Title: Robust Contextual Pricing
Venue: NeurIPS 2025
Accepted: yes
Keywords: pricing

contextual search

corruptions

online learning
Abstract: We provide an algorithm with regret $O(C d \log \log T)$ for contextual pricing with $C$ corrupted rounds, improving over the previous bound of $O(d^3 C \log^2(T))$ of Krishnamurthy et al. The result is based on a reduction that calls the uncorrupted algorithm as a black-box, unlike the previous approach that modifies the inner workings of the uncorrupted algorithm. As a result, it leads to a conceptually simpler algorithm.

Finally, we provide a lower bound ruling out a $O(C + d\log \log T)$ algorithm. This shows that robustifying contextual pricing is harder than robustifying contextual search with $\epsilon$-ball losses, for which it is possible to design algorithms where corruptions add only an extra additive term $C$ to the regret.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies a version of contextual pricing robust to corruption. The buyer's valuation is represented by $\\langle v*, x_t\\rangle$ and the purchase decision depends on whether $v_t \\geq p_t$ - the price - or not. The purchase decision represents the learner's available feedback. In the setup here, the corruption budget $C$ represents how many times the buyer can deviates from the expected purchase decision. To prove their upper-bound, that outperforms the state-of-the-art, the authors rely on $C+1$ copies of an uncorrupted contextual pricing algorithm that they update specifically in order to handle corruption. They also show that the uncorrupted algorithm proposed by Liu et al. 2021 satisfies the required assumptions and can be used as a subroutine. Finally, they extend their setup to encompass the unknown-corruption setting and prove a lower bound.\n\nSTRENGTHS AND WEAKNESSES:\nI believe that the main strengths are:\n\n- the paper is very clear and easy to read.\n\n- the proposed method based on replicata of an uncorrupted pricing algorithm to deal with the corruption is elegant and efficient.\n\n- their method outperforms the existing bounds for this setup.\n\nThe main weaknesses are:\n\n- the setup studied in the paper is not new and already exists, which diminishes the novelty of the work.\n\n- the question tackled by this work is quite niche because it consists of a method that only applies to corrupted contextual pricing and very close scenarios. It is interesting but quite narrow.\n\nQUESTIONS:\nI understand that the paper is mostly theoretical but do the authors know if there is empirical evidence for the use of corrupted pricing algorithm?\n\nLIMITATIONS:\nAs said in the weaknesses, the limitation is mostly about the broad impact of the work, which appears as quite niche - although interesting - to me.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of robust contextual pricing. At round $t$, the seller can propose a price $p_t$, and is reported whether $v^* \\cdot x_t \\ge p_t$ (in which case the seller gets revenue $p_t$) or not (in which case the seller gets revenue $0$). However, there will be $C$ rounds where the feedback to the seller is corrupted.\n\nVia a reduction to the uncorrupted case (specifically, to algorithms in the uncorrupted case that exhibit \"slowdown-able\" and \"many-sales\" properties), this paper was able to improve the following upper bounds:\n- $O(C d \\log \\log T)$ regret in the known-corruption setting (from the SOTA $O(C d^3 \\log^2 T)$).\n- $O((C + \\log U) d \\log \\log T)$ regret in the unknown-corruption-with-known-upper-bound setting (from the SOTA $O((C + \\log U) d^3 \\log^2 T)$).\n\nFinally, the paper showed that any algorithm must incur a regret of $\\Omega(C + d \\log \\log T)$, via a reduction to a lower bound for the related *online product design* problem.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n- The results are conceptually simple and clear, yet it improves the state-of-the-art by quite a large margin. Specifically, the reduction to the uncorrupted setting is easy to understand and only incurs an additional $C + 1$ loss in multiplicative factor, so one can simply verify-and-plug the state-of-the-art uncorrupted algorithm to obtain near-optimal bound.\n- The paper is well-written, and the proofs provided (seem to) have no issues.\n\n**Weaknesses**:\n- A very minor weakness of this paper would be that the lower bound and upper bound is still not matching, but I would not put too much weight on this at all.\n\nQUESTIONS:\n1) Do you think the right answer for this model would be $\\Theta(C d \\log \\log T)$ or $\\Theta(C + d \\log \\log T)$, or somewhere in between? In other words, do you think the $C$ multiplicative factor incurred for robustifying is necessary, or are there different ways to reduce the corrupted case to the uncorrupted case, or other techniques that one can do to mitigate this loss?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies (noise-free) contextual pricing under adversarial corruptions. Specifically, in at most $C$ corrupted rounds the adversary can show to the learner an arbitrary feedback. The paper provides an algorithm with regret $O((C+1)d\\log\\log T)$ in the case in which $C$ is known upfront, and an algorithm with regret $O((C+\\log U)d\\log\\log T)$ for the case in which only an upper bound $U$ on $C$ is known. The paper also provides a negative result showing that no deterministic algorithm can guarantee a regret of $O(C+\\log\\log T)$ in this setup (even when knowing $C$ in advance). Finally, the paper provides an algorithm guaranteeing regret $O(C+\\log T)$ in the case in which the adversary is only allowed to change the feedback of no-sales to sales.\n\nSTRENGTHS AND WEAKNESSES:\nI think the paper potentially has a set of interesting results but should be improved along several directions. \n\nFirst, the relation of the corruption model with those of prior works is not entirely clear. For instance, the paper repeatedly compares its results to those of Krishnamurthy et al.. However, the corruption model in Krishnamurthy et al. seems to be different from what is considered here: in their paper the adversary can manipulate the valuation, while here the adversary can only manipulate feedback. I would appreciate a more detailed discussion of the corruption model of Paes Leme et al. 2022 as well. Moreover, guarantees of Krishnamurthy et al. hold in high probability with respect to the realised losses, while from my understanding the guarantees provided in this paper hold only in expectation. \n\nRelated to the last point above, I think the paper could benefit from a rewriting in which details of the results are formally spelled out. As it is, it feels a bit handwavy in certain points (e.g., in theorem 3.1 what’s the notion of regret you’re actually bounding?). \n\nThe lower bound provided in the paper is for deterministic algorithms, but the algorithms presented are all randomised. It would be nice to have a characterisation for the same class of algorithms (or at least some discussion about that). I guess part of the problem here is using the result on online product design as a blackbox. \n\nSection 6 is lacking many details: does the learner need knowledge of $C$ in Theorem 6.1? What is the algorithm mentioned at line 282 supposed to do? It would also be nice to add some further comments on why one sided errors make the problem fundamentally easier. \n\nThe reference for Krishnamurthy et al. should be updated since it is referring to an old version of the paper.\n\nQUESTIONS:\ncan you clarify the relation between the corruption model you consider and that of Krishnamurthy et al., and comment on the precise notion of regret you’re considering?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies contextual pricing under adversarially corrupted binary feedback.  \nIn each round \\\\( t \\\\in [T] \\\\), a seller observes a feature vector \\\\( x_t \\\\in \\\\mathbb{R}^d \\\\) that describes the product, posts a price \\\\( p_t \\\\in [0,1] \\\\), and observes only whether the buyer purchases (\\\\( \\\\sigma_t = 1 \\\\)) or not (\\\\( \\\\sigma_t = 0 \\\\)).  \nThe buyer’s latent valuation is assumed linear: \\\\( v_t = \\\\langle v^\\\\star, x_t \\\\rangle \\\\), but the feedback can be arbitrarily corrupted in at most \\\\( C \\\\) rounds.  \nThe goal is to minimize regret — the cumulative revenue loss relative to the clairvoyant seller.\n\n**Known-\\\\( C \\\\)** case.  \nThe authors give a reduction that calls any uncorrupted contextual pricing algorithm as a black box.  \nInstantiating it with the best current algorithm of Liu et al. (2021) yields a regret bound:  \n\\\\[\nO\\\\left( (C+1)\\\\, d \\\\log \\\\log T \\\\right),\n\\\\]  \nimproving the previous  \n\\\\[\nO\\\\left( C\\\\, d^3 \\\\log^2 T \\\\right)\n\\\\]  \nbound of Krishnamurthy et al. (2020).\n\n**Unknown-\\\\( C \\\\)** (upper bound \\\\( U \\\\) known).  \nUsing a multi-layer update-probability schedule (à la Lykouris et al., 2018), they obtain:  \n\\\\[\nO\\\\left( (C + \\\\log U)\\\\, d \\\\log \\\\log T \\\\right)\n\\\\]  \nwith high probability.\n\nFor the non-contextual case (\\\\( d = 1 \\\\)), they prove that no deterministic algorithm can achieve \\\\( O(C + d \\\\log \\\\log T) \\\\) regret.  \nWhen \\\\( C = \\\\Theta(\\\\log \\\\log T) \\\\), any algorithm incurs at least  \n\\\\[\n\\\\Omega\\\\left( \\\\frac{C^2}{\\\\log C} \\right),\n\\\\]  \ndemonstrating that robustifying pricing is inherently harder than robustifying \\\\( \\\\varepsilon \\\\)-ball contextual search.\n\nFor a cautious buyer — who may refuse bargains but never overpays — they design an  \n\\\\( O(C + \\\\log T) \\\\) algorithm in the non-contextual case, matching their lower-bound order (up to \\\\( \\\\log \\\\log T \\\\) vs. \\\\( \\\\log T \\\\)).\n\n**Methodology**  \nThe key idea is to run \\\\( C + 1 \\\\) (or \\\\( \\\\lceil \\\\log U \\\\rceil \\\\)) parallel copies of a standard uncorrupted learner (e.g., the Liu–Leme–Schneider 2021 algorithm with \\\\( O(d \\\\log \\\\log T) \\\\) regret).\n\n1. On each context \\\\( x_t \\\\), all copies propose prices; the maximum is posted.  \n   This ensures that a sale implies every other copy’s price was also acceptable, allowing one to charge the loss of the master algorithm to (essentially) the best uncorrupted copy.\n\n2. Update rules are randomized so that, in expectation, at least one copy receives no corrupted feedback, yet every copy is slowed down by at most a factor \\\\( O(C) \\\\) (or \\\\( O(C / \\\\delta) \\\\) in the high-probability version).\n\nA potential-function analysis borrowed from Liu et al. (2021) then bounds the regret of each slowed-down copy, which transfers to the master algorithm.\n\nThe lower bound reduces online product design with \\\\( n \\\\approx C \\\\) items (Emamjomeh-Zadeh et al., 2021) to contextual pricing with \\\\( C \\\\) corruptions, showing that an additive \\\\( C \\\\) price for robustness is unachievable.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**  \nThe paper offers a clean and modular algorithmic framework that black-boxes any uncorrupted learner, leading to easy implementation and broad applicability.  \nIt significantly improves regret bounds by reducing their dependence on both dimension \\\\( d \\\\) and corruption level \\\\( C \\\\), and it sharpens theoretical understanding through matching lower bounds.  \nThe black-box reduction paradigm has potential to inspire similar techniques in other contextual decision problems with binary feedback.  \n\n**Limitation**  \nI greatly appreciate the paper’s clear presentation and its solid theoretical foundation. My main reservation concerns the model’s practical applicability. Could you please provide concrete examples or application domains in which this type of pricing problem naturally arises?\n\nIn addition, it would be very helpful to include numerical experiments that benchmark the proposed algorithms against established methods in the literature. Such results would illustrate the empirical effectiveness of your approach and strengthen the paper’s practical contribution.\n\nQUESTIONS:\nSame as weakness\n\nLIMITATIONS:\nSame as weakness", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}