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

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 $ σ^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( δ^{*2}, \varepsilon σ^2) \wedge d^2 $ for a star-shaped set with diameter $ d $ (set $d = \infty$ if the set is unbounded), with $ δ^* $ determined via the local entropy $ \log M^\mathrm{ loc }(δ,c) $ as \begin{align*} δ^*:= \sup\bigg{δ\geq 0: N\frac{δ^2}{σ^2}\leq \log M^\mathrm{ loc }(δ,c) \bigg}, \end{align*} where $ c $ is a sufficiently large constant. Crucially, we require that the sample size satisfies $N \gtrsim \mathop{ \sup }\limits_{δ\geq 0} \log M^\mathrm{ loc }(δ,c)$. We also show that the minimax rate is $ \max(δ^{*2},\varepsilon ^2σ^2) \wedge d^2 $ for known or sign-symmetric distributions, matching the rate achieved in the Gaussian case.

• The paper presents a nonasymptotic minimax risk characterization for robust mean estimation under star-shaped constraints with heavy-tailed noise and adversarial contamination.
• It develops matching lower and upper bounds using information-theoretic and robust tournament arguments that hinge on the local metric entropy of the constraint set.
• The study illustrates applications in sparse (ℓ0) and ℓ1-constrained frameworks, offering benchmarks for robust high-dimensional inference in challenging noise regimes.

Robust Mean Estimation under Star-Shaped Constraints with Heavy-Tailed Noise

Problem Formulation and Minimax Risk Characterization

The paper investigates the robust estimation of the mean parameter $\mu$ for a distribution on $\mathbb{R}^n$ where the observations are subjected to both adversarial contamination (fraction $\varepsilon <c$ for some constant $c<1/2$) and heavy-tailed additive noise with only a finite second moment $\sigma^2$. The estimation is further constrained by the requirement that $\mu$ resides within a known star-shaped subset $K\subseteq\mathbb{R}^n$, generalizing convex constraints. The statistical risk of interest is the minimax expected squared $\ell_2$ loss over all estimators, contamination schemes, and admissible distributions as

$$\mathfrak{M} = \inf_{\hat{\mu}} \sup_{\mu\in K} \sup_{\xi\in\Xi_{\sigma^2}} \sup_{\mathcal{C}} \mathbb{E}_{\mu}\left[\|\hat{\mu}(\mathcal{C}(\tilde{X})) - \mu\|^2\right],$$

where $\mathcal{C}$ denotes the adversary and $\mathbb{R}^n$0 denotes the family of mean-zero distributions with maximal covariance eigenvalue $\mathbb{R}^n$1.

The core technical contribution is the nonasymptotic characterization of $\mathbb{R}^n$2 as a function of: the geometry of $\mathbb{R}^n$3 (encoded by local metric entropy), contamination rate $\mathbb{R}^n$4, noise variance $\mathbb{R}^n$5, and sample size $\mathbb{R}^n$6. The main result is that, for sufficiently large $\mathbb{R}^n$7 (relative to the metric entropy), the minimax risk is

$\mathbb{R}^n$8

where $\mathbb{R}^n$9 and $\varepsilon <c$0 is a critical radius determined by the local metric entropy: $\varepsilon <c$1 For sign-symmetric noise (and, equivalently, for known noise laws via symmetrization), the dependence on contamination improves to $\varepsilon <c$2.

Lower and Upper Bounds: Packing, Contamination, and Algorithmic Rates

The minimax lower bound consists of two sharp contributions:

1. Information-theoretic bound: Via Fano's lemma and local metric entropy, the best possible risk absent contamination is bounded below by the $\varepsilon <c$3 above.
2. Adversarial contamination bound: The contamination imposes an irreducible error of order $\varepsilon <c$4 (or $\varepsilon <c$5 under symmetry).

Matching upper bounds are achieved by an information-theoretically optimal, albeit computationally intractable, estimator. The estimator operates by constructing a multiscale, local-packing-based directed tree on $\varepsilon <c$6 and performing robust tournaments at progressively finer resolutions. At each node/layer, the estimator selects the "best" candidate using robust hypothesis tests involving tournament-based trimmed means and medians (in the heavy-tailed regime) or Huber-type procedures (for symmetric noise).

The minimaxity is formally established by matching lower and upper bounds, managing both unbounded and bounded constraint sets, and precisely controlling error propagation through multiple layers of the estimation tree. For all $\varepsilon <c$7, the sample complexity is $\varepsilon <c$8.

Analysis in Heavy-Tailed and Symmetric Regimes

For general (potentially asymmetric) heavy-tailed noise, only finite variance is assumed, and the robust tests at the heart of the procedure combine trimming and median-based decision rules. The analysis leverages Cantelli inequalities to control one-sided deviations and establishes that, as long as the packing scales are chosen relative to the intrinsic local metric entropy of $\varepsilon <c$9, error control is feasible.

In the symmetric or known (via symmetrization) noise regime, Huber-type robust mean estimators are shown to achieve polynomial- or exponential-rate concentration, permitting improvement in the contamination term. The precise improvement is $c<1/2$0.

Examples and Implications for Structured Constraints

The paper systematically analyzes two canonical cases:

• $c<1/2$1-constraint ("sparse" mean estimation): $c<1/2$2, the set of $c<1/2$3-sparse vectors. The minimax rate is

$c<1/2$4

provided $c<1/2$5 is at least of order $c<1/2$6. This rate differs from results in the one-sample setting and emphasizes the need for large enough $c<1/2$7 in heavy-tailed scenarios.

• $c<1/2$8-constraint:

$c<1/2$9

with the precise transition point mapped via entropy calculations.

The results also clarify the role of local geometry (packing/covering structure) for nonconvex constraint sets, such as star-shaped but highly nonconvex sets, establishing that convexity is not essential for nonasymptotic optimality.

Theoretical and Practical Implications

The main theoretical implication is a tight, nonasymptotic minimax characterization for high-dimensional robust mean estimation under mild moment assumptions and very general geometric constraints (star-shaped, including nonconvex sets). The local metric entropy regime is critical, and the contamination-imposed error rate is sharp.

Practically, while the estimators described are not computationally efficient, the analysis suggests benchmarks for polynomial-time procedures and highlights the fundamental statistical price for treating contamination and heavy tails simultaneously. For special cases (e.g., symmetry), efficient procedures based on robust M-estimation may be available.

Potential directions for future work include:

• Extension to other functionals, including regression and density estimation under contamination and structure.
• Algorithmically efficient procedures achieving (nearly) optimal rates for more general classes of constraints (beyond convex/Type-2 bodies).
• Refinements in adaptivity to unknown $\sigma^2$0 and $\sigma^2$1, possibly via Lepskiĭ-type methods.
• Investigation of robust covariance (rather than only mean) estimation under analogous conditions.

Conclusion

The paper rigorously determines minimax rates for constrained robust mean estimation under star-shaped geometry and heavy-tailed, only-finite-variance noise, in the presence of adversarial contamination. The rates are expressed precisely in terms of local metric entropy and reveal a nuanced dependence on symmetry assumptions, constraint set geometry, and contamination fraction. The techniques advance understanding of robust estimation beyond convex or sub-Gaussian settings and set statistical benchmarks for both theory and practice (2604.05063).