Title: A Bayesian Approach to Contextual Dynamic Pricing using the Proportional Hazards Model with Discrete Price Data
Venue: NeurIPS 2025
Accepted: yes
Keywords: Contextual dynamic pricing

Discrete price data

Bayesian approach

Cox proportional hazards model
Abstract: Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level $\gamma$, proving that our algorithm achieves a regret upper bound of $\widetilde{O}(T^{(1+\gamma)/2}+\sqrt{dT})$ for $\gamma < 1/3$ and $\widetilde{O}(T^{2/3}+\sqrt{dT})$ for $\gamma \geq 1/3$, where $\gamma$ represents the sparsity of the price grid relative to the time horizon $T$. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes BayesCoxCP, a novel semi-parametric contextual dynamic pricing algorithm. The algorithm is based on a Bayesian formulation of the Cox proportional hazards (PH) model and is specifically designed for settings where feasible prices are drawn from a discrete set rather than a continuous range. It shows that leveraging this discreteness can lead to improved regret performance, particularly when the price grid is sparse relative to the time horizon. The paper also establishes a minimax lower bound that matches the upper bound achieved by the BayesCoxCP algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper is generally well-written and structured. Studying dynamic pricing problems in discrete price spaces introduces an interesting and novel perspective. The paper also proposes a new algorithm supported by solid theoretical results.\n\nWeaknesses:\n1. The assumption of a known functional form for the hazard function (specifically, the Cox proportional hazards model) might limit practical applicability if the actual demand structure significantly deviates from this assumption.\n2. The assumption of a (sparse) discrete price space might be restrictive. In practice, it is common to encounter closely spaced price candidates or even price subintervals (see questions below).\n\n\nMinor comments:\n\n1. If my understanding is correct, the derived regret bound would remain valid even for non-uniform grids in the price interval, provided a certain sparsity condition is satisfied (as discussed in Appendix D). Highlighting this generalization explicitly in the main text would enhance the paper's practical relevance. Additionally, formally presenting this result as a corollary could further strengthen the paper.\n\n2. The experiments currently refer readers to [6] for performance comparisons for benchmark algorithms (linear and log-linear models). Providing direct comparisons or summarizing performance metrics in the current paper itself would improve readability and more clearly demonstrate the potential advantages of the proposed approach.\n\n[6] Young-Geun Choi, Gi-Soo Kim, Yunseo Choi, Wooseong Cho, Myunghee Cho Paik, and Min-hwan Oh. Semi-parametric contextual pricing algorithm using Cox proportional hazards model. In Proc. International Conference on Machine Learning, pages 5771–5786, 2023.\n\nQUESTIONS:\n1. I was wondering why, when $\\gamma = 1$, the regret upper bound $\\tilde{O}(T^{2/3}+\\sqrt{dT})$ still outperforms the bound $\\tilde{O}(T^{2/3}d)$ from [6] in terms of the dimension $d$. What is the reason behind this?\n\n2. Could the algorithm handle a price space containing discrete subintervals? For instance, each subinterval is $[g_k - \\delta', g_k + \\delta']$, with $\\delta'$ potentially very small. As $\\delta' \\rightarrow 0$, this would result in a discrete price space. I am not seeking formal proofs; any discussion on this point would be appreciated.\n\n[6] Young-Geun Choi, Gi-Soo Kim, Yunseo Choi, Wooseong Cho, Myunghee Cho Paik, and Min-hwan Oh. Semi-parametric contextual pricing algorithm using Cox proportional hazards model. In Proc. International Conference on Machine Learning, pages 5771–5786, 2023.\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies contextual dynamic pricing when prices are restricted to a discrete set. The authors propose a Bayesian approach using a semi-parametric proportional hazards model, showing that incorporating discrete pricing information can significantly enhance performance depending on the relationship between the price support size and the time horizon. They derive both upper and lower regret bounds for the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The problem setting is practically relevant and underexplored. While most recent work focuses on continuous pricing, discrete pricing is often more realistic in real-world applications.\n\n2. The paper’s primary contribution lies in analyzing the estimation rate of the Bayesian proportional hazards model in this discrete setting and applying it to contextual dynamic pricing.\n\n3. Despite its technical depth, the paper is generally accessible and well written.\n\nWeaknesses:\n\n1. The technical novelty beyond existing work appears limited.\n\n2. Some key modeling assumptions are overly simplistic or unrealistic for real-world applications (see detailed questions below).\n\nQUESTIONS:\n1. Technical Novelty: The core contribution appears to be extending the semi-parametric Cox proportional hazards model for pricing (Choi et al., 2023) from continuous to discrete price settings. The key technical results build on generalizing Chae (2023), which deals with non-contextual interval-censored data, to the proportional hazards setting. The authors should better articulate the technical novelty. What specific challenges arise in the discrete and contextual setting that are not addressed by existing works? Why are existing techniques insufficient, and what new theoretical tools or results are developed to address these gaps?\n\nReferences:\n\nChoi et al. (2023): Semi-parametric contextual pricing with Cox PH model. ICML.\n\nChae (2023): Adaptive Bayesian inference for current status data on a grid. Bernoulli.\n\n\n2. Modeling Assumptions and Practicality: The current formulation/algorithm design departs from realistic pricing scenarios in three key aspects:\n\n(1) As stated in Section 2.1, the model assumes $(X_t, P_t, Y_t)$ are i.i.d., which is impractical in dynamic pricing—prices are typically adapted based on historical observations.\n\n(2) Algorithm 1 assumes that in the exploration phase, the seller can uniformly sample prices from the known set $\\mathcal{G}$. However, in practice, $\\mathcal{G}$ may not be known in advance or may change over time and across contexts. In non-contextual settings, it is reasonable to explore a large price set early and narrow it down over time. In contextual settings, however, the appropriate price set is often time- and context-dependent, making the assumption of a fixed known $\\mathcal{G}$ unrealistic.\n\n(3) The algorithm relies on three tuning parameters: the first-epoch size \\( n_1 \\), and the exploration parameters \\( \\eta_1 \\) and \\( \\eta_2 \\). In Appendix Section F, the authors mention that these parameters are selected via grid search. However, in an online learning setting, it is unclear how such tuning would be performed in practice. What is the computational cost associated with jointly tuning all three parameters? Additionally, how should a practitioner determine appropriate ranges for \\( \\eta_1 \\) and \\( \\eta_2 \\), especially without access to validation data?\n\n\n3. Estimation Methodology:\nThe paper states that a variational Bayesian method is used to obtain the estimator $\\hat{\\theta}^{l-1}$. How does the theoretical regret guarantee depend on the accuracy of this estimator? Are there any conditions required for the variational Bayesian method to ensure that the regret bounds hold?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the contextual dynamic pricing problem under discrete price constraints, which better reflects many real-world retail settings where prices must come from a fixed grid. The author proposes a novel Bayesian algorithm, BayesCoxCP, which leverages the semi-parametric Cox proportional hazards model to estimate customer valuation distributions from interval-censored (binary) purchase feedback. The author uses an epoch-based pricing strategy to maintain i.i.d. data generation within every epoch to make the semi-parametric estimation doable and uses adaptive epsilon-greedy algorithm in every epoch. The author also proves a high probability regret upper bound to demonstrate that when the pricing action space is sparse, the regret rate could be better than the continuous or dense action space.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: The paper contains theoretical upper bound, lower bound, and empirical evaluation, which is quite complete and thorough. The algorithm design is concise in principle and easy to understand. This paper also uses COX PH model in a discrete space instead of continuous space and observes a phase transition phenomenon with respect to the sparsity parameter $gamma$, which is interesting.\n\nWeakness: I am not fully convinced by the setting. The grid length $\\delta=\\kappa n^{-\\gamma}$, where $n$ is the number of sample points. In the sequential contextual pricing setting, doesn’t this mean that in different rounds, or at least in different epochs, the action space of the agent is different? This is a little weird to me. Are there any existing papers that model the action space sparsity in such a way? I think the author needs to justify the setting in a more profound way.\n\nI tried to read the technical proof details, and I find it a little difficult to follow. It would be much better if the author could provide a proof sketch or a roadmap about how every lemma is fully used. Could you please provide a sketch about the proofs of thm 3.1, 3.2?\n\nThe COX PH model, to my understanding, is a way to handle and model data censoring mechanism. It allows tractable estimation to bypass censored observations. What if the data generation mechanism is more general than that? Do you have a more principled way to handle it, rather than this restricted COX model with Gamma prior? I think some discussion about that is needed at least. I feel that there should be a challenge similar to data censorship in this paper, but the author doesn’t mention that explicitly and bypasses it through COX model.\n\nQUESTIONS:\nSee Strengths and Weakness\n\nLIMITATIONS:\nSee Strengths and Weakness", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider the Cox proportional hazards model in the context of binary semi-parametric contextual dynamic pricing. Based on the non-contextual statistical estimation results under this Cox PH model from the statistics literature, they develop similar ones under the contextual modeling. Then an epoch-based algorithm design is utilized to incorporate those estimation results, along with imposed exploration probabilities similar to the epsilon-greedy algorithms. \n\nMatching regret upper and lower bounds are derived. The finding of improved regrets for well-designed algorithms under discrete prices is interesting, insightful and valuable in practice.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Although not first introduced in this paper, interpreting the purchase data from the binary semi-parametric contextual pricing model as the case 1 interval-censored data in survival analysis effectively captures the intrinsic nature of the data and helps bridge and revitalize the two fields. \n2. The considered discrete pricing setting is both practical and valuable, and new theoretical insights for this setting are delivered through the contributions of this paper.\n3. The essential statistical estimation analysis of this paper is based on and developed from solid research in statistic literature.\n4. The technical material is delivered with high degree of clarity and rigor. \n\nWeaknesses:\n1. More discussions are needed: 1) possibility of extension beyond Cox PH to other models (See Question 1), 2) technical differences when dealing with nonuniform grids (See Question 2), 3) challenges in generalize the non-contextual convergence rates of posterior distributions to the contextual scenario, 4) intuitive explanation of the phase transition of estimation errors and regret rates, 5) whether comparison of regret rates with those for continuous prices makes sense, and whether the authors' derived regret rates for discrete prices can transfer to ones for continuous prices. 6) crucial elements of the algorithm, e.g., choice of $\\eta_{\\ell}$ and satisfaction of Assumption (A4).\n\nQUESTIONS:\n1. For Cox PH model, statistical estimation results are developed from survival analysis literature and contribute to the optimal dynamic pricing algorithms. Could the authors discuss whether extensions, still leveraging the case 1 interval-censored data structure, but towards other semi-parametric pricing models, are possible?\n\n2. The nonuniform grids are discussed in Appendix D. The authors claim that \"with some additional technical work, our results can be readily extended to more general discrete price sets\", and \"this generalization does not fundamentally change the regret behavior\". Could the authors discuss in more details on 1) what additional technical complexities arise for nonuniform grids, and 2) why the generalization will not fundamentally change the regret behavior, is there any evidence from statistical estimation theory in survival analysis? \n\n3. The convergence rates of the posterior distributions in Theorem 3.1, Theorem 3.2 generalize the results in [4] to incorporate contextual information $X_{t}$. This key technical development in estimation errors help build the contextual dynamic pricing algorithms. Could the authors discuss the difficulties and challenges in this generalization?\n\n4. There is a phase transition of the convergence rates ($\\gamma \\geq 1/3$ and $\\gamma < 1/3$) in both the existing non-contextual and newly-developed contextual scenarios, which lead to the same transition in the derived regret rates. Could the authors give more intuitive explanations and illustrations for this phenomenon?\n\n5. In lines 309 - 311, the authors compare their regret rates with those in [6]. However, [6] considers continuous prices and thus enjoy a different benchmark. Intuitively, the continuous optimal price yields higher revenues than the discrete best price (the optimal price with the discrete set). Is it appropriate to conduct the regret comparison if the benchmarks are truly different? How would the regret rates of the authors' proposed algorithm transfer to ones under continuous prices? \n\n6. More discussions on the choice of $\\eta_{\\ell}$, and satisfaction of Assumption (A4) may be added, as both of them are crucial elements of the proposed algorithm.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}