equivariant graph policy learning

This work addresses the challenge of simultaneously achieving symmetry breaking and high learning efficiency in graph neural network (GNN)-based learned planners. Methodologically, it introduces a symmetry-aware graph representation learning framework that unifies symmetry detection with GNN-guided planning learning for the first time. A differentiable graph structure encoder is designed to explicitly identify structural redundancies in the search space, while action pruning and state pruning mechanisms are jointly integrated to actively eliminate equivalent symmetric paths during planning. The approach builds upon graph-based modeling of planning problems and remains compatible with the Fast Downward planner. Evaluated on the latest International Planning Competition (IPC) learning track benchmarks, our method surpasses LAMA—the prior state-of-the-art—for the first time, achieving a +12.3% improvement in solution success rate and a 37.6% reduction in average expanded nodes. It establishes a scalable and interpretable paradigm for symmetry handling in learned planning.

In real-world multi-agent reinforcement learning (MARL), environmental asymmetries—arising from external forces, measurement errors, and systemic biases—severely degrade the sample efficiency and generalization of equivariant graph neural networks (EGNNs). To address this, we propose Partially Equivariant Graph Neural Networks (PEGNNs), the first framework to formally characterize partial equivariance across four dimensions: subgroups, features, spatial regions, and approximations—establishing a continuous spectrum from full equivariance to complete non-equivariance. PEGNNs unify EGNNs and standard GNNs within a single architecture via group-representation-driven differentiable subgroup masking, hierarchical feature disentanglement, and hybrid equivariant message passing. Evaluated on multiple MARL benchmarks exhibiting realistic asymmetries, PEGNNs achieve up to 42% higher sample efficiency than EGNNs and up to 67% higher than standard GNNs, while significantly improving generalization and robustness.

Diffusion-based policies suffer from poor generalization and low sample efficiency in robotic control. To address this, we propose a lightweight symmetry-incorporation method that avoids the complexity of fully equivariant networks. Our approach leverages two key insights: (i) theoretical proof that “eye-in-hand” perception combined with relative action parameterization is inherently SE(3)-invariant; and (ii) a synergistic fusion mechanism integrating frame averaging with an equivariant visual encoder, coupled with relative trajectory representation and pretrained feature extraction. The resulting architecture remains computationally lightweight and conceptually simple, yet matches or surpasses fully equivariant baselines in performance. Empirically, it achieves significant improvements in both generalization—across diverse robot poses and unseen environments—and sample efficiency—requiring fewer environment interactions to reach comparable policy performance.

Equivariant networks strictly preserve input symmetries, rendering them ill-suited for generative tasks requiring *active symmetry breaking*—e.g., reconstructing asymmetric structures from highly symmetric latent representations. To address this, we establish the first necessary and sufficient representation theorem for equivariant conditional distributions and propose SymPE: a method that achieves *controllable symmetry breaking* via learnable stochastic normalized positional encodings, while preserving the group-equivariant inductive bias. SymPE unifies probabilistic symmetry breaking, positional encoding, and equivariant graph neural networks, and naturally integrates with diffusion-based generative frameworks. Empirically, it significantly improves performance on graph diffusion modeling, graph autoencoding, and lattice spin system generation. Theoretically, we prove that SymPE’s generalization bound is strictly superior to that of conventional equivariant networks.

Reinforcement learning (RL) often encounters approximate symmetries—structural invariances that hold only approximately—not exact symmetries—posing challenges for existing equivariant RL methods designed for strict symmetry. Method: This paper introduces the Approximate Equivariant Markov Decision Process (AE-MDP) framework, the first formal integration of approximate equivariance into RL theory and algorithm design. We propose relaxable group convolution and direction-aware convolutional networks to adaptively model symmetry deviations. Theoretically, we characterize how approximate equivariance affects the optimal Q-function and prove that our method achieves optimal performance under both exact and approximate symmetry. Results: Experiments on continuous control (SAC/PPO-based) and real-world stock trading demonstrate that our approach matches state-of-the-art equivariant methods under exact symmetry, significantly outperforms them under approximate symmetry, and improves robustness to test-time observation noise.

Existing equivariant networks typically support only fixed symmetry groups, limiting their ability to flexibly handle multimodal data with diverse symmetries. This work proposes the ASEN model, which achieves simultaneous equivariance to multiple permutation subgroups within a single architecture by incorporating symmetry-breaking auxiliary input features and leveraging an approximate symmetry-breaking mechanism together with an efficient 2-closure fast algorithm. Built upon a fully permutation-equivariant basis model and employing equivariant MLP emulation techniques, ASEN overcomes the rigidity of conventional equivariant networks. Experiments demonstrate that ASEN outperforms both specialized equivariant models and non-equivariant baselines across tasks involving graph and image symmetry selection, as well as sequence-based multitask and transfer learning scenarios.

本文提出了一种基于群胚的强化学习框架，通过动态发现局部对称性来提高样本效率和收敛速度，解决了传统方法在处理局部、上下文依赖规律时的局限。

This work addresses the combinatorial explosion inherent in searching for minimally rigid graphs with a maximal number of realizations within rigidity theory, a challenge that renders traditional exhaustive methods infeasible at scale. The authors propose a novel progressive construction framework that integrates deep reinforcement learning with graph isomorphism networks, leveraging Henneberg operations to iteratively build candidate graphs. Their policy network combines a graph isomorphism encoder with a permutation-equivariant action head and is optimized via the deep cross-entropy method to maximize an invariant proxy for the number of realizations. The approach successfully reproduces known optimal results in the planar case and establishes new records on the sphere by discovering previously unknown minimally rigid graphs with exceptionally high realization counts, thereby substantially overcoming the scalability limitations of conventional search strategies.

研究通过揭示深度神经网络中的对称性（纤维化和覆盖），利用这些特性进行模型压缩，并通过打破对称性提高持续学习性能，从而解决AI模型的不透明性和效率问题。

This work addresses the insufficient modeling of complex symmetric structures in deep learning—such as non-invertible symmetries and higher-order relations beyond graphs—by introducing Order-Equivariant Neural Networks (OENN) and Category-Equivariant Neural Networks (CENN). Grounded in equivariant bundles, face posets, and category theory, this study provides the first complete characterization of all linear order-equivariant maps and establishes a Universal Approximation Theorem (UAT), thereby filling a critical theoretical gap: the absence of a UAT for layered neural architectures. Furthermore, it generalizes the UAT for graph neural networks to a broader equivariant framework. The proposed architectures unify message-passing mechanisms across graph and layered models, demonstrating empirical effectiveness and offering a cohesive theoretical foundation with guaranteed approximation capabilities for equivariant deep learning.