Subequivariant Morphology-Behavior Co-Evolution in 3D Environments

We sincerely appreciate the reviewer’s kindness and honesty in acknowledging their unfamiliarity with the field, yet still providing a fair and positive assessment of our work. Thank you for your time and consideration in evaluating our paper, and for bringing this to the attention of the area chairs.

We sincerely thank the reviewers for their encouraging comments and thoughtful suggestions. We appreciate the opportunity to further improve the clarity and completeness of our work, and we will address each of these points in detail.

W1: The paper implicitly assumes some prior knowledge of subequivariant graph neural networks on the reader’s side. Some of the mathematical notations and formulae (such as the stack of equations on page 5) might need some further clarification so that they are more understandable.

Thank you for your suggestion. We will provide further clarification on the notations and formulae to improve readability.

W2: I find that the authors did not mention the architectures of the morphology transform policy and “conventional neural networks” involved in behavior control and value estimation. The exact process of morphology transform (such as the actions taken and attributes being transformed) is not explained either. If these design choices are identical to some previous work, I suggest that the authors cite the related papers so that readers could refer to them for more details.

Thank you for your feedback. We have detailed the architectures of the morphology transformation policy and the neural networks for behavior control and value estimation in Sec. 4.2 "Implementations" and Table 2. We have enhanced the Sec. 3.2 with additional explanations and relevant citations for clarity.

Q1.Regarding Figure 5, is it that each pair of agents are trained simultaneously and competing against each other? There are some cases where the win rates of both opponents decrease with the training process. Could you further explain how this should be interpreted?

Thank you for your question. Yes, in Figure 5, each pair of agents is trained simultaneously and competes against each other. The observed decrease in win rates for both opponents during training can be attributed to the presence of draw outcomes in the sumo environment, as neither agent wins in these cases. We mention the draw bonus in the reward settings for sumo, which accounts for these situations. Thank you for pointing this out. We have revised the paper to include a special note on draw outcomes in the Termination section for the sumo environment, clarifying how draws impact win rates during training.

Q2.The authors did not mention any publicly accessible code repository in the paper or supplementary material. Do you have any plan to open source your code and environment?

We plan to open-source our code and environment on GitHub upon the paper's acceptance.

Q3.The baseline algorithms seem a bit simple, more like ablated versions of your proposed method. I wonder how your RL approach compares with evolutionary algorithms (EAs), such as Neural Graph Evolution, in terms of morphological evolution. From my understanding, the control algorithm that you propose could be readily combined with any of those EAs, serving as inner-loop fitness evaluation. Did you carry out any comparison like this?

Thank you for this insightful question. We did consider comparing our method with evolutionary algorithms (EAs), such as Neural Graph Evolution, but ultimately chose not to include them in this study for several reasons. First, our focus is on introducing subequivariance within a reinforcement learning (RL) framework, where symmetry is directly leveraged to improve sample efficiency in continuous control tasks. In contrast, EAs typically focus on population-based search and often require a significantly larger number of samples, which may not align with our emphasis on sample efficiency. Additionally, integrating EAs would shift the focus of our work and complicate the analysis, as EAs follow a different paradigm for morphological evolution. However, we agree that combining our control algorithm with EAs could be an interesting direction for future work and may serve as a robust inner-loop fitness evaluation.

Dear Reviewer 5q96,

Thank you for your thoughtful and constructive review. In response, we have made specific revisions to address your concerns, including clarifying notations and formulae and adding implementation details. We look forward to any further comments you might have before the discussion period concludes.

Best regards,

Authors

We sincerely thank the reviewers for their constructive feedback and positive assessment of our work. We appreciate the insights provided and will carefully address each point to improve the clarity and robustness of our study.

W1.The main algorithmic contribution of this work is the introduction of geometric symmetry into the formulation of co-evolution. However, geometric symmetry is a widely recognized concept for enhancing the sample efficiency of interactive learning methods, such as reinforcement learning, making the contribution appear modest.

We agree that geometric symmetry is a widely recognized concept for improving sample efficiency in interactive learning methods like reinforcement learning. Rather than focusing solely on behavior, we leverage subequivariance in a way that impacts both behavior and morphology. The morphology network’s learning signal is derived from rewards generated through behavior-environment interactions. Through this interaction-driven mechanism, subequivariance indirectly influences morphology evolution, contributing to an integrated framework for co-evolution.

Furthermore, the evolution of morphology is fundamentally shaped by environmental interactions. The geometric symmetries present in the environment and its dynamics profoundly influence both morphology and behavior. Previous methods have often overlooked environmental symmetries or imposed direct constraints on morphology, such as bilateral symmetry, without allowing for natural, autonomous development through environmental interaction. Our framework introduces an environment that embodies these geometric symmetries, facilitating the autonomous co-evolution of morphology and behavior. This aligns with the principles of embodied intelligence, where agents learn and evolve through interactions with their environment, leading to more natural and effective adaptations.

Please refer to our General Response. We have revised the first paragraph of the introduction, with the modifications highlighted in blue text, and have re-uploaded the PDF to emphasize the importance and challenges of this problem setup.

W2: The paper claims to address the lack of spatial geometric information consideration in tasks, yet the navigation environments used in the experiments are relatively simple and lack significant geometric complexity. Real-world navigation tasks typically involve obstacles. The reviewers would appreciate seeing ablation studies in navigation scenarios with obstacles to further demonstrate the proposed method's effectiveness.

We appreciate the reviewer's insightful comments regarding the geometric complexity of our navigation environments. In real-world scenarios, navigation tasks often involve obstacles, leading to exponential growth in the system's geometric transformations. To manage this complexity, navigation can be decomposed into simpler, obstacle-free local environments through planning. Our current work focuses on leveraging geometric equivariance to evolve morphologies that enhance behavioral efficiency in such symmetric environments. Exploring how morphology-behavior co-evolution can facilitate planning capabilities in more complex settings is a valuable direction for future research. Notably, certain organisms, such as slime molds, lack a central nervous system yet effectively navigate simple environments through morphology-behavior co-evolution.

Dear Reviewer ymVo,

We deeply appreciate your detailed and constructive review. In response to your suggestions, we have made specific revisions, including clarifications in the Introduction regarding the importance of co-evolution, and detailed explanations in the supplementary material. As the discussion deadline approaches, your perspective has already been invaluable in refining our work, and any additional feedback would help us ensure its value.

Best regards,

Authors

We sincerely appreciate the reviewer's constructive feedback and thoughtful suggestions, which have greatly helped us improve the clarity and presentation of our paper. We are committed to addressing each of the concerns raised in detail and have incorporated relevant clarifications and citations into our revised version. Below, we provide detailed responses to each point to further highlight the motivations and contributions of our work.

W1: The exposition of the motivations, the methods and the results should be improved. I provide some examples.

In the summary of the contributions. 3DS-MB is defined as a "benchmark", in the rest of the paper as a "method".

I could not find where the authors define which symmetries the network and the morphology are invariant to.

From the figures it seems that the ant evolves in a radially-symmetric way (figure 2), but then in figure 10 it seems to be laterally symmetric.

W4: In the introduction, the authors write "An essential aspect of our approach is ensuring that the evolution of the morphology remains invariant to geometric transformations". At this point, one would expect the morphology transformations to be constrained by some kind of symmetry.

Figure 1 and 2 also reinforce this misunderstanding: the morphology on the right (with subequivariance) looks symmetric (radially and laterally), while the one on the left is asymmetric. Instead, it seems to be that, while the representation is symmetric due to the subequivariance of the graph, the morphology transformations are not (figure 10). The authors should make these aspects clearer.

We appreciate the reviewer’s suggestion regarding consistency in terminology. In our work, "3DS-MB" refers to the proposed benchmark setup, whereas "EquiEvo" denotes our specific method. To improve clarity, we have updated the title of the methods section, "Setup and method", to clearly distinguish between the benchmark setup and the specific method.

• Regarding the symmetries in morphology and network invariance, we did not impose specific symmetry constraints on the morphology itself. Instead, we applied equivariance constraints to the network to leverage symmetry in the state representation. In this way, we ensure that the network’s output is equivariant to transformations applied to the input. We described our Eg(3)-equivariant setup in Section 3.1.
• The evolved morphology is determined by both the objectives of the task and the effectiveness of the optimization. Figures 1 and 2 show the evolved results for the navigation task. In Ant Navigation task setting, radially symmetric morphologies are more effective for omnidirectional movement. With the use of equivariant networks, our method successfully evolved a radially symmetric morphology, while the baseline without equivariant networks evolved an asymmetric morphology due to lower optimization efficiency.
• In Figure 10(b) and (c), we compare the evolution processes under different task (reward) settings. For the task with a forward reward, the evolved morphology exhibits a laterally symmetric structure with stronger front legs and weaker hind legs, while the task without a forward reward leads to a radially symmetric morphology. This shows that the evolution of morphology is fundamentally shaped by environmental interactions and task demands, rather than being a predefined goal. The comparison between Figures 10(a) and (b) shows that methods without integrating geometric equivariance, due to lower sample efficiency, fail to fully evolve a radially symmetric morphology that meets task requirements. This highlights the necessity of considering geometric equivariance in the co-evolution of morphology and behavior frameworks.

W2: Maybe the authors mean that the studying the role of symmetry was not the focus of the paper, but their wording suggests something else. I recommend to clarify how the author's approach to symmetry in co-evolution differs from or extends beyond the bilateral symmetry used in the cited papers, and if the symmetry they impose is only in the state representation or also in the morphology transformations.

We appreciate the reviewer's query about symmetry. In our work, we address the co-evolution of morphology and behavior by incorporating the geometric symmetries inherent in real-world dynamics. The evolution of morphology is fundamentally shaped by environmental interactions and task demands. Previous methods have often either overlooked the role of geometric symmetries of the environment, resulting in reduced effectiveness, or imposed rigid constraints on morphology, such as bilateral symmetry, which restricts the natural development of morphology through interaction with the environment. Our framework introduces a novel problem setup that embeds geometric symmetries, facilitating the autonomous co-evolution of morphology and behavior. This aligns with the principles of embodied intelligence, where agents learn and evolve through interactions with their environment, leading to more natural and effective adaptations.

Furthermore, geometric symmetries present in the environment and its dynamics profoundly influence both morphology and behavior. Rather than focusing solely on behavior, we leverage subequivariance in a way that impacts both behavior and morphology. The morphology network’s learning signal is derived from rewards generated through behavior-environment interactions. Through this interaction-driven mechanism, subequivariance indirectly influences morphology evolution, contributing to an integrated framework for co-evolution.

Please refer to our General Response. We have revised the first paragraph of the introduction, with the modifications highlighted in blue text, and have re-uploaded the PDF to emphasize the importance and challenges of this problem setup.

W3: It is unclear when reading the paper which components of the design are novel and which have been taken from previous work.

For example, in the section "Morphology transform", no previous paper is mentioned, but several papers cited in the introduction define similar or identical transformations. Instead, the authors write "We divide the morphology transform stage into two sub-stages", implying that is an idea of this paper.

In general, it is very difficult to evaluate this paper because the design choices are not introduced in view of the related works. Similarly, how does the subequivariant graph neural network described in this paper differ from the ones used in [3] and [4]?

Thank you to the reviewer for the insightful feedback. We will carefully review our citations and ensure that prior work is properly referenced to clarify our novel contributions.

• Morphology Transformation: For the morphology transformation stage, we adopted the two-stage design introduced in Transform2Act. A core aspect of our approach is that morphology evolution is inherently invariant to spatial transformation, which led us to incorporate an equivariant network to achieve invariance in morphology evolution and equivariance in behavior control, improving the efficiency of the overall co-optimization process.
• Subequivariant Graph Neural Network: Both our approach and [3][4] implement equivariance based on the core function described in Equation (2). However, there are key differences in the design and application. [3] employs a transformer architecture with a fully connected topology, whereas both [4] and our approach utilize GNNs. The main distinction lies in the graph construction: [4] models multi-agent relationships, requiring an entity assignment to define the graph's topology, while our method focuses on single-agent morphology evolution, directly using the morphology's inherent topology to build the graph. This design aligns with our goal of co-optimizing morphology and behavior efficiently within a single-agent framework. We have already addressed these differences in the Related Work section.

Thanks again for your constructive comments, which has significantly contributed to enhancing our paper!

I1: Could the authors clarify whether the morphology value is a function of the state or not? Does it take as input only the structural features of the morphology (e.g., weight, thickness, ...) or also the initial/current pose? The authors say that it depends on the "initial state distribution" - how does the initial state distribution change for a given morphology? Or do you rather mean "the initial state"? I maintain that the value of a morphology should not depend on something that it is not one of its structural properties, and therefore I do not see why a subequivariant representation of the state would make a difference.

Thank you for your insightful question. The objective in Equation (1) is given by:

$G_{m} = \text{argmax} J(\pi_\mathcal{G}, \mathcal{G}_{m})$

$\text{s.t.} \quad \pi_\mathcal{G} = \text{argmax} J(\pi, \mathcal{G}_{\mathrm{m}}).$

where $J(\pi, G_{m})$ represents the cumulative reward for a given morphology $G_{m}$ and policy $\pi$. From this two-layer objective, it is clear that the optimization of morphology depends on the behavior. It is important to note that the objective $J$ is defined as an expectation, meaning that in the expectation sense, all states are integrated out. Therefore, the morphology’s policy and value functions ultimately depend only on the morphology $G_{m}$ and the behavior policy $\pi_G$.

In practice, the bi-level optimization is implemented as a co-optimization process, where the inner and outer optimizations alternate incrementally rather than fully optimizing inner layer. Since the evaluation of $J$ relies on finite samples of the behavior policy, if the behavior policy is not equivariant, the limited sampling will cause $J$ to vary under sampled initial state. Geometric symmetry reduces the variability introduced by sampling, as equivariant networks ensure equivalent actions and consistent values, providing consistent feedback across state transformations——such as translations, rotations, or reflections.

Besides, the "initial state distribution" refers to the set of possible initial configurations for the environment and the agent at the start of a task, such as the initial pose, position, or orientation of the agent, as well as any task-specific initialization parameters. This distribution is fixed for a given task. For example, in navigation tasks, the initial state distribution might include random positions and orientations of the agent within a defined space. The cumulative reward $J(\pi, \mathcal{G}_{\mathrm{m}})$ is evaluated under this fixed initial state distribution, which implies that morphologies are optimized to generalize across the different initial states.

We apologize for any confusion in our earlier rebuttal and hope this provides a clearer and more intuitive understanding of the relationship between morphology and state.

I2: Maybe the authors are implying that, as the agent's behavior is the same in every direction, it is easier to learn a value for the morphology, because the task basically becomes 1d instead of 2d/3d, reducing its complexity. In this case I would agree, but if this is the point, I find the explanation overly complicated.

Your explanation is indeed intuitive and aligns closely with the concept of rotational equivariance. Our goal is to utilize geometric equivariance of transformations such as translations, rotations, and reflections in Euclidean space to enhance the efficiency of co-evolution. This approach corresponds to what van der Pol et al. (2020) described as "a reduction in an MDP’s state-action space under an MDP homomorphism."

The complexity of our explanation arises from the nature of the co-evolution process, which is framed as an MDP that involves both policy and value equivariance. Furthermore, the intertwined evolution of morphology and behavior adds another layer of interaction, where the optimization of one impacts the other. We appreciate your perspective and will consider including a simple case similar to your example to provide researchers with a more intuitive explanation.

Elise van der Pol, Daniel E. Worrall, Herke van Hoof, Frans A. Oliehoek, and Max Welling. MDP homomorphic networks: Group symmetries in reinforcement learning. In  Advances in Neural Information Processing Systems 33, NeurIPS 2020, December 6-12, 2020, virtual, 2020.

We appreciate the reviewer’s detailed feedback and recognition of our work’s originality. We understand that some points may stem from misinterpretations of our methodology. We are preparing responses to clarify these aspects and address the questions raised. Thank you for your comments, which will help us further improve the clarity and rigor of our presentation.

W1: The primary contribution of this work is a substitution of the reinforcement learning policy module from the Transform2Act approach with an equivariant GNN. Compared to the work cited by authors (Chen et al., 2023; Chen et al., 2024), this submission appears to be a repackaging of an existing subequivariant reinforcement learning idea with a different problem setup.

Thank you for your feedback regarding the novelty of our work. We would like to clarify that adapting existing methods to address significant problem setups is a common and valuable approach in impactful research. For instance, AlphaGo combined MCTS with DRL to tackle the complex game of Go; AlphaFold series utilized existing Transformer and diffusion model to predict protein folding; GPT-3 scaled up transformer models with larger datasets and parameters for natural language processing tasks; and DALL-E employed transformers for text-to-image generation. These examples demonstrate that applying established techniques to new domains can yield significant advancements by addressing unique challenges within those domains.

• In our work, we address the co-evolution of morphology and behavior by incorporating the geometric symmetries inherent in real-world dynamics. The evolution of morphology is fundamentally shaped by environmental interactions and task demands. Previous methods have often either overlooked the role of geometric symmetries of the environment, resulting in reduced effectiveness, or imposed rigid constraints on morphology, such as bilateral symmetry, which restricts the natural development of morphology through interaction with the environment. Our framework introduces a novel problem setup that embeds geometric symmetries, facilitating the autonomous co-evolution of morphology and behavior. This aligns with the principles of embodied intelligence, where agents learn and evolve through interactions with their environment, leading to more natural and effective adaptations.
• Furthermore, geometric symmetries present in the environment and its dynamics profoundly influence both morphology and behavior. Rather than focusing solely on behavior, we leverage subequivariance in a way that impacts both behavior and morphology. The morphology network’s learning signal is derived from rewards generated through behavior-environment interactions. Through this interaction-driven mechanism, subequivariance indirectly influences morphology evolution, contributing to an integrated framework for co-evolution.

Please refer to our General Response. We have revised the first paragraph of the introduction, with the modifications highlighted in blue text, and have re-uploaded the PDF to emphasize the importance and challenges of this problem setup.

W2: Although the symmetry enhancement is the core claim, improved symmetry was only explored within a single basic agent skeleton -- the ant morphology. There was no metric or quantitative analysis to test the consistency of symmetry improvement across varied morphologies, or how the improvement of implementing LRF transformation in the policy module affects the optimization algorithm. Without visualizations of generated morphologies from additional environments, and without a symmetry-based metric to measure the improvement across different setups, it is difficult to assess the claims of the submission. ... In two of the three explored benchmark environments (the humanoid and sumo) the morphology transformation was omitted and only attribute transformations were possible.

We thank the reviewer for highlighting concerns about symmetry and generalizability across varied morphologies. Our primary goal is not to enforce symmetry in morphology per se, but rather to inject geometric symmetry into the policy module to leverage the symmetry inherent in real-world environments and dynamic systems. This enables a more efficient co-optimization of morphology and control. The resulting symmetry in morphology is an emergent property, not the intended objective of our approach.

• To evaluate the effectiveness of our method, we provided visualizations (Fig. 10 for Ant, Fig. 6 for Humanoid) to qualitatively analyze the symmetry in evolved morphologies. Quantitatively, we used the “reward obtained by training duration” metric (shown in Figs. 7, 8, and 9) as it directly reflects the effectiveness of morphology-control co-evolution, which aligns with our motivation.
• We designed the Ant-Navigation task starting from a simple “atomic morphology” (a torso body) as the initial point. This choice allows the agent to autonomously co-optimize morphology and control based on task requirements and environmental interaction, focusing on the effects of injected symmetry. Examining different starting morphologies was not within our scope, as our objective was to understand the impact of geometric symmetry on co-optimization.
• Regarding different initial skeletons, like prior work such as CompetEvo, we did not focus on skeleton evolution. Instead, we investigated the impact of morphological attributes on task performance. To this end, we further explored two complex tasks, Humanoid-Navigation and Ant-Sumo, to assess the effect of geometric symmetry injection on morphology-control co-optimization.

We have already reiterated our motivation in the introduction to clarify this.

W3: The only metric that was considered was reward. … The submission claims ‘integrating morphology evolution with sub-equivariance yielded a synergistic effect’ … which is not supported by the experiments and could simply be the policy network producing better control under LRF transformation that are independent of morphology

Q1: Can you provide a concrete analysis of the contribution of subequivariant policy in morphology evolution?

Q2: Can you provide morphology metrics figures over time? Such as the length of limbs, and compute the symmetric error of design by comparing the difference of limb length divided by total limb length?

Thank you for the thoughtful feedback. We would like to reiterate that our approach focuses on embedding geometric symmetry in the policy module to exploit the inherent symmetry of real-world environments and dynamic systems, thus enhancing the co-optimization of morphology and control. Symmetry in morphology is an outcome of the optimization process rather than an explicit requirement.

• As reward is the primary metric for evaluating sample efficiency, which aligns with our goals, the plotted reward curves in our results clearly demonstrate that the contribution of the subequivariant policy in morphology evolution lies in improving sample efficiency.
• To provide additional insights, we have included the evolving processes of Ant in Fig. 10 and compared the morphology changes over time using different methods. These observations are intended to demonstrate that the development of morphology is fundamentally shaped by environmental interactions (states) and task demands (reward), rather than being a predefined goal.

We appreciate the reviewer’s suggestion to further analyze the characteristics of the evolved morphologies. We will consider incorporating such analyses to provide readers with additional insights into the results.

Dear Reviewer Zyvk,

Thank you for your detailed review, which has been crucial in refining our paper. We've made specific revisions to address your concerns, particularly around presentation clarity. With the discussion deadline approaching, your final insights would be invaluable to perfecting our work and would be greatly appreciated.

Best regards,

Authors

Dear Reviewer Zyvk,

Thank you for taking the time to provide feedback and for raising your score for presentation—we deeply appreciate your recognition of our efforts to improve the clarity of the paper. Your comments and encouragement are invaluable, which have helped us improve the paper!

Morphological Evolution Analysis.

We acknowledge your suggestion regarding analyzing the features of evolved morphologies over time. To address this, we have incorporated additional visualizations and discussions in the supplementary material to provide more insight into the co-evolution process:

• Step-by-Step Visualization: As shown in Figure 10 (main text), we present a step-by-step visualization of the morphology transformation within a single episode. This illustrates the trajectory of morphology transformations during one complete episode of training.
• Training (Evolution) Progress Analysis (Supplementary Section D): In Figure 11 (Section D: The Role of Geometric Symmetry in Morphology Evolution), we have included a visualization of morphological evolution across the training process. This figure compares two settings:
  • Policy Without Geometric Symmetry (left): Morphological evolution appears highly asymmetric and incomplete. Structural transformations often get trapped in local optima early in the process, as reflected in the lower reward curves. This is likely due to the reinforcement learning framework, where abundant exploration is required to optimize both morphology and behavior. In such cases, the interplay between morphology and behavior becomes more prone to local minima, complicating optimization within the vast search space.
  • Policy With Geometric Symmetry (right): The integration of geometric symmetry fosters more intricate and comprehensive morphology evolution, as evidenced by higher reward curves. This improvement supports the claim that leveraging geometric symmetry reduces search space redundancy in a lossless manner, leading to better optimization and higher co-evolution efficiency. Additionally, the training progress demonstrates that morphological symmetry is not explicitly enforced but naturally emerges and varies throughout the training process, evolving in response to interactions with the environment and task demands.

These visualizations aim to highlight how injecting geometric symmetry not only improves reward outcomes but also fosters a more efficient and complete co-evolution of morphology and behavior by better reflecting the symmetry inherent in the environment and task demands, all while addressing the challenging optimization dynamics of reinforcement learning.

Please review the updated anonymous paper, where updates are highlighted in red, including the caption for Figure 10 and Section D: "The Role of Geometric Symmetry in Morphology Evolution" (featuring Figure 11). We hope these updates address your concerns about how the policy module influences evolution and offer a more comprehensive view of the evolving morphologies, both within a single episode and throughout different training epochs.

Thank you again for your support! We look forward to any further comments you might have before the discussion period concludes.

Best regards,

Authors

We also appreciate the reviewers for their thoughtful comments and concerns. Below we summarize three core aspects:

1. Main focus and contributions.

   In our work, we address the co-evolution of morphology and behavior by incorporating the geometric symmetries inherent in real-world dynamics. The evolution of morphology is fundamentally shaped by environmental interactions and task demands. Previous methods have often either overlooked the role of geometric symmetries of the environment, resulting in reduced effectiveness, or imposed rigid constraints on morphology, such as bilateral symmetry, which restricts the natural development of morphology through interaction with the environment. Our framework introduces a novel problem setup that embeds geometric symmetries, facilitating the autonomous co-evolution of morphology and behavior. This aligns with the principles of embodied intelligence, where agents learn and evolve through interactions with their environment, leading to more natural and effective adaptations.

   Furthermore, geometric symmetries present in the environment and its dynamics profoundly influence both morphology and behavior. Rather than focusing solely on behavior, we leverage subequivariance in a way that impacts both behavior and morphology. The morphology network’s learning signal is derived from rewards generated through behavior-environment interactions. Through this interaction-driven mechanism, subequivariance indirectly influences morphology evolution, contributing to an integrated framework for co-evolution.

2. What's the symmetry in our paper?

   Our primary objective is not to enforce symmetry of the morphology but rather to inject geometric symmetry constraints within the policy module, enabling a more efficient co-optimization of morphology and control. Specifically, in our co-optimization framework, the quality of a morphology is captured by the morphology value, defined as the expected cumulative reward for a given morphology, initial state distribution, and policy $\pi$. For a fixed initial state distribution, the morphology value is independent of environment states such as position or velocity, but its updates still rely on feedback from agent-environment interactions. This feedback has a key property: for environment states and actions that are equivariant, such as rotated or translated configurations, the feedback should be naturally invariant. By integrating subequivariant neural networks into our co-evolution framework, we ensure consistent feedback for morphology value updates, eliminating the need for extensive sampling to approximate invariance.

   Notably, any symmetric structure observed in the resulting morphology is the outcome of optimization under the given task setting rather than an explicit requirement of our method; if a symmetric morphology arises, it is because such a structure achieves better performance within that specific task.

3. How would morphology evolution be affected?

   The obtained morphology is the outcome of co-optimization, shaped jointly by environmental interactions and task demands. Figures 1 and 2 show the evolved results for the navigation task. In Ant Navigation task setting, radially symmetric morphologies are more effective for omnidirectional movement. With the use of equivariant networks, our method successfully evolved a radially symmetric morphology, while the baseline without equivariant networks evolved an asymmetric morphology due to lower optimization efficiency.

   To provide additional illustration, in Figure 10(b) and (c), we compare the evolution processes under different task (reward) settings. For the task with a forward reward, the evolved morphology exhibits a laterally symmetric structure with stronger front legs and weaker hind legs, while the task without a forward reward leads to a radially symmetric morphology. This shows that the evolution of morphology is fundamentally shaped by environmental interactions and task demands, rather than being a predefined goal.

   The comparison between Figures 10(a) and (b) shows that methods without integrating geometric equivariance, due to lower sample efficiency, fail to fully evolve a radially symmetric morphology that meets task requirements. This highlights the necessity of considering geometric equivariance in the co-evolution of morphology and behavior frameworks.

In response to the reviewer’s valuable suggestion regarding the analysis of evolved morphologies over time, we have incorporated new visualizations and discussions to provide deeper insights into the co-evolution process. Specifically:

• Step-by-Step Visualization: As previously included in the main text (Figure 10), we provide a step-by-step visualization of morphology transformation within a single episode. This visualization already illustrates the trajectory of morphology transformations throughout a complete episode, offering a detailed view of the evolution process. We would like to highlight this existing content to emphasize its role in demonstrating the morphology changes within an episode.
• Training Progress Analysis: In Section D (Supplementary Material), we have introduced Figure 11, which visualizes the morphological evolution across different stages of the training process. This figure compares two settings—one without the integration of geometric symmetry and one with it—highlighting how geometric symmetry fosters more efficient and complete co-evolution. We show how symmetry naturally emerges and evolves over time in response to task demands and environmental interactions, further improving the co-evolution of morphology and behavior.

These updates, particularly Figure 11, emphasize the impact of geometric symmetry in facilitating better optimization and more robust morphology evolution, while reflecting the challenges posed by reinforcement learning dynamics.

We hope these new additions help clarify how the leverage of geometric symmetry via neural networks influences morphology evolution over time and provide a more comprehensive understanding of the co-evolution process. The updated anonymous paper, including these changes, has been re-uploaded with updates highlighted in red.

We sincerely thank all reviewers for their insightful comments, which have significantly contributed to the enhancement of this paper.