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Auditors can continuously bet on their own features to prove that black-box quantile forecasts are miscalibrated, with anytime-valid evidence that grows against feature-aligned failures.

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2026-07-14 04:05 UTC pith:OOUG4CE5

load-bearing objection Solid anytime-valid, feature-aware quantile audit with a real information hierarchy and non-i.i.d. regret-to-power; soft spots are scope and experiment hygiene, not the core math. the 2 major comments →

arxiv 2607.11653 v1 pith:OOUG4CE5 submitted 2026-07-13 cs.LG stat.ML

Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters

Ivane Antonov , Sohom Mukherjee , Richard Pibernik , Yo Joong Choe This is my paper
classification cs.LG stat.ML
keywords conditional quantile calibrationanytime-valid inferencetesting by bettinge-processesfeature-aware auditingonline convex optimizationnon-i.i.d. streamsblack-box forecasting
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Black-box quantile forecasts are used for sequential decisions under asymmetric costs, yet once deployed they must be watched continuously as streams drift. Standard fixed-horizon backtests lose validity under optional stopping and ignore that calibration itself is information-dependent: a forecast can look fine to an auditor with coarse data while being systematically wrong once richer features are seen. This paper builds a distribution-free sequential betting game in which the auditor stakes on hit errors using only the information available before each outcome. Under the calibration null relative to that information, every legal bet yields an anytime-valid evidence process. Power appears only when the auditor can bet on predictable structure visible in its own features. Linear contextual bets learned online from a feature dictionary convert pathwise regret bounds into finite-time detection guarantees against non-i.i.d. alternatives that retain a persistent feature-aligned edge. The resulting wealth paths are readable at the feature level and, on Chronos-2 forecasts for synthetic and store-sales data, expose promotion- and calendar-driven miscalibration that marginal audits miss.

Core claim

Conditional quantile calibration is indexed by the auditor's monitoring information, and coarser valid audits can be powerless against richer violations. When full-feature miscalibration has a persistent linear predictable edge, any online learner with a pathwise regret bound produces an anytime-valid wealth process that detects the violation in finite time and yields explicit stopping-time bounds, without independence or stationarity assumptions.

What carries the argument

Contextual betting over a predictable feature dictionary: the auditor forms linear stakes λ_t(θ) = ⟨θ, φ_t⟩ inside a no-bankruptcy interval and updates θ online; under the monitoring null the product wealth is a test martingale (hence an e-process), and a regret-to-power theorem converts any pathwise OCO bound into finite-time detection against linear full-feature edge alternatives.

Load-bearing premise

Finite-time detection is guaranteed only when miscalibration has a lasting linear edge along some fixed direction in the auditor's chosen feature dictionary; if the true failure is intermittent, nonlinear, or invisible in those features, the power bounds do not apply.

What would settle it

On a stream where the true conditional hit probability equals α given the auditor's features, check whether the feature-aware wealth process still exceeds the Ville threshold 1/γ with frequency higher than γ; or, under a known linear edge of size κ, check whether empirical detection times exceed the paper's certification bound n^ctx_γ(r).

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Continuous monitoring of deployed quantile models can keep Type-I control under optional stopping while still detecting feature-specific failures.
  • An external auditor who sees only coarse aggregates may correctly fail to reject, even when an internal auditor with promotions or calendar flags rejects quickly.
  • Learned betting weights serve as diagnostics that name which features expose the forecaster's conditional bias.
  • Any online convex optimizer with sublinear regret can be plugged in as a contextual skeptic and inherits the finite-time power guarantee against linear-edge alternatives.
  • The same betting construction extends, in principle, to other elicitable functionals once a bounded identification function and a no-bankruptcy stake interval are available.

Where Pith is reading between the lines

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

  • If the feature dictionary is incomplete relative to the true drivers of error, the audit remains valid yet silent; practitioners may therefore treat non-rejection as 'no evidence under this view' rather than 'the forecaster is calibrated'.
  • Nonlinear or kernelized betting classes would be the natural next step for high-dimensional or interaction-driven miscalibration while keeping the same martingale validity argument.
  • Head-to-head comparison of two forecasters could be cast as a relative betting game on the same hit stream, reusing the same anytime-valid machinery.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops an anytime-valid, distribution-free framework for auditing black-box conditional quantile forecasters under non-i.i.d. streams. Calibration nulls are indexed by the auditor’s monitoring information H, yielding a hierarchy P^ϕ_0 ⊆ P^w_0 ⊆ P^marg_0. Predictable no-bankruptcy bets produce test martingales / e-processes for P0(H) (Thm. 2.5, Cor. 2.6–2.7). Validity transfers from coarser to richer nulls, while coarser audits can be powerless against richer violations (Prop. 2.8). For power, the authors introduce contextual linear bets λ_t(θ)=⟨θ,ϕ_t⟩ learned by OCO over a predictable feature dictionary, and prove finite-time detection and stopping-time bounds against linear full-feature predictable-edge alternatives (Def. 3.1, Thm. 3.2; cumulative-edge extension Thm. C.8). Experiments show Type-I control under a Negative-Binomial oracle and feature-specific rejections for Chronos-2 on synthetic and Rossmann data that marginal audits miss.

Significance. The contribution is timely and useful: continuous monitoring of black-box quantile forecasts under asymmetric costs is a real operational need, and making the auditor’s information set explicit cleanly separates validity from power. The technical core is solid and standard in the right way—bounded MDS + Ville for validity, pathwise OCO regret + Hoeffding–Azuma for power—without i.i.d. assumptions. Strengths include an explicit null hierarchy, interpretable feature-level evidence, a cumulative-edge generalization in the appendix, and careful Type-I checks under a correctly specified oracle. If accepted, the paper would give practitioners a practical, anytime-valid audit toolkit and give theorists a clean information-indexed formulation of conditional quantile calibration.

major comments (2)
  1. The finite-time power claim (Thm. 3.2) is proved only for the linear full-feature predictable-edge class (Def. 3.1 / cumulative version Def. C.7): a fixed comparator θ* ∈ K with quadratic edge b^Q_t(θ*) ≥ κ a.s. for all t ≥ t0 (or cumulative envelope D_T). This is a genuine scope limitation, not a hidden flaw—the paper already flags it (Rem. 3.4, conclusion)—but the abstract and introduction state “finite-time detection guarantees for such alternatives” without always foregrounding that “such” means linear feature-aligned edge. Please state the alternative class explicitly in the abstract/intro and, if space allows, add one synthetic experiment with intermittent or nonlinear miscalibration to illustrate when the guarantee does and does not apply.
  2. Empirical power claims for Chronos-2 (Fig. 5 and App. D figures) rest on hyperparameter choices (lr, l1, shrink) and sliding-window context lengths that are selected by mean rejection on the full skeptic (App. D). While Type-I validity under the oracle is clean (Fig. 4), the real-data rejections could be sensitive to this selection. Please report a fixed-hyperparameter or pre-specified protocol (or a sensitivity summary) so that the feature-aware advantage over marginal audits is not confounded by post-hoc tuning.
minor comments (5)
  1. Typo: “Corrolary 2.6” in the contributions list (p. 4) should be “Corollary 2.6”.
  2. Figure 1 caption and body text refer to “promotion- and Saturday-aware audits”; ensure the corresponding feature names (promo, is_saturday) are consistent across Fig. 5 and the appendix panels.
  3. Notation: Z_t is defined as the centered hit, but some places write “hit process” for B_t and Z_t interchangeably; a one-sentence clarification early in §2 would help.
  4. Table B.1 and the OCO regret rates are useful; a short pointer in the main text to which algorithm is used in each main figure would improve readability.
  5. Related work is thorough; a brief sentence distinguishing the present information-indexed null from the full-filtration setting of Casgrain et al. (2024) and Wang et al. (2025) already appears but could be moved earlier for readers coming from the e-backtesting literature.

Circularity Check

0 steps flagged

No significant circularity: validity follows from MDS of centered hits under the defined nulls, and power is an implication from explicitly defined linear/cumulative edge alternatives plus pathwise OCO regret, not a tautology or fitted prediction.

full rationale

The paper's load-bearing chain is self-contained and non-circular. Nulls P0(H) are defined by the conditional hit probability equaling alpha (Def. 2.1); the centered hits Zt form a bounded MDS under those nulls by direct calculation (Prop. C.3). Predictable no-bankruptcy bets then yield test martingales/e-processes by the product construction and conditional mean-1 multipliers (Thm. 2.5, Cor. 2.6), with the hierarchy and validity-transfer following from the tower property (Prop. 2.2, Cor. 2.7) and the powerlessness of coarser bets under richer violations following from the same martingale property (Prop. 2.8). Finite-time power (Thm. 3.2) is proved only against the explicitly defined alternative class Q^{phi,lin}_{kappa,t0} (Def. 3.1) that assumes a fixed comparator with uniform quadratic edge; the proof converts any pathwise regret bound into log-wealth growth via the elementary inequality log(1+x) >= x - x^2 on the no-bankruptcy domain, then applies a conditional Hoeffding/Azuma bound (Lem. C.2, Thm. C.8). This is a standard implication, not a reduction of the conclusion to its inputs. Empirical Type-I checks use an oracle whose quantiles match the true conditional law by construction, so rejections are genuine false alarms; Chronos-2 results are observational detections, not parameters fitted to force a theoretical claim. Related-work citations (Casgrain et al., Wang et al., standard OCO) are external and not load-bearing uniqueness/ansatz imports. No self-definitional loop, fitted-input-as-prediction, or self-citation chain forces the central claims.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 2 invented entities

The central claims rest on standard martingale and OCO machinery plus modeling choices that define the monitoring game and the alternative class against which power is proved. Free parameters appear mainly in the experimental OCO configurations and the feature bound used to keep bets inside the no-bankruptcy set. No new physical entities are postulated; the invented objects are definitional constructs of the testing game.

free parameters (4)
  • OCO hyperparameters (learning rate, l1, shrink)
    Chosen per pipeline and auto-selected by mean rejection on the full skeptic; they affect reported power curves in §4 and Appendix D.
  • Feature radius R and feasible set K = {θ : ||θ||₂ ≤ 1/(2R)}
    Hand-set bound that keeps contextual bets inside a strict subset of the no-bankruptcy interval; enters the wealth process and regret analysis in §3.
  • Test level γ_test = 0.05 and horizon T
    Fixed experimental thresholds for rejection (τ_γ ≤ T); not estimated from data but control the reported Type-I and power rates.
  • Auditor feature dictionary Φ
    User-chosen predictable features (promo, Saturday, lags, seasonality, etc.); power is only guaranteed relative to this dictionary.
axioms (5)
  • standard math Ville’s inequality for nonnegative supermartingales
    Used to convert test martingales into anytime-valid Type-I control (Cor. 2.6).
  • domain assumption Under H-conditional quantile calibration, the centered hits Z_t form a bounded martingale-difference sequence w.r.t. H
    Definition 2.1 / Prop. C.3; the exact hit-coverage null is slightly stronger than the general quantile inequality unless atoms have probability zero.
  • standard math Pathwise sublinear regret bounds for projected OGD, Euclidean FTRL, and ONS on the exp-concave betting losses
    Invoked via Table B.1 and standard OCO references to obtain finite certification times in Thm. 3.2.
  • standard math Tower property for nested monitoring filtrations H^marg ⊆ H^w ⊆ H^ϕ
    Drives the null hierarchy (Prop. 2.2) and the validity-transfer / power-loss statements.
  • ad hoc to paper Linear full-feature predictable-edge alternative (uniform or cumulative edge along some θ* in K)
    Definition 3.1 and Def. C.7; this is the modeling restriction under which finite-time power is proved.
invented entities (2)
  • Information-indexed calibration nulls P0(H) no independent evidence
    purpose: Make the auditor’s monitoring filtration part of the null so validity and power become information-relative.
    Definitional construct; no independent physical evidence required beyond the game protocol.
  • Contextual betting wealth process M^ctx and linear predictable-edge alternative class no independent evidence
    purpose: Turn OCO over a feature dictionary into an e-process with finite-time detection guarantees.
    Methodological objects introduced to state Thm. 3.2; falsifiable only through the empirical audits.

pith-pipeline@v1.1.0-grok45 · 36498 in / 3619 out tokens · 38415 ms · 2026-07-14T04:05:00.497213+00:00 · methodology

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read the original abstract

Black-box conditional quantile forecasts are widely used for sequential decisions under asymmetric costs, such as inventory planning in supply chain management. Once deployed, such forecasters must be monitored continuously as data streams drift and regimes change; this invalidates standard, fixed-horizon backtests for calibration. Further, existing backtests do not take into account that the notion of calibration is, in fact, information-dependent: forecasts can look calibrated to an auditor with coarse information while being miscalibrated to an auditor with richer information. We develop a distribution-free and game-theoretic testing framework for continuously auditing black-box conditional quantile forecasters with non-i.i.d. losses, such that the resulting evidence process is powerful against predictably chosen alternatives specified by the features available to the auditor. We first formalize notions of conditional quantile calibration when different sets of features are available to the auditor, establishing that the coarseness of the auditor's information set determines the hardness of the testing problem. We then identify the sets of alternatives for which the auditor can achieve power, and focusing on contextual bets linear in the features, we derive finite-time detection guarantees for such alternatives, all without an i.i.d. assumption. The resulting evidence processes are interpretable at the feature level, as they quantify fine-grained, "feature-aware" evidence for miscalibration. We empirically validate these methods on simulated and real data, finding that a popular time series forecaster (Chronos-2) is highly miscalibrated w.r.t. multiple relevant features.

Figures

Figures reproduced from arXiv: 2607.11653 by Ivane Antonov, Richard Pibernik, Sohom Mukherjee, Yo Joong Choe.

Figure 1
Figure 1. Figure 1: Chronos-2 quantile forecasts on Rossmann store-sales data can pass marginal audits [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: one round of the conditional quantile calibration game with information asymmetry [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean rejection time τ¯ over the (α, δ) grid for the AdaGrad OCO skeptic (N=50 runs, T=1,460, γtest=0.05). Censored cells, where no rejection occurs by time T, appear dark blue; faster rejections appear light blue; grey marks invalid parameter pairs. The marginal skeptic is censored across the valid region, while the Zt−1-aware skeptic rejects quickly. the oracle issues the corresponding Negative-Binomial f… view at source ↗
Figure 4
Figure 4. Figure 4: Type-I rejection rate versus quantile level [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Per-feature rejection rate versus quantile level [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Type-I rejection rate vs. quantile α for the SGD and FTRL Skeptics (N=1,000 oracle series, T=1,460, αtest=0.05). Grey band: one-sided 99% MC tolerance. Companion to [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Per-feature rejection rate vs. quantile α for Chronos-2 with a 30-day sliding window (FTRL betting on both pipelines). Companion to [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Per-feature rejection rate vs. quantile α for Chronos-2 with a 90-day sliding window (Rossmann: AdaGrad; Synthetic: FTRL). Companion to [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Per-feature rejection rate vs. quantile α for Chronos-2 with a 180-day sliding window (FTRL betting on both pipelines). Companion to [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Per-feature rejection rate vs. quantile α for Moirai-2 with a 365-day sliding window (Rossmann: AdaGrad; Synthetic: FTRL). Companion to [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Rossmann HP sweep: AdaGrad, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Rossmann HP sweep: SGD, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Rossmann HP sweep: FTRL, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Synthetic HP sweep: AdaGrad, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p037_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Synthetic HP sweep: SGD, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Synthetic HP sweep: FTRL, Chronos-2 with a 365-day sliding window, [PITH_FULL_IMAGE:figures/full_fig_p038_16.png] view at source ↗

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