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Catoni-style heavy-tailed confidence sequences

Mathematical Methods AIMech:CatoniStyleHeavyTailedConfidenceSequences DRAFT

Aliases: Catoni Supermartingale Confidence Sequence for Heavy-Tailed Means, Infinite-Variance p-Catoni Confidence Sequence, Stitched Catoni Law-of-the-Iterated-Logarithm Confidence Sequence, Dubins-Savage and Delyon Self-Normalized Heavy-Tailed Confidence Sequences

A family of nonasymptotic, anytime-valid sequential confidence sequences (Wang & Ramdas 2023, building on Catoni 2012) for the conditional mean mu of potentially heavy-tailed real-valued stochastic processes (X_t)_{t>=1} knowing only an upper bound sigma^2 on the conditional variance E[(X_t - mu)^2 | F_{t-1}] <= sigma^2 (or more generally knowing only a p-th central moment bound E[|X_t - mu|^p | F_{t-1}] <= v_t for 1 < p <= 2 when the variance is infinite), constructed by applying Ville's maximal inequality to the nonnegative Catoni supermartingales M_t^C = prod_{i=1}^t exp(phi(lambda_i(X_i - mu)) - lambda_i^2 sigma^2 / 2) and N_t^C = prod_{i=1}^t exp(-phi(lambda_i(X_i - mu)) - lambda_i^2 sigma^2 / 2) for any influence function phi satisfying -log(1 - x + x^2/2) <= phi(x) <= log(1 + x + x^2/2) (or the p-Catoni influence function -log(1 - x + |x|^p/p) <= phi_p(x) <= log(1 + x + |x|^p/p)) with predictable tuning sequence (lambda_t)_{t>=1}, yielding connected root-finding intervals CI_t^C = {m in R : |sum_{i=1}^t phi(lambda_i(X_i - m))| <= sum_{i=1}^t lambda_i^2 sigma^2 / 2 + log(2/alpha)} (and stitched intervals CI_t^{Cstch}) that achieve logarithmic dependence O(log(1/alpha)) in the error level alpha, match the optimal law-of-the-iterated-logarithm rate O(sqrt(t^{-1} log log t)) for finite variance, and attain width O_tilde(t^{-(p-1)/p} log(1/alpha)) for infinite-variance distributions with 1 < p < 2.

Identity Notes & Concept Scope No normalized identity match for Catoni-style heavy-tailed confidence sequences across all 442 pinned AIO records and 577 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only lines in Round 182 discovery artifacts where Wang & Ramdas (2023, arXiv:2202.01250) was queued as a pending frontier lead. Distinct from AIMech:SubPsiUniformBoundaryConfidenceSequences (Howard, Ramdas, McAuliffe, & Sekhon 2021), which requires light-tailed sub-psi / sub-gamma / empirical-Bernstein conditions or bounded increments to control exponential processes exp(lambda S_t - psi(lambda) V_t), and from AIMech:BettingConfidenceSequencesForBoundedMeans (Waudby-Smith & Ramdas 2024), which requires observations bounded in [0, 1], whereas CatoniStyleHeavyTailedConfidenceSequences transforms raw unbounded observations inside the exponent via Catoni's M-estimator influence function phi(lambda_i(X_i - m)) (and its p-Catoni generalization phi_p for 1 < p < 2) so that only a 2nd or p-th central moment bound is required without any moment generating function or boundedness assumption.

Claims & Grounded Evidence

4 Claims

Mechanism #catoni-influence-supermartingale-and-tightness-necessity

For any sequence of random variables (X_t)_{t>=1} satisfying E[X_t | F_{t-1}] = mu and E[(X_t - mu)^2 | F_{t-1}] <= sigma^2, any predictable positive sequence (lambda_t)_{t>=1}, and any influence function phi satisfying -log(1 - x + x^2/2) <= phi(x) <= log(1 + x + x^2/2) (such as the narrowest choice phi(x) = log(1 + x + x^2/2) for x >= 0 and -log(1 - x + x^2/2) for x < 0), the inequality exp(+-phi(lambda_i(X_i - mu))) <= 1 +- lambda_i(X_i - mu) + lambda_i^2(X_i - mu)^2 / 2 implies that M_t^C = prod_{i=1}^t exp(phi(lambda_i(X_i - mu)) - lambda_i^2 sigma^2 / 2) and N_t^C = prod_{i=1}^t exp(-phi(lambda_i(X_i - mu)) - lambda_i^2 sigma^2 / 2) are nonnegative supermartingales with initial value 1 (Lemma 8) and that CI_t^C = {m in R : |sum_{i=1}^t phi(lambda_i(X_i - m))| <= sum_{i=1}^t lambda_i^2 sigma^2 / 2 + log(2/alpha)} is a connected (1 - alpha)-confidence sequence (Theorem 9); conversely, Proposition 16 proves this variance condition is tight because if E[(Y_t - mu)^2 | F_{t-1}] = v_t > (1 + 2 kappa) sigma^2 on an event S in F_{t-1} of positive probability, there exists lambda_t > 0 for which E[exp(phi(lambda_t(Y_t - mu)) - lambda_t^2 sigma^2 / 2) | F_{t-1}] > 1 on S.

supports Source: Catoni-style confidence sequences for heavy-tailed mean estimation · Locator: Section 6, lines 99–114, Equations 26–33, Lemma 8, Theorem 9; Appendix B.3, lines 650–670, Proposition 16
supports Source: Challenging the empirical mean and empirical variance: A deviation study · Locator: Abstract and Section 2 (Influence function bounds -log(1 - x + x^2/2) <= phi(x) <= log(1 + x + x^2/2))

Mechanism #comparison-with-dubins-savage-and-delyon-self-normalized-confidence-sequences

While the Dubins-Savage confidence sequence CI_t^{DS} = [sum_{i=1}^t lambda_i X_i / sum_{i=1}^t lambda_i +- (2/alpha - 1 + sigma^2 sum_{i=1}^t lambda_i^2) / sum_{i=1}^t lambda_i] (Theorem 2, derived from Lemma 1's linear supermartingale inequality P(exists t : S_t >= a + b V_t) <= 1/(1 + a b)) suffers polynomial width dependence O(1/alpha) (or O(alpha^{-1/2}) when tuned to alpha), and the Self-Normalized confidence sequence CI_t^{SN} (Lemma 5, Theorem 7, derived from Delyon's inequality exp(x - x^2/6) <= 1 + x + x^2/3 via M_t^{SN} = prod_{i=1}^t exp(lambda_i(X_i - mu) - lambda_i^2(X_i - mu)^2 / 6 - lambda_i^2 sigma^2 / 3)) achieves O(log(1/alpha)) scaling by placing the quadratic empirical-variance term -(lambda_i^2/6)(X_i - mu)^2 in the exponent at the cost of producing a disconnected three-component union of intervals L_t^{SN} U M_t^{SN} U U_t^{SN} (complement of two anticonfidence intervals when the quadratic discriminants D^+, D^- are positive), the Catoni-style confidence sequence CI_t^C simultaneously preserves connected single-interval topology [u_t, l_t] and logarithmic O(log(1/alpha)) width scaling.

supports Source: Catoni-style confidence sequences for heavy-tailed mean estimation · Locator: Section 1, lines 28–39, Table 1; Section 3, lines 53–68, Lemma 1, Theorem 2; Section 5, lines 77–98, Lemmas 4–6, Theorem 7; Section 6, lines 99–114, Theorem 9; Appendix B.1–B.2, lines 616–648

Mechanism #nonasymptotic-quadratic-width-bound-and-stitched-iterated-logarithm-rate

Using the cubic upper bound phi(x) <= x - x^2/2 + x^3/3, Theorem 10 proves that whenever the weighted empirical variance sigma_hat_i^2 = (X_i - mu_hat)^2 around mu_hat = sum_{i=1}^t lambda_i X_i / sum_{i=1}^t lambda_i satisfies the discriminant condition Delta_t = (sum_{i=1}^t lambda_i)^2 - 2(sum_{i=1}^t lambda_i^3)(sum_{i=1}^t lambda_i^2 sigma_hat_i^2 + sum_{i=1}^t lambda_i^2 sigma^2 / 2 + log(2/alpha)) >= 0, the roots [u_t, l_t] of CI_t^C exist and have half-width bounded deterministically by the smaller quadratic root rho_t = (sum_{i=1}^t lambda_i - sqrt(Delta_t)) / sum_{i=1}^t lambda_i^3 (Appendix B.4, Equations 34 and 72–76), which yields O(sqrt(t^{-1} log t)) shrinkage under predictable plugin tuning lambda_t = Theta(1/sqrt(t log t)) (Corollary 10.1) and attains the optimal law-of-the-iterated-logarithm rate O(sqrt(t^{-1} log log t)) via geometric epoch stitching CI_t^{Cstch} = cap_{s=1}^t I_s in Section 6.2 (Corollary 10.2, Lemma 17).

supports Source: Catoni-style confidence sequences for heavy-tailed mean estimation · Locator: Section 6.1–6.2, lines 115–144, Theorem 10, Corollaries 10.1–10.2, Equations 34–38; Section 8, lines 155–175, Figures 4–5; Appendix B.4–B.5, lines 672–758, Lemma 17
supports Source: Time-uniform, nonparametric, nonasymptotic confidence sequences · Locator: Section 3.1, lines 221–248, Theorem 1 (Geometric epoch stitching over LIL boundaries)

Mechanism #infinite-variance-p-catoni-supermartingale-confidence-sequences

When the conditional variance is infinite but the conditional p-th central moment is bounded by E[|X_t - mu|^p | F_{t-1}] <= v_t for known 1 < p < 2, replacing the quadratic Catoni influence function with the p-Catoni influence function satisfying -log(1 - x + |x|^p/p) <= phi_p(x) <= log(1 + x + |x|^p/p) makes M_t^{Cp} = prod_{i=1}^t exp(phi_p(lambda_i(X_i - mu)) - lambda_i^p v_i / p) and N_t^{Cp} = prod_{i=1}^t exp(-phi_p(lambda_i(X_i - mu)) - lambda_i^p v_i / p) nonnegative supermartingales (Lemma 13) and yields the connected confidence sequence CI_t^{Cp} = {m in R : |sum_{i=1}^t phi_p(lambda_i(X_i - m))| <= sum_{i=1}^t lambda_i^p v_i / p + log(2/alpha)} (Theorem 14), whose half-width is bounded via the inequality |x + y|^p <= |x|^p + 4|y|^p + p y |x|^{p-1} sgn(x) (Lemma 21) and root analysis of g(r) = 4 A r^p - B r + C (Lemmas 19–20) to contract at rate O_tilde(t^{-(p-1)/p} log(1/alpha)) when lambda_t = Theta_tilde(t^{-1/p}) (Theorem 15, Corollary 15.1).

supports Source: Catoni-style confidence sequences for heavy-tailed mean estimation · Locator: Section 9, lines 176–196, Equations 47–52, Lemma 13, Theorems 14–15, Corollary 15.1; Appendix C, lines 759–803, Lemmas 19–21

⚠️ Curated Limitations & Negative Results

Constructing valid Catoni-style (or p-Catoni) confidence sequences requires a known a priori upper bound sigma^2 on the conditional variance E[(X_t - mu)^2 | F_{t-1}] <= sigma^2 (or a known upper bound v_t on the p-th central moment E[|X_t - mu|^p | F_{t-1}] <= v_t); by Bahadur-Savage-style impossibility and Proposition 16, if the true variance exceeds (1 + 2 kappa) sigma^2 on a positive-probability event, the Catoni process ceases to be a supermartingale and anytime coverage can fail (Section 2, Section 7, Appendix B.3).
At very small sample sizes t before the discriminant condition Delta_t >= 0 of Theorem 10 is satisfied (for example when a single extreme outlier makes sum_{i=1}^t lambda_i(X_i - m) fail to cross the threshold within the quadratic regime or whenphi is chosen from a bounded influence family), root-finding must be performed numerically via bisection/Newton-Raphson and bounded influence functions phi can yield infinite confidence intervals at small t (Section 6, Remark 2, Theorem 10).
Under adversarial contamination where an epsilon-fraction of observations come from an arbitrary corrupting distribution with unbounded moments, standard Catoni-style confidence sequences can be biased unless modified to incorporate Huber-robust contamination budgets (Section 10, Wang & Ramdas 2023 AISTATS).
All 804 lines of Wang & Ramdas (2023, arXiv:2202.01250) were read. Constructed numerical checks in research/discovery/catoni-style-heavy-tailed-confidence-sequences-boundaries.py verify Catoni influence bounds, Lemma 8 supermartingale expectations on heavy-tailed Student-t_3 and Proposition 16 tightness failure under underestimated variance, Delyon's inequality and the disconnected 3-component topology of Self-Normalized CS vs. connected O(log(1/alpha)) Catoni CS and O(1/alpha) Dubins-Savage CS, Student-t_3 anytime coverage vs. naive CLT peeking failure and Theorem 10's quadratic width bound rho_t, and Section 9's infinite-variance p-Catoni (p = 1.5) Lemma 21 inequality and anytime coverage on symmetric Pareto(alpha_tail = 1.8) streams.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Catoni-style and p-Catoni confidence sequences guarantee nonasymptotic time-uniform coverage across all stopping times under heavy-tailed distributions with only finite variance or finite p-th central moment (1 < p < 2), preventing false-positive inflation from continuous peeking in sequential experiments (AIO:DataDredgingBias).
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	Catoni-style supermartingales track conditional means across dependent sequential processes (X_t)_{t>=1} adapted to filtrations (F_t)_{t>=0} with time-varying predictable variance bounds v_t (AIO:TimeSeriesAnalysis).

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`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…
`catoni2012-deviation-study`	Challenging the empirical mean and empirical variance: A deviation study Olivier Catoni	primary paper abstract_only	ID: https://doi.org/10.1214/11-AIHP454 sha256:af6e53dac76c…
`howard-ramdas-mcauliffe-sekhon2021-subpsi`	Time-uniform, nonparametric, nonasymptotic confidence sequences Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, Jasjeet Sekhon	primary paper full_text	ID: https://doi.org/10.1214/20-AOS1991 sha256:aef352a6e271…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T12:50:00Z

Updated: 2026-10-17T12:50:00Z

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

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