Title: BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Combinatorial Auctions

Continuous Flow Matching

Optimization

Dominant-Strategy Incentive Compatible

Revenue-Maximizing
Abstract: Differentiable economics—the use of deep learning for auction design—has driven progress in multi-item auction design with additive and unit-demand valuations. However, there has been little progress for combinatorial auctions (CAs), even in the simplest and yet important single bidder case, due to exponential growth of the bundle space with the number of items. We address this challenge by introducing a deep network architecture for a menu-based CA, which supports the first dominant-strategy incentive compatible (DSIC), revenue-optimizing single-bidder CA. Our idea is to generate a bundle distribution through an ordinary differential equation (ODE) applied to a tractable initial distribution. The BundleFlow method learns suitable ODE-based transforms, one for each menu element, to optimize expected revenue. BundleFlow achieves up to 2.23$\times$ higher revenue than baselines on standard CA testbeds and scales up to 500 items. Compared with other menu-learning baselines, BundleFlow also reduces training iterations by 3.6-9.5$\times$ and cuts training time by about 80% in settings with 50 and 100 items.


Peer reviews:
comments: SUMMARY:
This paper attempts to solve the combinatorial auction problem and proposes bundflow, a menu-based mechanism that represents the bundle distribution of each menu option via a continuous normalizing flow. This method avoids the exponential blow-up in explicit bundle representations and preserves exact dominant-strategy incentive compatible. Experiments and theoretical analysis are conducted.

STRENGTHS AND WEAKNESSES:
**Strengths**

1. Using the normalizing flow to solve the combinatorial auction problem seems new in the literature.

2. The sample codes are provided to help the reviewer verify independently. 


**Weaknesses**

1. The presentation of this paper could be improved. For example, the proof of Theorem 1 is not self-contained. Based on the current version, Theorem 1 seems to be a simple extension/corollary of results in [17].

2. It would be better to include the wall-clock training and inference times (and GPU type) for all baselines to support the 80 % saving claim clearly.

QUESTIONS:
Please see the weaknesses section.

Based on the current presentation and considering the novelty, I tend to recommend acceptance. However, I'm not an expert in this domain and am open to revising my decision according to future discussions.

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-25

comments: SUMMARY:
The paper deals with single-bidder combinatorial auctions. The paper proposes a new method inspired by diffusion which consists of two stages. In the first stage a diffusion model is trained so that the entire space of bundles is in its support. The second stage uses this distribution for optimization by manipulating the initial distributions in order to maximize the revenue.

STRENGTHS AND WEAKNESSES:
### Strengths:
- Well written and clearly motivates the single bidder setting of combinatorial auctions whereas most methods I am familiar with only discuss the multi-bidder case. Figure 1 is very didactic and explains the method beautifully.
- Strong empirical results, clearly demonstrating the method's superior performance compared to baselines.

### Weaknesses:
- Can you clarify why your method only scales to 150 items? Do you see a degradation in performance or does it become too computationally expensive? 
- Going from D=1 to D=2 seems lead to most of the performance improvement which I didn't fully understand. Can you clarify why increasing D further only provides marginal benefits?

QUESTIONS:
-

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-28

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies menu-based auctions for single-bidder combinatorial auctions. The authors proposed a diffusion-based computational techniques for generating menus. Experimental results show that the technique scales really well. It can handle 150 items with ease (less than 1 hour). The achieved revenue is significantly better a few baselines.\n\nSTRENGTHS AND WEAKNESSES:\nSince the authors focused on single-bidder menu-based auctions. Strategy-proofness and individual rationality are essentially non issues. As long as the menus are presented to the bidder (with the option of not paying and get nothing), and the bidder picks the best menu option, then SP and IR are achieved. Even when restricted to a single bidder, the setting is well motivated and has significance in practise.\n\nThe authors presented the main computational challenge via an example. What does [0.3, 0.6] mean for two items? As these can be achieved via different ways to group the deterministic bundles. The main compuational challenge is there are exponential number of ways to achieve a given item vector. To address this, the authors' main technique is to use an ODE to describe how the bundle breakdown is generated, and this ODE is characterized via a vector field. The first step of the proposed computational technique is to learn this vector field so that it generates feasible determinsitic bundle breakdowns.  The second step is to maximize revenue via menu optimization. Again this step is technically sophisticated, relying on ODE manuveour for the actual training.\n\nThe main downside of the proposed approach is its practical significance. Technically, the proposed technique makes a lot of sense. It is sophisticated and it does what is designed to do well, i.e., proposing good solutions for *very large* number of items. When we have very large number of items, we cannot use AMA/VVCA (and other heuristic methods for revenue maximization), the authors used baselines like menu+, menu-, grand, rochetnet, which are designed for scalability. The proposed method convincingly beat all existing baselines for large number of items. However, when the number of items is small, say, 10 items, the proposed method does not perform better than all baselines, as shown in Table 1. I totally understand that for this diffusion-based method, its advantage is to handle large number of items as it handles scalability really well. My main concern is whether single-bidder 50 item CA is an important or even a practically realistic setting. We are talking about 2^50 bundles. What is a realistic situation for this kind of scale? The main contribution of this paper is a technique to push up scalability. My question is whether scalability is actually an issue for the studied model.\n\nQUESTIONS:\nCan you give some practical scenarios where we need to solve single-bidder CA auction for 50-150 items? I am happy to change my rating if I see a good practical scenario. As I mentioned earlier, I think the techniques used in this paper are of high quality and original.\n\nLIMITATIONS:\nThis paper studies a theoretical auction model. The data are synthetic. I don't see any potential negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper presents BUNDLEFLOW, the first DSIC and revenue-optimizing deep learning method for single-bidder combinatorial auctions (CAs). It overcomes the exponential bundle space challenge by generating bundle distributions using ODE-based transforms, one per menu element. The method significantly outperforms baselines in revenue (up to 2.23×), reduces training iterations by up to 9.5×, and cuts training time by ~80% for large item sets (e.g., 50–150 items).

STRENGTHS AND WEAKNESSES:
Strength:
1. Scalable and efficient: Achieves superior performance and faster training in large-scale CA settings.

2. Novel technique: Introduces an ODE-based randomized menu approach that balances flexibility and tractability.

Weakness:
1. The usage of ODE training and hyperparameter tuning may pose a barrier for practical deployment.

QUESTIONS:
NA

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-06