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Huber contamination modulus lower bound for tail risk

Statement

Assume the Axiom of Choice. Fix a contamination level $0 \leq ϵ \leq 1$ , an integer sample size $N \geq 1$ , and a nonempty statistical model ${P_{θ} : θ \in Θ}$ on a common measurable sample space, and give $Θ$ a measurable structure as the parameter action space (Measurable spaces and measurable sets). Let the target-first loss $L : Θ \times Θ \to [0, + \infty]$ of Loss functions and risk have measurable sections $a \mapsto L (θ, a)$ . Suppose a fixed $0 < κ \leq 1$ satisfies

\max {L (θ_{0}, a), L (θ_{1}, a)} \geq κ L (θ_{0}, θ_{1})

for all $a, θ_{0}, θ_{1} \in Θ$ . Suppose also that $0 \leq M_{0} < \infty$ , $0 < c_{0} \leq 1$ , and

\inf_{\hat{θ} : X^{N} \to Θ m e a s u r a b l e} \sup_{θ \in Θ} P_{θ}^{\otimes N} {L (θ, \hat{θ}) \geq M_{0}} \geq c_{0} .

If $ω (ϵ, Θ) < \infty$ , then with

M_{ϵ} = \frac{κ}{4} (M_{0} \lor ω (ϵ, Θ))

one has

\inf_{\hat{θ} : X^{N} \to Θ m e a s u r a b l e} \sup_{θ \in Θ} \sup_{Q} P_{ϵ, θ, Q}^{\otimes N} {L (θ, \hat{θ}) \geq M_{ϵ}} \geq \min (c_{0}, 1 / 2) .

Thus the lower-bound threshold is comparable to $M_{0} \lor ω (ϵ, Θ)$ . If $ω (ϵ, Θ) = + \infty$ , the same supremum is at least $1 / 2$ at every finite threshold. In particular, if $c_{0} \geq c_{*} > 0$ for a fixed constant $c_{*} \leq 1 / 2$ independent of the model, the finite-modulus conclusion has the universal probability constant $c_{*}$ . A metric loss has the separation property with $κ = 1 / 2$ ; its square has it with $κ = 1 / 4$ .

Facts & Assumptions

Given: AC; $N \geq 1$ ; a nonempty statistical model; a measurable parameter loss with the stated separation property; the clean tail-risk lower bound $(M_{0}, c_{0})$ ; and $0 \leq ϵ \leq 1$ .

[A1]

The Axiom of Choice: AC is used through the density construction in [F4].

[F1]

Huber contamination model: one observation has law $P_{ϵ, θ, Q} = (1 - ϵ) P_{θ} + ϵ Q$ and the $N$ observations are iid from this law.

[F2]

Huber contamination modulus: $ω (ϵ, Θ)$ is the supremum of parameter loss over clean pairs with $T V (P_{θ_{0}}, P_{θ_{1}}) \leq ϵ / (1 - ϵ)$ ; at $ϵ = 1$ all pairs are allowed.

[F3]

Contamination-robust expected risk and tail risk: the tail criterion is the worst-case probability of the event $L (θ, \hat{θ}) \geq t$ , distinct from expected risk.

[F4]

TV-close laws have identical Huber mixtures at the contamination threshold: under AC, clean laws within the stated TV threshold admit contaminating laws that make their Huber mixtures equal.

[F5]

A common data law forces a two-point tail-loss bound: if two parameter values have the same data law, the maximum of their two tail-error probabilities at the pairwise separation threshold is at least $1 / 2$ .

[F6]

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique: a finite product of sigma-finite factor laws is the unique measure with its rectangle values.

Proof

1.1givenF1algebra

Suppose $ω (ϵ, Θ) \leq M_{0}$ . For any estimator $\hat{θ}$ , the clean premise says its supremum over $θ$ of the clean probability at threshold $M_{0}$ is at least $c_{0}$ . For each $θ$ , the Huber supremum may choose $Q = P_{θ}$ , giving $P_{ϵ, θ, Q} = P_{θ}$ by [F1]. Since $M_{ϵ} = (κ / 4) M_{0} \leq M_{0}$ , the event at $M_{ϵ}$ contains the clean event at $M_{0}$ . Taking suprema therefore gives at least $c_{0}$ for every estimator.

1.2F2givenchoose

Suppose $ω (ϵ, Θ) > M_{0}$ and is finite. Then $ω > 0$ , so the definition of supremum supplies $θ_{0}, θ_{1}$ with $L (θ_{0}, θ_{1}) > ω / 2$ ; if $ϵ < 1$ this pair satisfies $T V (P_{θ_{0}}, P_{θ_{1}}) \leq ϵ / (1 - ϵ)$ , while if $ϵ = 1$ every pair is allowed.

2.1F2givenstep 1.1

If $ω (ϵ, Θ) \leq M_{0}$ , step 1.1 gives the asserted bound.

2.2A1F1F4F6step 1.2chooseconstruct

By [F4] choose $Q_{0}, Q_{1}$ making the two one-observation Huber laws equal to a common law $R_{1}$ . By [F1], each sample law is the product of $N$ copies of $R_{1}$ . The finite product sigma-algebra and measure are formed recursively; at each stage the existing partial product is sigma-finite, so [F6] supplies the next factor product. Both parameters therefore give the same recursively constructed sample law $R : = R_{1}^{\otimes N}$ .

3.1F2F3F5step 1.2step 2.2algebra

The loss-separation hypothesis gives $\max_{i} L (θ_{i}, a) \geq κ L (θ_{0}, θ_{1}) > κ ω / 2 > M_{ϵ}$ for every $a$ . Thus the events at threshold $κ L (θ_{0}, θ_{1})$ in [F5] are contained in those at threshold $M_{ϵ}$ . Apply [F5] under $R$ to get $\max_{i} R {L (θ_{i}, \hat{θ}) \geq M_{ϵ}} \geq 1 / 2$ . The supremum over $θ, Q$ is at least this maximum for each estimator.

4.1F3givenstep 2.1step 3.1algebra

If $ω \leq M_{0}$ , step 2.1 gives $c_{0}$ ; if $ω > M_{0}$ , step 3.1 gives $1 / 2$ . Taking the infimum over estimators proves the stated $\min (c_{0}, 1 / 2)$ bound. The explicit formula for $M_{ϵ}$ gives comparability to $M_{0} \lor ω$ . If $c_{0} \geq c_{*} > 0$ for a fixed $c_{*} \leq 1 / 2$ , then $\min (c_{0}, 1 / 2) \geq c_{*}$ , yielding a universal probability constant under that additional clean-bound normalization.

5.1A1F2F4F5choosealgebra∎

If $ω = + \infty$ , fix any finite threshold $T$ . The supremum definition gives a pair with $L (θ_{0}, θ_{1}) > 2 T / κ$ satisfying the TV constraint (or any pair when $ϵ = 1$ ). Equalize their mixtures by [F4]. Pairwise separation makes the events at threshold $κ L (θ_{0}, θ_{1})$ in [F5] subsets of the corresponding events at threshold $T$ , so [F5] gives a maximum tail probability at threshold $T$ of at least $1 / 2$ . This holds for every estimator and hence for the minimax supremum.

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber’s Contamination Model, §5, Theorem 5.1, and proof in the supplement (standard reference, not scraped)
Philippe Rigollet, 18.S997: High-Dimensional Statistics, Lecture Notes, Chapter 5, §§5.1–5.3 (standard reference, not scraped)