Title: Learning to price with resource constraints: from full information to machine-learned prices
Venue: NeurIPS 2025
Accepted: yes
Keywords: online learning

dynamic pricing

resource constraint

informed price
Abstract: Dynamic pricing with resource constraints is a critical challenge in online learning, requiring a delicate balance between exploring unknown demand patterns and exploiting known information to maximize revenue. We propose three tailored algorithms to address this problem across varying levels of prior knowledge: (1) a Boundary Attracted Re-solve Method for the full information setting, achieving logarithmic regret without the restrictive non-degeneracy condition; (2) an online learning algorithm for the no information setting, delivering an optimal $O(\sqrt{T})$ regret; and (3) an estimate-then-select re-solve algorithm for the informed price setting, leveraging machine-learned prices with known error bounds to bridge the gap between full and no information scenarios. Moreover, through numerical experiments, we demonstrate the robustness and practical applicability of our approaches. This work advances dynamic pricing by offering scalable solutions that adapt to diverse informational contexts while relaxing classical assumptions.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses dynamic pricing with resource constraints by developing three algorithms tailored to different levels of prior information about the demand function. For the full-information setting, where the demand function is known, the authors propose a Boundary Attracted Re-solve Method that achieves O(log T) regret without requiring conditions found in prior work. In the no-information setting, an online learning algorithm is presented that attains optimal regret by balancing exploration and exploitation. For the intermediate informed-price setting, where a machine-learned price estimate with a known error bound (ϵ_0) is available, the paper introduces an estimate-then-select algorithm. This algorithm achieves a regret of O(min{sqrt(T),(ϵ_0)2T}+log T), effectively bridging the performance gap between the full and no-information scenarios. The paper supports these theoretical findings with numerical experiments.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**  \nThe paper overall presents sound and novel results relating to the studied dynamic pricing problem, and is well-written.\nI appreciate the examination of improvements that are attainable when the seller has access to an initial machine-learned estimate of the price-demand, including the impossibility result when the error bound $\\epsilon_0$ is unknown.\n\n**Weaknesses**  \nAs far as I can tell, the numerical study does not compare against any baselines from prior work, and the instances are relatively small and randomly generated, which limits their appeal somewhat.  Moreover, the figures in the experiment section are quite hard to read — axes, legends, and labels should all be resized for this format.\n\nQUESTIONS:\nFor NeurIPS, I think it would be valuable to give a little bit of practical context about the dynamic pricing problem studied here — i.e., what are some applications that this problem models (or approximately models)?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe main contribution is an algorithm for dynamic pricing, where the authors augment the re-solve method from prior work with \"boundary attraction\". In the problem the authors consider, demand is stochastic and the goal is to set prices such as to maximize the revenue subject to budget constraints. The re-solve method updates prices based on the expected demand and the remaining budgets. Prior work only applies to the case where the re-solve solution is not degenerate; the authors do not require this assumption due to the via boundary attraction that was added to the re-solve algorithm.\n\nFor this algorithm, a (linear) demand function is assumed to be known. The authors extend their work by introducing an algorithm that does not require knowledge of the (linear) demand function and instead approximate it via linear regression during the execution of the algorithm.\n\nA third contribution is a learning-augmented algorithm that takes as input an initial price and estimate of the price. The resulting linear regression is then \"anchored\" in this pair, which helps to introduce offline knowledge into the online problem.\n\nLastly, the authors provide an experimental evaluation on synthetic data.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths \n\n1. The introduction of boundary-attraction is potentially novel.\n2. The authors provide a wealth of results where they cover a known, unknown, and partially known demand function. Especially the latter setting is currently very relevant. \n3. The authors can argue that their results for each setting are the strongest attainable. \n\n### Weaknesses\n\n1. I am unsure about the motivation for the \"boundary attraction\". It would be good if the authors could explicitly state how exactly it helps to deal with non-degeneracy, and how earlier work struggled with it. It could be that I'm just unfamiliar with such an approach, but typically I would expect one would introduce a regularizer to make the objective strongly concave. The authors only provide very high-level intuition in the paragraph in Line 149, but from how the boundary attraction is implemented I do not understand how either of the two points are achieved. \n2. The presentation is somewhat lacking. For instance, the related work (in the main body) is only an accumulation of papers, an equation is missing in Line 189, many references are (??) and some equations in the appendix go over the page. Also, concepts in the preliminaries such as the fluid benchmark are missing references. Overall, this makes understanding the paper more difficult (e.g. also Line 545: should be “zeta times”, not “zeta divided by”).\n\nQUESTIONS:\nCould you please provide a thorough explanation and intuition on the boundary attraction you are using. What exactly is the challenge in prior work that stems from degenerate solutions, and how does the boundary attraction allow us to overcome this?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers dynamic pricing with resource constraints under a linear demand model in a variety of settings:\n- Full-information: the expected demand $f(p) = \\mathbb{E}[d \\mid p] = \\alpha + Bp$ function is known to the algorithm up front.\n- No-information: $f(p)$ (i.e., $\\alpha$ and $B$) is unknown to the algorithm.\n- ML-informed price setting: the algorithm is given a triple $(p^0, d^0, \\epsilon^0)$ which satisfies $\\||f(p^0) -d^0\\|| \\leq \\epsilon^0$.  This models the idea that in practice we have past estimates/predictions with known error.\nThe objective is to construct a sequence of feasible pricing decisions $p^1,\\ldots, p^T$ which minimize the regret against the revenue of a fluid benchmark representing an upper bound on the expected revenue of any feasible pricing policy.  Additionally, the sequence of pricing decisions should respect the resource constraints and not serve demand that would exceed remaining capacity.  The main contributions of the paper are then as follows:\n- A method for the full-information setting based on resolving the fluid relaxation which then rounds demands to zero if they are close to 0 in order to avoid degeneracy conditions.  This method achieves $O(\\log T)$ regret.\n- A method for the no-information setting achieving $O(\\sqrt{T})$ regret based on estimating the demand model using the observed data and resolving the estimated fluid model in blocks of length $n$ (the number of distinct items being sold).\n- A method for the ML-informed price setting which uses the prediction as an \"anchor point\" for regression to estimate the demand model which either achieves $O(\\rho \\sqrt{T})$ regret or $O((\\epsilon^0)^2 T)$ regret, depending on how $\\epsilon^0$ compares to $\\rho$, a tunable parameter.  It is also shown that knowledge of the error parameter $\\epsilon^0$ is necessary.\nThe paper concludes with a simulation study of the three proposed algorithms, validating the theoretical bounds.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper considers a fundamental problem in dynamic pricing in standard settings (full information and no-information) as well as a new setting (ML-informed price) which is motivated by the access to potentially accurate predictions from ML models which we may have up-front.  Theoretical analyses are given for all proposed methods, and the presentation is fairly clear.\n\nThe weakest aspect of the paper is the simulation study, which seems to be a token inclusion in the paper.  Without a compelling baseline to compare to, I'm not sure what to take-away from the simulations that wasn't already communicated by the theoretical results (which are definitely interesting)\n\nQUESTIONS:\n- In Algorithms 2 and 3, what is meant by \"reject demands in ${\\cal I}_r^t$\"?\n- In light of Proposition 4.1, why is it not possible to estimate $\\epsilon^0$ by first trying the price $p^0$ once or a small number of times?  Adding a bit more on this to the main paper could help with the exposition.\n\nAdditionally, I have some minor comments and suggestions:\n- Please use a more typical capitalization style in the title and section titles.  For example, \"Learning to price with resource constraints: from full information to machine-learned prices\" -> \"Learning to Price with Resource Constraints: from Full Information to Machine-Learned Prices\" and \"Algorithm and logarithmic regret with full information\" -> \"Algorithm and Logarithmic Regret with Full Information\".\n- Line 105: \"Related Literatures\" -> \"Related Literature\" or \"Related Work\"\n- In the appendix there are some ?? where references should be, e.g., on lines 663 and lines 668.\n- Some display equations in the appendix don't use \\left( and \\right) to size brackets/parentheses appropriately.\n- Consider including the literature on Algorithms with Predictions/Learning-augmented algorithms in your discussion of related work as the ML-informed price setting seems very related and some of the ideas are similar (e.g., algorithm switching as is done in Algorithm 3).\n\t- The webpage at https://algorithms-with-predictions.github.io/ could be helpful with this.\n\t- Particularly relevant to regret minimization is the line of work on online linear optimization with hints, see the following, for example:\n\t\t- https://arxiv.org/abs/2010.03082\n\t\t- https://proceedings.mlr.press/v202/bhaskara23a.html\n\t\t- https://arxiv.org/abs/2111.05257\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers dynamic pricing with resource constraints in three settings:  (1) a full information setting where the demand function is assume known, (2) no information setting we're learning must be done on the fly, and (3) a mixed ``informed pricing\" setting some information is known about the part about the demand function apriori. In setting (1), the authors prove a $O(\\log T)$ regret bound that improves upon previous bounds that required non-degeneracy assumptions. In setting (2), the authors develop an online learning algorithm that combines the previous approach with linear regression and prove $O(\\sqrt{T})$ regret. Finally, in the mixed setting the authors provide an algorithm that leverages the prior knowledge in the regression and achieves an interpolation regret result between the previous two extremes. Numerical simulations validate the regret bounds.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. In my view this is a moderately high quality submission in its current state. The results are mostly technically sound. Although I did not read the appendix in detail, the claims appear well supported by extensive proofs.\n2. The results of this paper are significant, interesting, and have potential for impact. I think the topic is of importance to the NeurIPS community as it considers challenging and relevant online learning problems that form a nice bridge between ML and OR/MS.\n3. I quite like the presentation and style of analysis, starting with the full information setting, then the no information setting, then the informed pricing setting that blends the previous two. The regret bounds are all logical to me, and they are supported by corresponding lower bounds and numerical validation. All in all, this is a nice set of contributions presented in a logical and compelling presentation.\n4. The authors do a solid literature review and explain the related works well. Some ideas appear to be original enough, particularly the boundary attracted re-solve idea (Section 2) and anchoring idea (Section 4). \n\nWeaknesses:\n1. Clarity is the biggest dimension of weakness for this paper, and the related issues also hurt the quality in some ways too. Although the paper is well organized and articulates some ideas well, many of the ideas are not explained very well and in some places the paper appears sloppily written. Please see my first major question/comment in the Questions section below for more details.\n\n2. Commenting further on the dimension of originality, my biggest question mark is on the overall idea of regression (Section 3), which seems quite related/adapted from previous works to me. Some works I am thinking of in particular are:\n\nSimchi-Levi, David, and Yunzong Xu. \"Bypassing the monster: A faster and simpler optimal algorithm for contextual bandits under realizability.\" Mathematics of Operations Research 47.3 (2022): 1904-1931.\n\nXu, Yunbei, and Assaf Zeevi. \"Upper counterfactual confidence bounds: a new optimism principle for contextual bandits.\" arXiv preprint arXiv:2007.07876 (2020).\n\nTo my knowledge, both of these papers rely on similar regression learning ideas in online (contextual) bandit problems. (I'm sure there may be others too; please see the references therein and those more recent papers that have cited these.)\n\nMy two main related questions for the authors are:  (1) Given that these papers were not originally cited by you, can you please elaborate and clarify the connections to these and other papers in this space? (2) What difficulties arise and to what extent is it non-trivial to adapt these ideas to your particular problem structure?\n\nQUESTIONS:\nRegarding the issue of clarity, I list in linear order here the following issues/questions that I could not easily resolve when reading the paper:\n- What is the precise issue with non-degeneracy / non-uniqueness? You seem to be mixing up the primal/dual in some discussions. On the one hand, the paragraph on line 78 mentions \"the restrictive non-degeneracy condition (unique fluid optimal dual prices)\" but also \"fluid LP has multiple optima (a common occurrence.\" In my understanding, non-degeneracy of the dual implies the uniqueness of the primal optimal solution and vice versa. Another example is the paragraph on line 149, which seems to be discussing intuition relating to uniqueness of the primal solution. However, isn't the primal solution always unique by strict convexity due to Assumption 1.3? Please clarify.\n- Related, I really struggled to follow the intuition in the paragraph on line 149. I would appreciate some further elaboration or revision of this paragraph, to clarify how the boundary attraction thresholding idea helps. Maybe a simple figure would be helpful?\n- Two other confusions I I have with the boundary attraction:  (1) Although unlikely for realistic $\\alpha$ and $B$, in theory it's possible for $d = \\alpha + Bp$ to be negative; so why does thresholding $d$ close to zero and positive justified? (2) When you reset $p^t$ to match $f(p^t) = \\tilde{d}_t$ do you also need to be concerned with feasibility of $p$ in $[L, U]^n$?\n- Many details are missing about Algorithm 2 and the \"rejection\" scheme included in it, etc. In fact, you seemed to have a discussion on line 189 (\"by setting the desired price as\") that was completely cut off and deleted.\n\nOther Questions:\n1. Can you comment on the connection to RL? For example, one could think of applying a policy gradient method, actor-critic, or other RL method to your problem. In my understanding, your problem is a lot more structured which allows for specialized algorithms and theoretical regret bounds. However, much of the audience of NeurIPS would probably first think of these RL algorithms and so some commentary is justified. Furthermore you might even consider comparing to RL as a baseline in your experiments?\n2. Can you elaborate more on what \"anchoring\" the regression means? Do you solve a least squares problem where $\\alpha, B$ are constrained to satisfy $\\|d^0 - f(p^0)\\|_2 \\leq \\epsilon^0$?\n3. Could the results of Section 3 be extended or even improved with more information? For example, what if you had an estimation error bound on $B$ (and $\\alpha$) like $\\|B^0 - B^true\\|_2 \\leq \\epsilon^0$? Or maybe multiple samples:  $\\|d^i - f(p^i)\\|_2 \\leq \\epsilon^i$ for $i = 0, \\ldots, m$?\n\nMinor:\n1. Equation (1):  $T(p)$ looks like a function and not multiplication as you intended.\n2. You mention that (1) is concave before you state the assumptions that justify this.\n3. Line 73:  Since there is not necessarily a one-to-one correspondence between products and resources, I'm not sure what lines 72-73 mean. Does the whole process just stop?\n4. Lemma 3.2 states quadratic growth but you wrote linear growth.\n5. Several other typos throughout that are too many to list.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}