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An algorithm for robust contextual pricing achieves regret O(Cd + d² log T), separating corruption cost from horizon length.

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2026-07-14 06:15 UTC pith:VQVFLW6V

load-bearing objection Clean additive O(Cd + d^{2} log T) for robust contextual pricing; closes the GGLS25 gap with a modular density-plus-evidence argument that holds up.

arxiv 2607.11210 v1 pith:VQVFLW6V submitted 2026-07-13 cs.GT

Decoupling Corruption and Horizon in Robust Contextual Pricing

Matteo Castiglioni , Francesco Emanuele Stradi This is my paper
classification cs.GT
keywords contextual pricingcorruption-robust learningdensity updatesevidence setsregret boundsonline learning
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.

In repeated contextual pricing a seller sees a feature vector each round, posts a price, and learns only whether a sale occurred. An adversary may flip that binary feedback on at most C rounds. Earlier algorithms paid a multiplicative price for corruption: their regret grew like Cd times a slowly growing function of the horizon T. This paper gives the first algorithm whose regret is O(Cd + d² log T). The additive form means that once the seller has paid a linear price for the corruptions, further rounds cost only a mild logarithmic term in T. The result is obtained by first solving a cleaner “proximity-feedback” model that also reports whether the posted price was close to the true value, then reducing ordinary sale feedback to that model by an optimistic inflation of the corruption budget. A sympathetic reader cares because the same separation was already known in the non-contextual case and for pure search; pricing had remained open.

Core claim

There exists an algorithm that, given a known upper bound C on the number of corrupted sale bits, achieves regret O(Cd + d² log T) for linear contextual pricing. This is the first guarantee in which the corruption term and the horizon term appear additively rather than multiplicatively.

What carries the argument

A density maintained over the parameter ball together with 2Ć+1 disjoint evidence sets of near-value observations; the algorithm commits only when every set can linearly explain the current context, then posts the (Ć+1)-st largest lower endpoint of the resulting intervals.

Load-bearing premise

The seller must be given a correct upper bound C on the number of corrupted rounds; if that bound is missing or too small the commit rule no longer guarantees a safe price.

What would settle it

Exhibit a family of linear contextual pricing instances with known C for which every algorithm suffers regret ω(Cd + d² log T), or show that any algorithm achieving additive separation must use super-polynomial time.

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

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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

0 major / 5 minor

Summary. The paper studies repeated contextual pricing with linear valuations vt = ⟨θ⋅, ut⟩ under binary sale feedback that an adversary may corrupt in at most C known rounds. It designs Algorithm 4 that maintains a density μt over the parameter ball together with 2Ō+1 disjoint evidence sets and posts either a robust commit price (the (Ō+1)-st largest lower endpoint of the induced intervals) or the ε-window median. The main claim (Theorem 4.2) is the regret bound RT = O(Cd + d^{2} log T). The proof first obtains the cleaner O(Cd + d log T) guarantee under an auxiliary proximity-feedback model (Theorem 3.8) via a density-potential argument (Lemma 3.5) and a determinant-growth argument that limits each evidence set to O(d) insertions (Lemma 3.6); the reduction to ordinary sale feedback inflates the perceived corruption budget to Ō = ⌈5C + 20d log T⌉ and re-uses the same lemmas.

Significance. If correct, the result is the first additive separation of the corruption budget C from the horizon T for robust contextual pricing, closing the open question left by Gupta et al. (GGLS25) and improving their O(Cd log log T) bound whenever C = ω(log T). The modular architecture cleanly extends the density-update framework of LPS22/26 to the pricing setting (where a safe price strictly below valuation is required) by introducing evidence sets and a commit rule; this answers an explicit open question of LPS26. Complete pseudocode, explicit constants, and a transparent reduction from proximity to sale feedback are strengths that make the contribution reusable.

minor comments (5)
  1. [Abstract] Abstract and first sentence of the introduction: “valuations depends” should be “valuations depend”.
  2. [Table 1] Table 1 caption and surrounding text: the comparison is useful, but the entry for [KSCM26] is non-contextual; a short footnote clarifying that the O(C + log T) bound is one-dimensional would prevent misreading.
  3. [Lemma 3.5] Section 3.2.2, Lemma 3.5: the constant 20 in the density-potential bound is obtained by a crude estimate log(3/2) ≈ 0.4; a tighter numerical constant (or an explicit remark that any constant > 1/log(3/2) works) would improve readability without changing the asymptotic claim.
  4. [Section 4.1] Section 4.1: the inflated budget is written Ō = ⌈5C + 20d log T⌉; the factor 5 is slightly larger than the 4 appearing in Lemma 3.5. A one-line remark that any constant strictly larger than 4 is admissible would make the padding transparent.
  5. [Section 5] Open Problems §5: the first bullet correctly flags the extra d factor as an artifact of the reduction; it would be helpful to state whether the authors conjecture that O(C + d log T) is achievable or whether a matching lower bound of Ω(Cd) already exists.

Circularity Check

0 steps flagged

No significant circularity: self-contained algorithmic analysis with independent potential and determinant arguments.

full rationale

The paper's central claim (Theorem 4.2) is an existence result for an algorithm achieving additive regret O(Cd + d^{2} log T). The derivation proceeds by first analyzing an idealized proximity-feedback model (Algorithm 1, Theorem 3.8) via two independent potentials—density mass near θ⋆ (Lemma 3.5, multiplicative 3/2–1/2 updates) and maximum determinant of evidence-set contexts (Lemma 3.6, volume growth by factor > T^{2}/(16d))—then lifting to ordinary sale feedback by optimistic dual updates and an explicitly inflated budget Ō = ⌈5C + 20d log T⌉ (Algorithm 4, Lemma 4.1). Both potentials are defined from first principles on the maintained density µt and the stored pairs; their growth bounds follow by direct calculation from the update rules and Hadamard’s inequality, without reference to the target regret expression. Self-citations (KSCM26 for the commit idea, LPS22/26 for density medians) supply reusable subroutines whose statements are independent of the new additive separation; they are not invoked as uniqueness theorems that force the result. No parameters are fitted to data and re-labeled as predictions, no ansatz is smuggled via citation, and the regret definition itself is the standard clairvoyant benchmark. The known-C modeling assumption is stated up front and does not create a definitional loop. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 3 invented entities

The central claim is a worst-case regret bound for an explicit algorithm. It rests on the standard linear contextual pricing model, a known corruption budget, and several algorithmic design choices (ε, density multipliers, number of evidence sets) that are fixed by the analysis rather than fitted to data. No new physical entities are postulated; the auxiliary proximity-feedback model is an intermediate proof device, not an ontological claim.

free parameters (3)
  • ε = T^{-3}
    Accuracy radius used for proximity windows, evidence correctness, and commit intervals; chosen by hand so that residual error is 1/(8T) and determinant growth yields O(d) insertions.
  • Ō = ⌈5C + 20d log T⌉
    Inflated corruption parameter that absorbs both true corruptions and far-median rounds; the concrete coefficients 5 and 20 come from the density-potential calculation and are fixed by the proof, not data.
  • density multipliers 3/2 and 1/2
    Multiplicative update factors on the two tails of the ε-window median; standard choice from LPS22/26 that keeps total mass well-defined and yields constant-factor potential growth.
axioms (5)
  • domain assumption Buyer valuation is exactly linear: v_t = ⟨θ⋆, u_t⟩ with θ⋆, u_t ∈ B_d(0,1).
    Section 2.1; the entire geometric and density analysis relies on this exact linear structure.
  • domain assumption At most C rounds have corrupted sale (and proximity) feedback; C is known to the algorithm.
    Section 2.2; the 2Ō+1 evidence-set majority vote fails if the true number of bad samples exceeds Ō.
  • domain assumption Contexts may be chosen adversarially; no distributional assumption.
    Section 2.1; regret is worst-case over sequences.
  • standard math ε-window median of a continuous density is always well-defined (no atoms).
    Definition 3.2 and surrounding text; used every exploration round.
  • standard math Hadamard inequality: |det| of unit-norm columns ≤ 1.
    Lemma 3.6 Step 3; caps the determinant potential.
invented entities (3)
  • 2Ō+1 disjoint evidence sets H_j storing (u_i, p_i) pairs no independent evidence
    purpose: Enable robust commit by majority vote over linear reconstructions of the current context.
    Core algorithmic object introduced in Section 3.1; no external existence claim beyond the algorithm.
  • ε-window median of a density in direction u_t no independent evidence
    purpose: Provide a stable cut that leaves a buffer around the query so density near θ⋆ is never decreased on close uncorrupted rounds.
    Definition 3.2; generalization of the ordinary median used by LPS22/26.
  • Proximity-feedback model (auxiliary) no independent evidence
    purpose: Intermediate model that reveals whether |p_t − v_t| ≤ ε, used to isolate the commit and potential arguments before the optimistic reduction.
    Section 3; not claimed as a real feedback channel, only a proof device.

pith-pipeline@v1.1.0-grok45 · 19705 in / 3234 out tokens · 46208 ms · 2026-07-14T06:15:20.179628+00:00 · methodology

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Cite this review

Pith. "Pith review of Decoupling Corruption and Horizon in Robust Contextual Pricing." pith.science (2026). https://pith.science/paper/VQVFLW6V

@misc{pith2026260711210,
  author       = {Pith},
  title        = {Pith review of: Decoupling Corruption and Horizon in Robust Contextual Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQVFLW6V}},
  note         = {Machine review of arXiv:2607.11210}
}
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read the original abstract

We study robust repeated contextual pricing, where valuations depends linearly on the features. At each round $t\in[T]$, a seller observes a context, posts a price, and receives only a possibly corrupted binary sale feedback. The seller knows an upper bound $C$ on the number of corrupted rounds. We design an algorithm with regret $\mathcal O(Cd+d^2\log T)$, where $d$ is the context dimension. This is the first guarantee for robust contextual pricing that separates the dependence on the corruption budget $C$ from the horizon $T$, closing the problem left open by Gupta, Guruganesh, Paes Leme, and Schneider (2025).

Figures

Figures reproduced from arXiv: 2607.11210 by Francesco Emanuele Stradi, Matteo Castiglioni.

Figure 1
Figure 1. Figure 1: Geometric view of the proximity-feedback algorithm. The dashed parallel lines delimit the ϵ-window around the median price mt in the direction ut . Left: when θ ⋆ is outside this window, the density is updated toward the side containing θ ⋆ . Middle: when θ ⋆ is inside the window, mt is a near-value observation and the context-price pair is stored in an evidence set. Right: once the 2C + 1 evidence sets ex… view at source ↗

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Reference graph

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