped set with diameter d, where δ* is defined as the supremum of δ such that N δ²/σ² ≤ log M^loc(δ, c) for a large constant c, provided the sample size N is at least the supremum of log M^loc(δ, c) over δ. For known or sign-symmetric distributions the rate becomes max(δ*², ε² σ²) ∧ d².

What carries the argument

The local entropy log M^loc(δ, c) which determines the critical radius δ* at which the statistical complexity balances the sample size.

If this is right

  • The rate matches the Gaussian case for symmetric noise distributions.
  • For unbounded star-shaped sets the diameter bound is removed.
  • The result requires N to be at least the max local entropy.
  • The contamination must be less than some fixed constant.

Where Pith is reading between the lines

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

  • Similar entropy-based rates might apply to other robust estimation problems like covariance estimation under star-shaped constraints.
  • Practitioners could use this to choose sample sizes based on the entropy of their feasible set for robust means.
  • This framework could be extended to unbounded star-shaped sets without diameter cap by focusing on local behavior.

Load-bearing premise

The contamination level ε must remain below an unspecified constant, and the number of samples must exceed the supremum of the local entropy log M^loc(δ, c) over all scales δ.

What would settle it

Generate data from a heavy-tailed distribution with mean in a known star-shaped set, contaminate a fraction ε of samples adversarially, apply a robust estimator, and verify whether the squared error scales as predicted by max(δ*², ε σ²) where δ* is computed from the set's local entropy.

read the original abstract

We study the problem of robust mean estimation with adversarially contaminated data under star-shaped constraints in a heavy-tailed noise setting, where only a finite second moment $ \sigma ^2 $ is assumed. For a contamination level $ \varepsilon$ below some constant, we show that the minimax rate of the squared $ \ell_2 $ loss is $ \max( \delta ^{*2}, \varepsilon \sigma ^2) \wedge d^2 $ for a star-shaped set with diameter $ d $ (set $d = \infty$ if the set is unbounded), with $ \delta ^* $ determined via the local entropy $ \log M^\mathrm{ loc }(\delta ,c) $ as \begin{align*} \delta ^*:= \sup\bigg\{\delta \geq 0: N\frac{\delta ^2}{\sigma ^2}\leq \log M^\mathrm{ loc }(\delta ,c) \bigg\}, \end{align*} where $ c $ is a sufficiently large constant. Crucially, we require that the sample size satisfies $N \gtrsim \mathop{ \sup }\limits_{\delta \geq 0} \log M^\mathrm{ loc }(\delta ,c)$. We also show that the minimax rate is $ \max(\delta^{*2},\varepsilon ^2\sigma ^2) \wedge d^2 $ for known or sign-symmetric distributions, matching the rate achieved in the Gaussian case.

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

1 major / 0 minor

Summary. The manuscript claims to establish minimax rates for robust mean estimation under star-shaped constraints with heavy-tailed noise (finite second moment σ²). For contamination level ε below a constant, the rate is max(δ*², ε σ²) ∧ d², where δ* is the supremum of δ such that N δ²/σ² ≤ log M^loc(δ, c) for large c, and this requires the sample size N to satisfy N ≳ sup_δ log M^loc(δ, c). A variant rate max(δ*², ε² σ²) ∧ d² is shown for known or sign-symmetric distributions.

Significance. If the results hold, this work provides a general entropy-based characterization of robust estimation rates for star-shaped sets, which include many non-convex and unbounded geometries. The explicit dependence on local entropy allows for adaptive rates in structured problems, and the heavy-tailed assumption (only second moment) is weaker than sub-Gaussian. The matching to Gaussian rates in the symmetric case is a positive feature. However, the applicability hinges on whether the sample-size condition can be satisfied for interesting sets.

major comments (1)
  1. [Abstract and main theorem statement] Abstract (and main theorem): The sample size condition N ≳ sup_{δ ≥ 0} log M^{loc}(δ, c) is likely unsatisfiable for non-trivial star-shaped sets. For sets with positive dimension (e.g., balls, cones, or unbounded sets in R^d with d ≥ 1), the local covering number M^{loc}(δ, c) typically diverges as δ → 0, causing log M^{loc}(δ, c) → ∞ and the supremum to be infinite. No finite N can satisfy the hypothesis, rendering the theorem inapplicable to most cases of interest. If the body restricts the supremum (e.g., over δ ≥ δ* or where the inequality holds) or redefines M^{loc}, this needs to be clarified and the claim restated accordingly. This condition is load-bearing for the central result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The concern about the sample size condition is well-taken and points to an imprecise formulation in the abstract that we will correct.

read point-by-point responses

  1. Referee: [Abstract and main theorem statement] Abstract (and main theorem): The sample size condition N ≳ sup_{δ ≥ 0} log M^{loc}(δ, c) is likely unsatisfiable for non-trivial star-shaped sets. For sets with positive dimension (e.g., balls, cones, or unbounded sets in R^d with d ≥ 1), the local covering number M^{loc}(δ, c) typically diverges as δ → 0, causing log M^{loc}(δ, c) → ∞ and the supremum to be infinite. No finite N can satisfy the hypothesis, rendering the theorem inapplicable to most cases of interest. If the body restricts the supremum (e.g., over δ ≥ δ* or where the inequality holds) or redefines M^{loc}, this needs to be clarified and the claim restated accordingly. This condition is load-bearing for the central result.

    Authors: We agree that the global supremum over all δ ≥ 0, as written in the abstract, would be infinite whenever local entropy diverges as δ → 0, which occurs for any set of positive dimension. This is a misstatement. In the body (Theorem 1 and its proof), δ* is defined as the supremum of the set of δ satisfying the inequality; because log M^{loc}(δ, c) → ∞ as δ → 0 while the left-hand side → 0, the set is always non-empty for any finite N and δ* is always well-defined and finite. The sample-size requirement is intended only to guarantee that this δ* lies below the diameter d (when finite), so the rate is not trivially capped at d². We will revise the abstract and the theorem statement to remove the global-supremum phrasing, replace it with the self-consistent requirement N ≳ log M^{loc}(δ*, c), and add a short remark explaining that this condition is satisfied for all sufficiently large but finite N in standard examples (balls, cones, etc.). revision: yes

Circularity Check

0 steps flagged

No circularity; standard critical-radius definition from local entropy

full rationale

The claimed minimax rate is expressed as max(δ*^2, ε σ²) ∧ d² where δ* is defined by the fixed-point equation δ* := sup{δ ≥ 0 : N δ²/σ² ≤ log M^loc(δ, c)}. This is the conventional construction of a critical radius from the entropy integral of the constraint set and does not reduce the output to any fitted parameter, self-citation, or input quantity by construction. The auxiliary sample-size requirement N ≳ sup_δ log M^loc(δ, c) is an explicit hypothesis on the problem parameters rather than a tautological re-statement of the rate; the entropy function itself is an independent geometric property of the star-shaped set. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on the definition of local entropy, the star-shaped geometry, and a sample-size lower bound expressed in terms of that entropy; no new entities are introduced.

free parameters (1)
  • c
    Sufficiently large constant used inside the local-entropy threshold that defines δ*.
axioms (3)
  • domain assumption The feasible set for the mean is star-shaped
    Invoked throughout the problem setup and rate derivation.
  • domain assumption Noise has finite second moment σ²
    The only moment assumption placed on the heavy-tailed noise.
  • domain assumption Contamination level ε lies below some constant
    Required for the stated form of the minimax rate to hold.

pith-pipeline@v0.9.0 · 5574 in / 1525 out tokens · 72648 ms · 2026-05-10T18:46:43.118961+00:00 · methodology

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