IO-LVM: Inverse optimization latent variable models with applications to inferring and explaining paths

Dear Reviewer, thank you for your feedback. We hope the following responses clarify your points of concern:

Q: "The coefficient β is presented as ‘introduced’ to balance data reconstruction with a Gaussian latent distribution."

A: This is not our intention, and we will be glad to clarify better the writing if this is the interpretation. Our intention is to claim that we generalize the β parameter for a constrained optimization scenario.

Q: "The comparison with baselines is hard to interpret. How significant are IO-LVM results compared to PO?"

A: Unlike PO (and similar state-of-the-art methods), IO-LVM can model multiple cost functions. This capability is unique to our approach, this is why we bring this comparison.

Q: "Are latent spaces always 2-dimensional in the paper? What conclusions can be drawn from using low-dimensional spaces?"

A: The main challenge in our problem is path structuring, which is addressed by Dijkstra’s explicit constraints. Unlike traditional VAEs, which require higher-dimensional spaces to reconstruct complex structures (e.g., images), our VAE only needs to reconstruct a simpler, unconstrained edge cost space. This is why 2 / 3 dimensions are generally enough. More dimensions would not me harmful though (as we have observed), with a good choice of beta.

Dear Reviewer, thank you for your constructive feedback. We hope the following responses clarify some key aspects of our work:

Q: "The paper leans too heavily on qualitative analysis over quantitative analysis. Only one table of results compares against baselines, and IO-LVM’s performance gain is only meaningful on the synthetic dataset’s Spearman metric."

A: One of the focus of this paper was indeed to provide more qualitative analysis on the latent space, which we believe it is not a weakness. While there are many paper providing tables and numbers, we preferred to bring a more insightful perspective in the unsupervised setting, together with the reconstruction table results.

Q: "It is unclear what different Jensen-Shannon divergence values tell us about a model’s predictive and reconstructive abilities. Additional quantitative metrics would be beneficial."

A: Selecting a reconstruction metric for paths is challenging. Previous works (e.g., https://arxiv.org/abs/2002.08676 and https://arxiv.org/abs/1912.02175) often use “full path match” metrics (1 if the path fully match, 0 otherwise, and then average), but these are unreliable for larger graphs with noisy paths, where full path matching would be near 0% despite of the method chosen. Instead, we chose JS divergence for its symmetry and ability to quantify probability distribution distance between inferred and actual edges/paths choice. One could think of Euclidean metrics also, but they would not make much sense in our case because we wanted to show a "discrete" quantification, e.g., closer nodes in distance are as wrong as distant nodes. We are open for suggestions in this regard.

Q: "The proposed experiments are relatively simple. Given that IO-LVM’s latent space clusters agent ID significantly better on simpler synthetic datasets than on the Ships dataset, I question its scalability."

A: As the reviewer mentioned, the ship dataset is a complicated dataset. We believe the ship dataset is complicated enough to test the scalability of IO-LVM. The reason why Figure 3c (showing the separability in the latent space in the ship data) is not as clean as Figure 2b (showing the separability for the agents) is because there might be many other variables impacting the path choice in the ship dataset, even variables (or noise) that we do not have access to for a possible evaluation.

Q: "For the Ships dataset, why do you think having just two latent dimensions was sufficient? Is this related to the problem’s simplicity?"

A: The main challenge in our problem is path structuring, which is addressed by Dijkstra’s explicit constraints (or potentially any other solver for different problems). Unlike traditional VAEs, which require higher-dimensional spaces to reconstruct complex structures (e.g., images), our VAE only needs to reconstruct a simpler, unconstrained edge cost space.

Q: "Do you know of any other metric aside from Jensen-Shannon divergence and Spearman’s rank correlation that can better highlight IO-LVM’s advantage?"

A: This is addressed in the response above.

Q: "What other problems aside from path explanation could IO-LVM be applied to with small modifications?"

A: With slight modifications, IO-LVM could apply to any problem where the COP can be formulated as an Integer Linear Program with an efficient solver available.

Dear Reviewer, we value your detailed feedback. We are open for improving our work, however, we think that there were some misunderstandings regarding the content of our proposed method and the topic in general, so we would like to clarify it better.

Q: "The method seems unappealing because of the complexity of needing to run a pre-defined planning algorithm (such as Dijkstra's)."

A: Many studies in this field incorporate these algorithms within neural network frameworks, i.e., they also have to run does algorithms during training and inference. We could list at least 50 papers in this area, but here are some of the important works: https://arxiv.org/abs/2002.08676 https://arxiv.org/abs/1710.08005 https://arxiv.org/abs/1912.02175 We hope you can reconsider whether this aspect is truly a weakness in the paper.

Q: "The paper doesn’t carefully validate the claim that VAEs and other generative models produce paths which violate constraints. I’m not fully convinced of this."

A: It is well known that neural networks cannot ensure COP feasibility. VAEs do not impose explicit constraints; instead, they would need substantial data to infer constraints, with no guarantee of generalization. By integrating a Dijkstra (or other structured algorithm) block, we ensure that the constraints are explicitly modeled, ensuring feasibility. This challenge is reflected in Table 2, where VAE results are significantly less effective due to their lack of path structure representation.

Q: "Is it possible to use the algorithm to learn optimal trajectories even when trained on suboptimal data? I think this aspect could be interesting."

A: Yes, and we explored this in our work. The ship paths in our dataset are not always optimal and include noise and therefore suboptimal characteristics. Similarly, in the synthetic data, we introduce three agents with different cost functions and inject noise into each function to generate suboptimal paths. Our model can either overfit to this noise or denoise (through variations in the beta parameter).

Dear Reviewer, we appreciate the time and effort you put into reviewing our paper. We hope the following clarifications will address your concerns:

Q: "If I understand correctly, the latent learned space has only 2 or 3 dimensions. The method would be more convincing if there were experiments with more dimensions, e.g., discrete decision problems."

A: The path reconstruction problem is inherently discrete, as the decision variables correspond to binary vectors in the edges’ space (i.e., selecting or not selecting each edge for the path). Our latent space (z) generates a continuous edge cost space. Because reconstructing edge costs is simpler than reconstructing paths, a low-dimensional latent space (2 or 3 dimensions) is sufficient. The complexity of path structure is managed by the shortest path algorithm (Dijkstra in this case), allowing the neural blocks to focus on the simpler task of decoding/reconstructing edge costs.

Q: "Quantifying how well the model captures the ship width would be interesting. I'm curious whether you can learn a mapping from z to the weight of the ship if that data is available."

A: We are not sure if you meant "width" or "weight," but one of our focuses here is on unsupervised learning insights. We demonstrate that latent space vectors correlate with variables (e.g., ship width) that were not part of the training data, providing evidence that the learned latent space is meaningful without explicit supervision. Having contextual features as part of the training data is not the scope of this work. Accounting them transfers the variability of the observed paths in a context-conditioned model, which is the same as done in the PO baseline.

Q: "What could be other applications of this method apart from path sampling? Could this be applied to discrete decision problems from the Perturbed Optimizer paper?"

A: This is a great question. Yes, this could be applied to other discrete problems that are in the Perturbed Optimizer paper. We thought that giving an emphasis in path problems is more interesting and enough to show the capabilities of our method. It is not only interesting in a graphical/illustrative perspective, but also in a real-world application for path prediction/reconstruction/denoising.