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Huber-robust heavy-tailed confidence sequences

Mathematical Methods AIMech:HuberRobustHeavyTailedConfidenceSequences DRAFT

Aliases: Huber-Robust Confidence Sequence, TV-Contaminated Heavy-Tailed Robust Confidence Sequence, Robust Catoni Supermartingale Confidence Sequence, Infinite-Variance p-Moment Huber-Robust Confidence Sequence, Huber-Robust Kelly Betting Supermartingale Confidence Sequence

A family of nonasymptotic, anytime-valid sequential confidence sequences (Wang & Ramdas 2023, AISTATS / PMLR 206:9662–9679) for the mean mu(P) of an uncorrupted distribution P with bounded variance Var_P(X) <= sigma^2 (or bounded p-th central moment E_P[|X - mu(P)|^p] <= kappa for 1 < p <= 2) when observations X_1, X_2, ... are drawn sequentially from an adversarially contaminated distribution Q in the total-variation neighborhood B_TV(P, varepsilon) = {Q : D_TV(P, Q) <= varepsilon} (strictly generalizing Huber's gross-error contamination model Q = (1 - varepsilon) P + varepsilon N with no moment assumptions on N), constructed by pairing the bounded narrowest Catoni influence function phi(x) in [-log 2, log 2] (where exp(+-phi(x)) lies in [1/2, 2] of range 1.5, or phi_p(x) in [-log p, log p] with exp(+-phi_p(x)) in [1/p, p] of range p - 1/p) with the total-variation integral probability metric shift to form the nonnegative robust Catoni supermartingales M_t^{RC} = prod_{i=1}^t [exp(phi(lambda_i(X_i - mu(P)))) / (1 + lambda_i^2 sigma^2 / 2 + 1.5 varepsilon)] and N_t^{RC} = prod_{i=1}^t [exp(-phi(lambda_i(X_i - mu(P)))) / (1 + lambda_i^2 sigma^2 / 2 + 1.5 varepsilon)], whose Ville inversion CI_t^R = {m in R : |sum_{i=1}^t phi(lambda_i(X_i - m))| <= sum_{i=1}^t log(1 + lambda_i^2 sigma^2 / 2 + 1.5 varepsilon) + log(2/alpha)} under constant tuning lambda = sqrt(varepsilon)/(2 sigma) achieves the minimax-optimal high-probability diameter O(sigma sqrt(varepsilon)) (specifically <= 28 sigma sqrt(varepsilon) for varepsilon <= 1/7 and <= (14 p / (p - 1)) kappa^{1/p} varepsilon^{(p-1)/p} for 1 < p < 2) and non-trivial finite breakdown point varepsilon < 8/13.

Identity Notes & Concept Scope No normalized identity match for Huber-robust heavy-tailed confidence sequences across all 442 pinned AIO records and 578 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only lines in Round 183 artifacts where Wang & Ramdas (2023, AISTATS / PMLR 206:9662-9679, arXiv:2301.09573) was queued as a pending frontier lead. Distinct from AIMech:CatoniStyleHeavyTailedConfidenceSequences (Wang & Ramdas 2023, Stochastic Processes and their Applications 163:168-202, arXiv:2202.01250), which assumes uncontaminated observations E[X_t | F_{t-1}] = mu with time-decaying tuning lambda_t = Theta(1/sqrt(t log t)) and unclipped influence functions that blow up under a single adversarial outlier Distribution N with infinite moments, whereas HuberRobustHeavyTailedConfidenceSequences enforces bounded influence phi(x) in [-log 2, log 2] (so exp(+-phi) has bounded TV sensitivity 1.5), incorporates the explicit TV-contamination compensation +1.5 varepsilon in the supermartingale denominator, and uses constant tuning lambda_t = sqrt(varepsilon)/(2 sigma) to match the non-vanishing information-theoretic lower bound sigma sqrt(varepsilon) over B_TV(P, varepsilon).

Claims & Grounded Evidence

4 Claims

Mechanism #tv-contamination-minimax-width-lower-bound-and-robust-catoni-supermartingales

For the class M_{sigma^2}^2 of distributions P with Var_P(X) <= sigma^2 and corrupted data X_1, X_2, ... ~ Q in the total-variation ball B_TV(P, varepsilon) = {Q : D_TV(P, Q) <= varepsilon} (which strictly contains Huber's gross-error contamination neighborhood {(1 - varepsilon) P + varepsilon N}; Definition 1, Remark 1), Lemma 1 and Lemma 6 establish via the two-point witness P_1 = delta_0 and P_2 = (1 - varepsilon) delta_0 + varepsilon delta_{sigma / sqrt(varepsilon)} in M_{sigma^2}^2 (with |mu(P_1) - mu(P_2)| = sigma sqrt(varepsilon) and D_TV(P_1, P_2) = varepsilon) that any (varepsilon, 1 - alpha)-RCI or RCS with alpha in (0, 1/2) must have diameter >= sigma sqrt(varepsilon) with probability at least 1 - 2 alpha > 0; to achieve time-uniform validity over B_TV(P, varepsilon), Lemma 3 pairs Catoni's narrowest bounded influence function phi(x) in [-log 2, log 2] (satisfying -log(1 - x + x^2/2) <= phi(x) <= log(1 + x + x^2/2) and exp(+-phi(x)) in [1/2, 2] of range 2 - 1/2 = 1.5) with the TV integral probability metric bound sup_{f in [c_1, c_2]} |E_P f - E_Q f| = (c_2 - c_1) D_TV(P, Q) (Equation 5) to prove E_Q[exp(+-phi(lambda_t(X_t - mu(P))))] <= 1 + lambda_t^2 sigma^2 / 2 + 1.5 varepsilon, making M_t^{RC} and N_t^{RC} nonnegative supermartingales and CI_t^R a valid (varepsilon, 1 - alpha)-RCS via Ville's inequality (Theorem 1).

supports Source: Huber-Robust Confidence Sequences · Locator: Section 2, lines 25–64, Equation 5, Definition 1, Remark 1, Lemma 1; Section 3.1, lines 65–96, Equations 13–24, Lemma 3, Theorem 1; Appendix A, lines 764–790, Lemma 6 and Proof of Lemma 1
supports Source: Catoni-style confidence sequences for heavy-tailed mean estimation · Locator: Section 6, lines 99–114, Equation 12, Lemma 8 (Uncontaminated narrowest Catoni influence function and supermartingale)

Mechanism #constant-tuning-minimax-width-concentration-and-breakdown-point

Whereas uncontaminated confidence sequences use decaying weights lambda_t = Theta(t^{-1/2}), under varepsilon > 0 any polynomially decaying or growing schedule lambda_t = 0.5 varepsilon^{1/2} sigma^{-1} t^u with u != 0 causes the robust confidence sequence width to diverge as t grows (either because lambda_t -> 0 inflates the root scale for u < 0 or because sum_{i=1}^t log(1 + lambda_i^2 sigma^2 / 2 + 1.5 varepsilon) exceeds the saturation ceiling t log 2 for u > 0), whereas constant tuning lambda_1 = ... = lambda = sqrt(varepsilon)/(2 sigma) (lambda^2 = varepsilon / (4 sigma^2)) guarantees for all 0 < varepsilon <= 1/7 and t > 4 varepsilon^{-1} log(4 / (alpha delta)) that P_Q(diam(CI_t^R) <= 28 sigma sqrt(varepsilon)) >= 1 - delta (Theorem 2, proved via Lemma 7's bounds A(x) <= 6.5 x and B(x) <= 7 x on [0, 1/7]), matching the sigma sqrt(varepsilon) lower bound up to constant 28, strictly improving upon the >= 96 sqrt(2) sigma sqrt(varepsilon) RCI width simulated from Lugosi & Mendelson's (2021) sample-split trimmed mean RRCI via coupling (Definition 2, Lemma 4, Theorem 3), and achieving a non-trivial finite breakdown point varepsilon < 8/13 approx 61.5% (since log(1 + 13 varepsilon / 8) < log 2 iff varepsilon < 8/13).

supports Source: Huber-Robust Confidence Sequences · Locator: Section 3.3, lines 105–113, Theorem 2, Figure 3; Section 4, lines 114–136, Definition 2, Lemma 4, Theorem 3, Figure 4; Section 5.1, lines 137–142 (Breakdown point varepsilon < 8/13); Appendix A, lines 791–845, Lemma 7, Proof of Theorem 2, and Proof of Lemma 4
supports Source: Huber-Robust Confidence Sequences (PMLR v206:9662-9679 Proceeding Record) · Locator: Abstract (Constant tuning proportional to sqrt(varepsilon)/sigma and high-probability width O(sigma sqrt(varepsilon)))

Mechanism #huber-robust-sequential-hypothesis-testing-exponential-e-process-growth

For the sequential composite robust mean testing problem H_0 : P in M_{sigma^2}^2, mu(P) <= mu_0 versus H_1 : P in M_{sigma^2}^2, mu(P) > mu_0 under corrupted observations X_1, X_2, ... ~ Q in B_TV(P, varepsilon), the one-sided robust Catoni supermartingale M_t^{RC}(mu_0) = prod_{i=1}^t [exp(phi(lambda(X_i - mu_0))) / (1 + lambda^2 sigma^2 / 2 + 1.5 varepsilon)] with lambda = sqrt(varepsilon)/(2 sigma) controls the sequential Type-I error below alpha across all stopping times via Ville's inequality and, whenever the true mean exceeds the irreducible ambiguity margin mu(P) > mu_0 + 14 sigma sqrt(varepsilon) for varepsilon in (0, 1/7], grows exponentially fast with asymptotic rate at least varepsilon / 4: for any delta in (0, 1), P_Q(t^{-1} log M_t^{RC}(mu_0) >= varepsilon / 4 - t^{-1} log(2/delta)) >= 1 - delta/2 for all t >= 1 (Section 5.1, Corollary 1).

supports Source: Huber-Robust Confidence Sequences · Locator: Section 5.1, lines 143–152, Equation 34, Corollary 1; Appendix A, lines 846–851, Proof of Corollary 1

Mechanism #infinite-variance-p-moment-robust-cs-and-bounded-robust-kelly-betting

When P has infinite variance but finite p-th central moment E_P[|X - mu(P)|^p] <= kappa for known 1 < p < 2 (P in M_kappa^p), replacing phi with the truncated p-Catoni influence function phi_p(x) in [-log p, log p] (satisfying -log(1 - x + |x|^p/p) <= phi_p(x) <= log(1 + x + |x|^p/p) with exp(+-phi_p(x)) in [1/p, p] of range p - 1/p) makes M_t^{RC,p} = prod_{i=1}^t [exp(phi_p(lambda_i(X_i - mu(P)))) / (1 + lambda_i^p kappa / p + (p - 1/p) varepsilon)] and N_t^{RC,p} nonnegative supermartingales over B_TV(P, varepsilon) (Lemma 5), yielding the (varepsilon, 1 - alpha)-RCS CI_t^{R,p} (Theorem 4) whose constant-tuned width (lambda_t = (varepsilon / kappa)^{1/p}) satisfies P_Q(diam(CI_t^{R,p}) <= (14 p / (p - 1)) kappa^{1/p} varepsilon^{(p-1)/p}) >= 1 - delta for 0 < varepsilon <= (p - 1)/(7 p) and t >= varepsilon^{-1} log(4/(alpha delta)) (Theorem 5, Lemmas 8–11, matching the Omega(kappa^{1/p} varepsilon^{(p-1)/p}) lower bound up to constants); in the sibling case where both P and Q are supported on [0, 1], Theorem 6 proves that L_t = prod_{i=1}^t (1 + lambda_i(X_i - mu(P)) - varepsilon |lambda_i|) is a nonnegative Huber-robust Kelly betting supermartingale for any predictable lambda_i in [-(1 - mu(P) + varepsilon)^{-1}, (mu(P) + varepsilon)^{-1}].

supports Source: Huber-Robust Confidence Sequences · Locator: Section 5.2–5.3, lines 153–175, Equations 36–42, Lemma 5, Theorems 4–6; Appendix A, lines 853–927, Lemmas 8–11 and Proofs of Lemma 5 and Theorems 4–6

⚠️ Curated Limitations & Negative Results

By Lemma 1 and Lemma 6, no (varepsilon, 1 - alpha)-robust confidence interval or confidence sequence over M_{sigma^2}^2 under TV contamination radius varepsilon > 0 can shrink to zero width as t -> infinity: any valid RCS must have diameter at least sigma sqrt(varepsilon) with probability >= 1 - 2 alpha, and similarly Omega(kappa^{1/p} varepsilon^{(p-1)/p}) over M_kappa^p for 1 < p < 2 (Section 2, Section 5.2).
Achieving the minimax O(sigma sqrt(varepsilon)) width requires a known upper bound sigma^2 on the uncorrupted variance (or kappa on the p-th central moment) and a known upper bound varepsilon on the total-variation contamination fraction in order to set the constant tuning parameter lambda = sqrt(varepsilon)/(2 sigma) (or lambda = (varepsilon / kappa)^{1/p}); furthermore, under lambda = sqrt(varepsilon)/(2 sigma), the per-step threshold log(1 + 13 varepsilon / 8) exceeds the influence cap log 2 whenever varepsilon >= 8/13 (and Theorem 2's high-probability bound requires varepsilon <= 1/7 and t > 4 varepsilon^{-1} log(4/(alpha delta))) (Sections 3.3 and 5.1).
The sequential robust confidence sequence model (Definition 1) assumes each observation X_t is drawn online from a distribution Q_t in B_TV(P, varepsilon); it does not cover an offline post-hoc replacement adversary who inspects the entire completed sample (X_1, ..., X_t) before retroactively replacing up to varepsilon t points (Definition 2, Section 4).
All 938 lines of Wang & Ramdas (2023, arXiv:2301.09573) were read. Constructed numerical checks in research/discovery/huber-robust-heavy-tailed-confidence-sequences-boundaries.py verify Lemma 1/6 minimax lower bound witness, Lemma 3 bounded narrowest Catoni influence range [1/2, 2] and supermartingale expectation <= 1 under varepsilon = 1/9 outlier contamination at +1000 vs. unclipped Catoni blowup (176.21), Lemma 7 algebraic bounds, Theorem 1/2 0.0% anytime miscoverage and mean width 7.35 < 28 sigma sqrt(varepsilon) < 135.76 (Lugosi-Mendelson RCI) vs. 100% miscoverage for non-robust Catoni and sub-Gaussian CSs, Section 3.3 constant vs. polynomial tuning schedules and Corollary 1 exponential test martingale growth, and Section 5.2–5.3 infinite-variance p-moment RCS and bounded Kelly robust supermartingale expectations.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Huber-robust heavy-tailed confidence sequences guarantee nonasymptotic time-uniform coverage across arbitrary stopping times even under total-variation adversarial contamination and heavy tails, preventing false-positive inflation from continuous sequential peeking (AIO:DataDredgingBias).
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	Robust Catoni and Kelly supermartingales operate sequentially over filtrations (F_t)_{t>=0} where each observation X_t may be drawn from a history-dependent distribution in B_TV(P, varepsilon) (Section 3.1, line 83; AIO:TimeSeriesAnalysis).

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`wang-ramdas2023-huber-cs-ar5iv`	Huber-Robust Confidence Sequences Hongjian Wang, Aaditya Ramdas	primary paper full_text	ID: https://proceedings.mlr.press/v206/wang23b.html sha256:86ca019ece98…
`wang-ramdas2023-huber-cs-pmlr`	Huber-Robust Confidence Sequences (PMLR v206:9662-9679 Proceeding Record) Hongjian Wang, Aaditya Ramdas	primary paper abstract_only	ID: https://proceedings.mlr.press/v206/wang23b.html sha256:ddfe27568ae0…
`wang-ramdas2023-catoni-cs`	Catoni-style confidence sequences for heavy-tailed mean estimation Hongjian Wang, Aaditya Ramdas	primary paper full_text	ID: https://doi.org/10.1016/j.spa.2023.05.007 sha256:e80a2fc7a2a4…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T13:05:00Z

Updated: 2026-10-17T13:05:00Z

Source File: data/curated/HuberRobustHeavyTailedConfidenceSequences.yaml · Schema: curated_concept.yaml

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