Multi-Agent Reinforcement Learning (MARL) struggles with sample inefficiency and poor generalization [1]. These challenges are partially due to a lack of structure or inductive bias in the neural networks typically used in learning the policy. One such form of structure that is commonly observed in multi-agent scenarios is symmetry. The field of Geometric Deep Learning has developed Equivariant Graph Neural Networks (EGNN) that are equivariant (or symmetric) to rotations, translations, and reflections of nodes. Incorporating equivariance has been shown to improve learning efficiency and decrease error [ 2 ]. In this paper, we demonstrate that EGNNs improve the sample efficiency and generalization in MARL.

The proof is given by [11]. Eq. (13)is clearlyO(3)-subequivariant, but theO(3)-subequivariant function is unnecessarily the form like Eq. (13). Then there must exit functionss( Z,h) and s ( Z,h), satisfying ˆf( Z,h) = [ Z, g]s( Z,h)+ Z s ( Z,h). Note thatf by Eq. (14) can also be considered as a function of both Z and g, and it is universal accordingtoProposition1. When f reducestoafunctionof Z byfixing g,thenbyTheorem1,itis 4 still universal with respect tothe subgroup that leaves g unchanged.

Reasoning system dynamics is one of the most important analytical approaches for many scientific studies.

However,theperformance ofexisting GNNs would decrease significantly when they stack many layers, because of the oversmoothing issue. Node embeddings tend to converge to similar vectors when GNNs keep recursively aggregating the representations ofneighbors.

Equivariant Graph Neural Networks (GNNs) that incorporate E(3) symmetry have achieved significant success in various scientific applications. As one of the most successful models, EGNN leverages a simple scalarization technique to perform equivariant message passing over only Cartesian vectors (i.e., 1st-degree steerable vectors), enjoying greater efficiency and efficacy compared to equivariant GNNs using higher-degree steerable vectors. This success suggests that higher-degree representations might be unnecessary. In this paper, we disprove this hypothesis by exploring the expressivity of equivariant GNNs on symmetric structures, including $k$-fold rotations and regular polyhedra. We theoretically demonstrate that equivariant GNNs will always degenerate to a zero function if the degree of the output representations is fixed to 1 or other specific values. Based on this theoretical insight, we propose HEGNN, a high-degree version of EGNN to increase the expressivity by incorporating high-degree steerable vectors while maintaining EGNN's efficiency through the scalarization trick. Our extensive experiments demonstrate that HEGNN not only aligns with our theoretical analyses on toy datasets consisting of symmetric structures, but also shows substantial improvements on more complicated datasets such as $N$-body and MD17. Our theoretical findings and empirical results potentially open up new possibilities for the research of equivariant GNNs.