Large-Scale Contextual Market Equilibrium Computation through Deep Learning

Market equilibrium is one of the most fundamental solution concepts in economics and social optimization analysis. Existing works on market equilibrium computation primarily focus on settings with relatively few buyers. Motivated by this, our paper investigates the computation of market equilibrium in scenarios with a large-scale buyer population, where buyers and goods are represented by their contexts. Building on this realistic and generalized contextual market model, we introduce MarketFCNet, a deep learning-based method for approximating market equilibrium. We start by parameterizing the allocation of each good to each buyer using a neural network, which depends solely on the context of the buyer and the good. Next, we propose an efficient method to unbiasedly estimate the loss function of the training algorithm, enabling us to optimize the network parameters through gradient. To evaluate the approximated solution, we propose a metric called Nash Gap, which quantifies the deviation of the given allocation and price pair from the market equilibrium. Experimental results indicate that MarketFCNet delivers competitive performance and significantly lower running times compared to existing methods as the market scale expands, demonstrating the potential of deep learning-based methods to accelerate the approximation of large-scale contextual market equilibrium.

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Reviewer 5naV5/10 · confidence 4/52024-07-09

Summary

This paper studies to how to use deep learning to solve large-scale contextual market equilibrium. This paper proposes MarketFCNet, a deep learning method for approximating market equilibrium. The paper propose an unbiased training loss and a metric called Nash Gap to quantify the gap between the learned allocation and the market equilibrium. Experiments on a synthetic game validates its effectiveness.

Strengths

Originality: The paper propose a deep learning method to solve large-scale market equilibrium, which represents buyers and goods, and directly outputs the allocation. The application is novel. Quality: The paper theoretically derives the loss function, and does some experimental analysis to validates the effectiveness of the propose method. Clarity: The paper clearly defines the contextual market modeling problem. Significance: Experiments validates that MarketFCNet are competitive with EG and achieve a much lower running time compared with traditional methods.

Weaknesses

Quality: The paper does not prove the convergence of the training algorithm. The paper either does not show the training curve. The paper does not provide the implementation code of the algorithm. Clarity: The paper is hard to follow. It is quite to hard to understand the meaning of each proposition. Significance: The paper aims to solve the large scale contextual market equilibrium, and proposes a novel deep learning method to approximate the equilibrium efficiently. However, the importance of the large scale contextual market equilibrium is not clear. I do not know how to apply the proposed method in real life.

Questions

See the weakness.

Rating

5

Confidence

4

Soundness

2

Presentation

2

Contribution

3

Limitations

Yes

Reviewer Hy1m6/10 · confidence 3/52024-07-12

Summary

This paper studies the computation of market equilibrium where there are a large number of buyers and the buyers and goods are represented by their contexts. It proposes a deep-learning method, termed MarketFCNet, to approximate the market equilibrium. The method outputs the good allocation by taking in the context embedding. It is trained on unbiased estimator of the objective function of EG-convex program using ALMM and is evaluated using a metric called Nash Gap. The method is validated by experimental results.

Strengths

The paper is well-written and easy to understand. The motivation of the paper seems natural. The paper fills the gap of using deep learning for large scale market equilibrium computation, which can be promising for future study.

Weaknesses

1. The proof of the unbiasedness of $\Delta \lambda_j$ and Lagrangian estimators in Sec 4.2 seems to be a bit hand-wavy. For example, should $b_i$’s be independent of each other? For a fixed $i$, is $b’_i$ an independent copy of $b_i$? It would be great if the authors could provide a formal (and more detailed) proof of the unbiasedness. 2. What is the effect of $k$ on the method performance? For example, if the dimension $k$ is very large, would the method fail to comprehend the context? 3. How to determine the architecture of allocation network? For example, can one use a Transformer or CNN as the allocation network? Minor issues: Line 164: It would be better to define $U(B)$ when introducing uniformly sampling to latter use. Some equations are missing “.” or “,” at the end. Please fix those.

Questions

See Weakness.

Rating

6

Confidence

3

Soundness

3

Presentation

4

Contribution

3

Limitations

Yes.

Reviewer m3RG5/10 · confidence 1/52024-07-13

Summary

The submission is not in your area and extends beyond my current expertise (from theory and applications to specific tasks and methods).

Strengths

The submission is not in your area and extends beyond my current expertise (from theory and applications to specific tasks and methods).

Weaknesses

The submission is not in your area and extends beyond my current expertise (from theory and applications to specific tasks and methods).

Questions

The submission is not in your area and extends beyond my current expertise (from theory and applications to specific tasks and methods).

Rating

5

Confidence

1

Soundness

3

Presentation

3

Contribution

3

Limitations

The submission is not in your area and extends beyond my current expertise (from theory and applications to specific tasks and methods).

Reviewer 84Ug5/10 · confidence 3/52024-07-18

Summary

This paper proposes a deep learning-based method called MarketFCNet to efficiently compute market equilibrium in large-scale contextual markets, where buyers and goods are represented by their contexts. The key idea is to parameterize the allocation of each good to each buyer using a neural network, and optimize the network parameters through an unbiased estimation of the objective function. This approach significantly reduces the computation complexity compared to traditional optimization methods, making it suitable for markets with millions of buyers. Experimental results demonstrate that MarketFCNet delivers competitive performance and much faster running times as the market scale expands, highlighting the potential of deep learning for approximating large-scale contextual market equilibrium.

Strengths

The deep learning-based approach, MarketFCNet, can efficiently approximate the market equilibrium in large-scale contextual markets by parameterizing the allocation using a neural network. This significantly reduces the computation complexity compared to traditional methods. The ability to handle large-scale markets with millions of buyers makes this approach highly relevant for real-world scenarios, such as job markets, online shopping platforms, and ad auctions with budget constraints. The paper introduces a new metric called Nash Gap to quantify the deviation of the computed allocation and price pair from the true market equilibrium, providing a meaningful way to evaluate the approximated solutions.

Weaknesses

The deep learning-based approach is inherently less interpretable compared to traditional optimization methods. Exploring ways to improve the interpretability of the learned allocation function, such as incorporating domain-specific constraints or incorporating interpretable components, could enhance the practical usability of the method. The paper does not discuss potential overfitting issues that may arise when training the MarketFCNet model, especially in settings with a large number of parameters. Incorporating appropriate regularization techniques and cross-validation strategies could help mitigate overfitting and improve the generalization performance. The paper assumes that the contexts of buyers and goods are homogeneous and can be directly used as inputs to the neural network. Extending the approach to handle heterogeneous context representations, potentially by incorporating feature engineering or meta-learning techniques, could increase the applicability to more diverse market scenarios.

Questions

From a technical perspective, what are the main strengths of the proposed deep learning-based approach, MarketFCNet, for computing market equilibrium in large-scale contextual markets? What novel evaluation metric is introduced in this paper to assess the quality of the approximated market equilibrium solutions, and how does it contribute to the methodological advancements? Given the deep learning-based nature of the approach, how might the authors address potential issues like overfitting or the ability to handle heterogeneous context representations in a more robust manner?

Rating

5

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

N/A

Reviewer Hy1m2024-08-08

Thank you for your clarifications! Since I do not have access to the PDF file for the proof of unbiasedness, I cannot check its correctness directly. Other justifications are satisfactory. I will keep my rating (Weak Accept).

Program Chairsdecision2024-09-25

Decision

Reject

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