Title: Probing Neural Combinatorial Optimization Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neural Combinatorial Optimization

Probing

Vehicle Routing Problem
Abstract: Neural combinatorial optimization (NCO) has achieved remarkable performance, yet its learned model representations and decision rationale remain a black box. This impedes both academic research and practical deployment, since researchers and stakeholders require deeper insights into NCO models. In this paper, we take the first critical step towards interpreting NCO models by investigating their representations through various probing tasks. Moreover, we introduce a novel probing tool named Coefficient Significance Probing (CS-Probing) to enable deeper analysis of NCO representations by examining the coefficients and statistical significance during probing. Extensive experiments and analysis reveal that NCO models encode low-level information essential for solution construction, while capturing high-level knowledge to facilitate better decisions. Using CS-Probing, we find that prevalent NCO models impose varying inductive biases on their learned representations, uncover direct evidence related to model generalization, and identify key embedding dimensions associated with specific knowledge. These insights can be potentially translated into practice, for example, with minor code modifications, we improve the generalization of the analyzed model. Our work represents a first systematic attempt to interpret black-box NCO models, showcasing probing as a promising tool for analyzing their internal mechanisms and revealing insights for the NCO community. The source code is publicly available [here](https://github.com/123zhangzq/NeurIPS2025_probing).


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper uses probing techniques to analyze the inner workings of neural combinatorial optimization (NCO) methods: the authors suggest that while these methods have achieved remarkable success, we lack an understanding of the reasons for this success particularly in terms of model representation and reasoning rationale. Therefore, the paper introduces Coefficient Significance Probing (CS-probing) to investigate the embedding space of transformer-based NCO models. The key finding is that models with better generalization tend to consistently rely on the same embedding dimensions across tasks. In contrast, models whose knowledge becomes disorganized across dimensions during generalization tend to suffer performance degradation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths And Weaknesses:\n\nStrengths:\n\n1. This is the first work to apply probing techniques in the context of NCO, which helps reveal the internal mechanisms of these models.\n\n2. The paper is clearly written and well-structured.\n\nWeaknesses:\n\n1. The paper is purely empirical with no methodological innovation. While this is acceptable, the experimental setup is limited in scope—focused only on small-scale instances of TSP, CVRP, and JSSP. Broader empirical evaluation is necessary to draw robust conclusions.\n\n2. It is difficult to pinpoint where differences between model behaviors originate. Are they due to variations in neural architectures or differences in training algorithms?\n\n3. The conclusions drawn from CS-probing, while interesting, lack practical implications. It remains unclear how these insights could inform the design of improved NCO methods in future work.\n\n4. The benchmarked methods, such as POMO and AM, are relatively early approaches. It would be more compelling to evaluate with newer, state-of-the-art models.\n\nMinor comment: The paper relies heavily on the appendix. I understand that the page limit makes it difficult to keep main results in the main text. Nonetheless, and this is therefore I am keeping this as a minor comment that I hope the authors will find helpful, I suggest the authors attempt to restructure the paper to make key findings more accessible within the main text.\n\nQUESTIONS:\nRecent works have introduced diffusion models for combinatorial optimization [1,2]. Could the proposed CS-probing technique be applied to analyze such models as well?\n\n[1] DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization, NeurIPS 2023\n\n\n[2] A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization, ICML 2024\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes probing techniques to Neural Combinatorial Optimization (NCO) models, addressing a fundamental gap in the interpretability of these \"black-box\" systems. The authors investigate the representations learned by SOTA NCO solvers (AM, POMO, LEHD) on the TSP and CVRP. They achieve this through a series of well-designed probing tasks that test for both low-level (e.g., Euclidean distance perception) and high-level (e.g., myopia avoidance) knowledge. The primary methodological contribution is Coefficient Significance Probing (CS-Probing), which analyzes both the magnitude and statistical significance of probe coefficients to identify key embedding dimensions responsible for encoding specific knowledge. The findings reveal that NCO models learn both geometric and strategic information, that different architectures possess distinct inductive biases, and that superior generalization correlates with the consistent reuse of a sparse set of key dimensions.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths:\n\n1. The paper introduces CS-Probing, a novel and fine-grained analytical tool that moves beyond simply detecting the presence of information to identifying where it is encoded. The design of NCO-specific probing tasks is well-motivated and thoughtfully adapted from established practices in NLP and computer vision.\n2. This is the first systematic effort to apply probing to NCO models, addressing the critical barrier of model interpretability that currently limits their adoption in high-stakes applications.\n3. The experimental evaluation spans multiple models, tasks, and problem scales. The analysis across model layers, training epochs, and generalization scenarios provides a multi-faceted view of model behavior. The finding that just two dimensions can suffice for near-optimal performance is a powerful and actionable insight.\n4. The paper is well-written, organized, and effectively uses figures to communicate complex findings.\n\n# Weaknesses:\n1. The paper does not provide a theoretical justification for why linear probing is sufficient for analyzing NCO models, nor does it empirically validate this assumption against non-linear probes. Furthermore, the work is presented without comparison to other established interpretability techniques (e.g., gradient-based attribution, attention visualization, neuron ablation), making it difficult to assess the unique advantages of CS-Probing.\n2. The analysis involves simultaneously testing 256 dimensions for significance without discussing or applying corrections for multiple hypothesis testing. This raises concerns about the potential for a high false discovery rate, and the statistical validity of the identified key dimensions is not as rigorously established as it could be.\n3. The core claims are based primarily on three attention-based models for routing problems. While extensions are mentioned in the appendix, the paper would be far more compelling if it demonstrated the generality of the approach on a fundamentally different architecture (e.g., a GNN) or CO problem in the main text. Additionally, the sensitivity of the findings to the specific design of the probing tasks is not explored.\n4. While the insights are analytically interesting, the paper stops short of demonstrating their practical utility. A crucial missing piece is a demonstration of how these interpretability insights could be used to guide the design of a new or improved NCO model, for instance, by applying regularization to encourage sparsity or by building a model that explicitly leverages the identified key dimensions.\n\nQUESTIONS:\nMy score would increase if the authors provide a stronger comparative context for CS-Probing against other methods or demonstrate the practical utility of their insights by using them to improve a model. And I have the following questions:\n\n1. How does CS-Probing compare, empirically or theoretically, to other interpretability methods like gradient-based attribution or attention visualization for NCO models? What unique insights does it provide? And what is your justification for the sufficiency of linear probes, given that the models themselves are highly non-linear?\n2. Have you considered the impact of multiple hypothesis testing when analyzing the significance of 256 dimensions? How might methods to control the false discovery rate (e.g., Benjamini-Hochberg procedure) affect your conclusions about which dimensions are truly important?\n3. The finding that two dimensions are sufficient for high performance is striking. How robust is this finding to different initializations or training runs? To establish causality, have you performed ablation studies where you zero out these specific dimensions and measure the performance degradation?\n4. Can you provide a concrete example or a proposal for how the insights from CS-Probing could be used to design a better NCO model? For instance, could you design a regularization term that encourages the model to learn sparse, disentangled representations like those you observed in the LEHD model?\n\nLIMITATIONS:\nThe authors acknowledge practical limitations in the appendix.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Coefficient Significance Probing (CS-Probing) to enable deeper analysis of representations for neural combinatorial models. Existing neural combinatorial models are black-box models that lack interpretability. CS-Probing reveals that different NCO models introduce diverse inductive biases into the representations. The authors also try to uncover the mechanisms of generalization in NCO models.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper proposes a method to interpret the black-box neural combinatorial models. The task is meaningful in the field of combinatorial optimization.\n2. The paper offers insights into the representations, inductive biases and generalization mechanisms of the neural combinatorial models.\n\nWeaknesses:\n1. The authors focus on analyzing the representations of the neural combinatorial models. However, I encourage the authors to provides some representation learning methods to improve the performance and generalization ability of combinatorial solvers based on the discoveries.\n2. I find the paper uses instances with only 20, 100 and 200 nodes in the experiments. I wonder whether the results also hold in large scale combinatorial problems. In fact, we are more interested in the analysis and improvement in the large-scale problems.\n3. The details and the usage of linear probing model is not clear. The authors may explain the probing process.\n\nQUESTIONS:\n1. Diffusion models have shown good performance in combinatorial optimizations. What about the analysis in diffusion models? Do the discoveries also hold in diffusion-based solvers?\n\n2. LEHD uses key dimensions for decision-making. Although the optimality gap increase as we use the two-dimension embeddings. Can we use less embeddings, such as 8, 16, 32 or 64 dimensions, while the performance does not decrease?\n\nLIMITATIONS:\nThe paper does not propose an effective algorithm to enhance the representation learning or generalization of the neural combinatorial solvers.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper investigates internal representations of attention-based models used for combinatorial optimisation (AM, POMO, LEHD). It does it in two ways: a. by using internal representations to solve auxiliary tasks b. by looking at significant dimensions of these vectors utilised by the models.

STRENGTHS AND WEAKNESSES:
Pros:
The use of probing for optimisation models is interesting, since it can identify limitations and potential improvement opportunities for existing models. It was interesting to see myopia avoidance and very small number of significant dimensions in the internal representations.

The study is extensive and provides a large number of experiments and data 

Cons:
Large parts of the paper contained detailed discussions of internal embeddings (e.g. these are significant dimensions or the models are not purely greedy optimisers) rather than actionable conclusions. Table 3 is an example when this further step was taken based on CS-probing at least to reduce the model size, but the general probing idea with auxiliary tasks has unclear actionable implications.

QUESTIONS:
Q1. One of the outcomes of your study is that CS-probing can identify the most significant dimensions, which helps to build a smaller model with similar performance. However, the same outcome can be achieved by using hyperparameter tuning methods. What are the other practical implications you envision?

Q2. The second task focuses on the models ability to identify parts of the optimal solution. However, it isn't always unique. Moreover, there might be a drastically different solution with very similar optimisation cost. How do address this issue in your study?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-03