We extend our sincere gratitude to the reviewer for their expert insights and valuable suggestions regarding the incorporation of chemistry-specific metrics to enhance our evaluation. We address their points below.

1. Report results against chemistry-specific metrics

We evaluated our best-performing models, trained on the ZINC250k dataset, using the Synthetic Accessibility Score (SAS), Quantitative Estimation of Drug-likeness (QED), diversity and novelty metrics. Diversity was measured by calculating the pairwise Tanimoto similarity within a set of generated molecules. The table below summarizes the results, including baselines and measurements on ground truth data, and more detailed visual comparisons of the metric distributions between our models and the ground truth data are available at [1]. As the results indicate, the distribution of these metrics in the samples generated by LO-ARM closely resembles that of the ground truth data.

2. Add baselines of SMILES-based transformers

Consistent with our baselines (e.g., DiGress and CatFlow), we focus on graph representation of molecules in this work. For the reviewer’s interest, we included a SMILES-based baseline in our GuacaMol results (please see our reply to Reviewer Pa4s). Morever, LO-ARM is agnostic to data representations and could also be applied to SMILES strings. As suggested by the reviewer, SMILES would be more practically useful, adapting LO-ARM to SMILES generation is a promising direction for our future work.

3. Time complexity of inference

Regarding the cost of sampling, we first note that we can skip the edge sampling stage if the order policy determines it as a no-edge dimension. Second, the learned ordering within LO-ARM effectively separates the generation of non-existent bonds (no-edges) from atoms and real bonds. This separation allows us to generate all no-edges in a single inference step without compromising the chemical validity of the generated molecules, substantially reducing inference overhead. We present a comparison of generic and ordering-informed sampling approaches evaluated on the ZINC250k dataset below.

We plan to further explore this learned ordering to scale up inference in future work. A possible yet more principled direction could be leverating the equivalence between random order ARMs and masked diffusion models to enable parallel generation. Our preliminary experiment shows that we can obtain 90.0% validity in half of the number of steps using the masked diffusion formulation of our model.

4. Suitability of learning ordering for molecular graph generation

While generation order constraints reduce the permutation space, our results show a clear benefit to learning a data-dependent order. For example, LO-ARM significantly outperforms GraphARM [2], which unmasks a node and its edges in one step, across all metrics.

To ensure a fair comparison with our baseline models, we have only evaluated unconditional generation. To further tailor LO-ARM for molecular graph generation, one could integrate inductive biases into the backward sampling process during training, which we also recognize as a promising avenue for future development of LO-ARM.

5. Should the equation for qtheta at the start of page 6 have a $x$ instead of $x_{z<i}$?

Yes, this is a typo, thanks for examining our work thoroughly.

6. Is each component of z sampled autoregressively, or are all sampled at the same time?

To clarify: z is sampled autoregressively via the order-policy for generation, but for the variational distribution during training, all its components are generated in one pass by the q-network using the Gumbel top-k trick [3].

7. Enrich essential references

We will update the Related Work section with the references you’ve suggested in the final version of the paper.

We hope that our responses have addressed your concerns and questions. If so, we would be grateful if you would reconsider your decision in light of our clarifications and update your recommendation score accordingly. We remain available and eager to address any further concerns you may have.

[1] Chemistry-specific metrics: https://drive.google.com/file/d/1miSlh2vYYRfqHJ-e5QmYFmht5xMQcqEO/view?usp=sharing

[2] https://arxiv.org/pdf/2307.08849

[3] https://arxiv.org/pdf/1903.06059

We sincerely appreciate your positive feedback on the quality of our work. We now address each of your concerns as below.

1. Preliminary results on larger bioactive molecule dataset (ChEMBL)

To address your concern regarding the scalability of LO-ARM to larger datasets, we include a preliminary result on the ChEMBL dataset (also known as the GuacaMol benchmark [1]) without hyperparameter tuning. Specifically, we preprocessed this dataset using the utility provided in [2], a method consistently employed in our benchmarks [2, 3]. We have summarized the statistics of the three datasets used in our evaluation—namely QM9, ZINC250k, and ChEMBL/GuacaMol—as presented below. We believe this inclusion provides a more comprehensive assessment of LO-ARM's capabilities.

Our preliminary results show that LO-ARM still exceeds or matches performance of the current state-of-the-art-models.

To ensure consistency in our evaluation across the QM9 and ZINC250k datasets (as is common in the literature), we have converted the reported FCD scores from prior work, which are often presented on an exponential scale, back to their raw FCD values.

Specifically, the LO-ARM model trained on the GuacaMol dataset exhibits a preference for an atom-first generation order. This means it tends to generate atoms first, followed by the real bonds connecting them, and lastly, it fills in the non-existent or "imaginary" bonds. We illustrate an example of this generation process in the accompanying figure [4]. This learned ordering contrasts with the edge-first ordering observed in models trained on the ZINC250k and QM9 datasets. This difference likely arises because generating node-related dimensions first is potentially simpler for the GuacaMol samples, given that the number of edge dimensions increases quadratically with the number of nodes.

Our preliminary findings indicate that LO-ARM surpasses other graph-based methods in terms of validity and uniqueness, achieving performance levels close to the state-of-the-art SMILES-based approaches. However, we have observed a small gap in FCD compared to DiGress. We hypothesize that this difference arises because for datasets with higher dimensionality, a generation strategy operating at a coarser granularity (e.g., dimension blocks) than individual dimensions might be more effective. For example, tokenizing molecules into fragments, as discussed in our Discussion section, could potentially improve performance. We defer the task of integrating domain-specific tokenization into LO-ARM for future investigation.

2. Some aspects of the methodology and evaluation could be further discussed.

In addition to these new results of GuacaMol experiments, we have provided an in-depth analysis on training stability with REINFORCE in the thread of Reviewer Stjb.

3. Improve the presentation of the main tables and enrich related work

Thank you for your thorough review and your advice, and we will refine the main tables and reference the surveys of graph generation in the final version.

We hope that our responses have addressed your primary oncerns regarding the scalability of our algorithm. If so, we respectfully request that the reviewer reconsider their decision in light of our responses and update their recommendation score accordingly. We remain available and eager to address any further concerns the reviewer may have.

[1] https://arxiv.org/abs/1811.09621

[2] https://arxiv.org/abs/2209.14734

[3] https://arxiv.org/abs/2406.04843

[4] Generation trajectory of GuacaMol sample: https://drive.google.com/file/d/1a5HU4FZ98bqS_9JFK4Eb60QfhXiksaue/view?usp=drive_link

Thank you for the positive feedback! We are glad that you find our theoretical results sound, our improvement consistent and our ablations thorough. We address your concerns as below, especially your concern on training stability with REINFORCE.

1. Performance gains on QM9 and ZINC250k

We believe our improvements on QM9 and ZINC250k are substantial. Although baselines score well on validity/uniqueness, our results indicated that there is still large room for improvement in distributional similarity metrics such as FCD — highlighted by Reviewer Stjb as a recognized measure for drug-like molecules. Our LO-ARMs model achieved new state-of-the-art FCD scores, reducing them to 0.240 (QM9) and 3.229 (ZINC250k) from prior bests of 0.441 and 13.21, respectively.

2. Smaller and larger scale benchmarks (e.g., FreeSolv, Lipophilicity, HOPV)

Thank you for your suggestion, and we will certainly include the results on smaller benchmarks in the final version of our work. Given the time constraints of this rebuttal period, we made a strategic decision to prioritize the GuacaMol experiment, which is a much larger dataset than ZINC250k, to maximize the efficiency of our response. This is because we anticipate that training stability might pose a greater challenge for larger datasets compared to smaller ones, as their training processes tend to exhibit more stochasticity. Furthermore, the scalability of LO-ARM is a concern shared by several reviewers. The detailed results on GuacaMol dataset are presented in the response to Reviewer Pa4s. In short, our preliminary results show that LO-ARM learns an order-first generation ordering [1] and still exceeds or matches performance of the current state-of-the-art-models.

3. Analysis of variance and convergence of REINFORCE-based training

To provide a clearer understanding of the variance and convergence of REINFORCE and the effectiveness of our variance reduction method, we have visualized the following quantities during training course of the GuacaMol experiments [2]:

• Negative Evidence Lower Bound (ELBO)
• Maximum and minimum q-logits from the variational order policy network.

Specifically, we conducted an ablation study on the learning rate in two experiments, keeping all other settings constant. Given that the gradient variance primarily arises from the stochasticity of the variational order policy, the maximum and minimum q-logits can reflect this variance throughout the training.

As seen in the plot, most of the time the RLOO variance reduction keeps the optimization smooth -- still occasionally training instability is observed with a large learning rate (2e-5), indicated by loss spikes and discontinuities in the maximum q-logits. Fortunately, reducing the learning rate to 1.5e-5 effectively stabilizes training, as demonstrated by the green curves.

We hope that our responses have addressed your concerns and questions. If so, we respectfully ask that you reconsider your decision based on our responses and update your recommendation score accordingly. We are also eager to address any further concerns you may have.

[1] Generation trajectory of GuacaMol sample: https://drive.google.com/file/d/1a5HU4FZ98bqS_9JFK4Eb60QfhXiksaue/view?usp=drive_link

[2] Variance visualization: https://drive.google.com/file/d/19b2wQocvLJc82bAm6eUG5qbAcmVAra0V/view?usp=drive_link

Thank you for the time you’ve taken to review our work and for the positive and constructive feedback! We are glad that you found the problem we dealt with important and our approach novel and well-grounded. In response to the weaknesses and questions:

1. Generalization to other high-dimensional data (images, graphs)

We would like to clarify that the focus of this work is on designing the order learning mechanism in ARMs and its application to molecular graph generation as the main testbed. We chose this task because we believe it is more suitable – As the reviewer has pointed out, an important issue in graph autoregressive generation is the difficulty to determine an order for sampling. Indeed, our learning-order ARM was able to discover an autoregressive order that outperforms fixed or random order used in prior graph generation work. Our focus is not on images – we included MNIST only as a sanity check to confirm the model can learn a meaningful order that distinguishes between the digits and the background. While there are no practical constraints for applying LO-ARMs to higher dimensional natural images – it is unclear whether there is a similarly meaningful and interpretable ordering to learn there and therefore we decided not to pursue it in this work.

To demonstrate the scalability of our algorithm to higher dimensional graph datasets, we conduct an additional experiment on the GuacaMol dataset, which is a larger molecule dataset with 3906 input dimensions (which is larger than 1482 of ZINC250k and 90 of QM9). We have provided the comparison of the three molecule datasets in the discussion with Reviewer Pa4s. We believe that these three datasets can provide a comprehensive evaluation matrix to support our claims.

Performance-wise, our preliminary results show that the order-policy still yields an advantage, exceeding or matching the SOTA performance, as you can see from the details we have provided in the discussion thread with Reviewer Pa4s.

2. Application to 3d point cloud, pixels, voxels

Thank you for the suggestion. We are interested in exploring these modalities in future work.

3. Pipeline illustration figure

We will follow the reviewer’s suggestion to add an illustration figure in the final version.

We hope that our responses have addressed your concerns and questions. If so, we would kindly ask the reviewer to reconsider the decision in light of our responses and update their score accordingly. We are also eager to address any additional concerns the reviewer may have.