NeurIPS 2020

Contextual Reserve Price Optimization in Auctions via Mixed Integer Programming

Review 1

Summary and Contributions: This paper considers the problem of setting a reserve price for a second price auction. For simplicity, the reserve price is a linear function of the features. First, the authors prove a hardness result which states that, unless the exponential-time hypothesis fails, there is no poly time algorithm for the reserve price optimization problem. Next, the authors give an MIP formulation of the problem as well as an LP relaxation of the the MIP formulation. Finally, the authors present some experimental results to show the effectiveness of the MIP and the LP relaxation.

Strengths: The paper is written well and easy to understand and follow. The authors clearly explain what is difficult about the problem (namely that ETH implies that the problem is difficult in general). They also run some nice experiments to illustrate the different performances of the algorithms. To the best of my knowledge, this work is novel. It is intriguing to me that setting reserve prices with features has not really been dealt with much in the literature. This paper may be of interest to practitioners looking to design better algorithms for reserve pricing.

Weaknesses: I am not sure the paper is contributing much from an ML standpoint. They propose a MIP and its LP relaxation then runs some experiments to showcase the different quality of solutions amongst different methods. I think the present paper may be more suited for a more specialized venue for MIP or for auctions, like WWW.

Correctness: The authors assert some claims which all seem sound to me. However, I have not checked the proofs int he appendix.

Clarity: The paper is written clearly.

Relation to Prior Work: The paper clearly discusses how the present work differs from previous works.

Reproducibility: Yes

Additional Feedback: It is interesting that the LP does not really do better than constant prices for synthetic data. Is this just because of the nature of the synthetic data? LP seems to be clearly better than constant prices for ebay experiments though. Minor Comments: - I found the explanation of (2) confusing. In line 39, it says that the setting recovers first price auction and pure price-setting. Do you mean by setting artificial bids? I understood (2) as: "the revenue that one gets given a reserve price v, highest bid b^1 and second highest bid b^2". ===== Edit: I have looked at the author's response and the other reviews. I am happy to accept that there may be a good proportion of attendees that I may be interested in such work so I have increased my score.

Review 2

Summary and Contributions: In the second-price auction with reserve price, the publisher sets a reserve price before the auction, and the payment of the winner is defined as the maximum of the second highest bid and the reserve price. The paper deals with the problem of deciding the reserve price from the side information to maximize the payment from the winner. It considers using linear regression to infer the best reserve price. Then the problem is reduced to the optimization problem of optimizing the model parameters so as to maximize the payment. The main contribution of this paper is to discuss how to solve this optimization problem. The objective function of the optimization problem is complicated, and so the optimization seems hard. Indeed, the paper proves that the k-densest subgraph can be reduced to the problem, meaning that there is no polynomial-time algorithm for it under a suitable assumption. Then, the paper proposes a mixed-integer programming (MIP) formulation. MIP is an NP-hard problem, but several efficient solvers are available for it. A linear programming relaxation for the MIP formulation can be also used. These approaches are evaluated by computational experiments. The result show that the algorithms of using MIP are superior to other approaches using the LP relaxation or other algorithms proposed by previous studies.

Strengths: - It formulates a new interesting optimization problem, which naturally appears in realistic setting of auctions. - The proposed approach is efficient, both in its performance (as shown by the experiments) and in the usability (since several efficient MIP solvers are available). - It also revels the computational difficulty of the optimization problem. - The topic is closely relevant to the NeurIPS community. The paper deals with an application of the machine learning to the common auction setting. It must be also interesting to the optimization community.

Weaknesses: Although I do not agree, there might be an opinion that the contribution is not enough because the paper just formulates a hard optimization problem as MIP, which is another hard optimization problem.

Correctness: As far as I checked, the claims and method seems correct. Since proofs of several claims are not presented, I could not check all the details.

Clarity: The paper is written very well. It is easy to read. It would be better if important claims (e.g., Theorem 1 and Proposition 2) were presented.

Relation to Prior Work: The paper clearly dicuss the difference from the previous studies.

Reproducibility: Yes

Additional Feedback: L.63: is a very popular assumption is computational complexity -> is a very popular assumption in computational complexity === I read the authors' feedback and my opinion hasn't changed.

Review 3

Summary and Contributions: The paper introduces a new technique for modelling reserve price in second-price auctions. The idea is to use contextual information as input variables in a linear model, and in doing so improve the reserve price and increase the profit. The authors propose a MIP optimization approach to discover model, and conduct experiments with real-world and syntetic data.

Strengths: - using contextual information improves reserved price. - solid algorithmic approach. - computational complexity results.

Weaknesses: - Limited target audience as the method is meant to be used in second-price auctions. Still the approach may be useful in this particular application. - Comparison to [38] should have been more detailed as both use contextual information. How their model is different than yours?

Correctness: The methodology is solid. The evaluation is done over a temporal data (the actions occur over time), but the training and the test is done in a cross-validation style, so it opens up a door to a data leakage (=predicting current auction from a model trained using future auctions). This probably doesn't have that a strong effect.

Clarity: Yes.

Relation to Prior Work: A stronger discussion with [38] would have been more appropriate (though the page limit is an issue here). Are both linear models, how are the models differ?

Reproducibility: Yes

Additional Feedback: The paper can be improved further by a more thorough comparison to [38]. More careful evaluation to avoid data leakage (though this probably doesn't change the conclusions). Perhaps a more straightforward approach here is possible where one would approximate r with a smooth differentiable function.