# Equivariant Q Learning in Spatial Action Spaces

arXiv: https://arxiv.org/abs/2110.15443 · HTML: https://mlsanity.com/paper/2110.15443/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2021.

## Built on by (most cited first, 30 of 62)

- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Equivariant Diffusion Policy](https://mlsanity.com/paper/2407.01812/) (2024)
- [Theia: Distilling Diverse Vision Foundation Models for Robot Learning](https://mlsanity.com/paper/2407.20179/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Equivariant Transporter Network](https://mlsanity.com/paper/2202.09400/) (2022)
- [Edge Grasp Network: A Graph-Based SE(3)-invariant Approach to Grasp Detection](https://mlsanity.com/paper/2211.00191/) (2022)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3D](https://mlsanity.com/paper/2401.12046/) (2024)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [SEIL: Simulation-augmented Equivariant Imitation Learning](https://mlsanity.com/paper/2211.00194/) (2022)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Equi-Tuning: Group Equivariant Fine-Tuning of Pretrained Models](https://mlsanity.com/paper/2210.06475/) (2022)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [A General Theory of Correct, Incorrect, and Extrinsic Equivariance](https://mlsanity.com/paper/2303.04745/) (2023)
- [Leveraging Symmetries in Pick and Place](https://mlsanity.com/paper/2308.07948/) (2023)
- [Imagination Policy: Using Generative Point Cloud Models for Learning Manipulation Policies](https://mlsanity.com/paper/2406.11740/) (2024)
- [Equivariant Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2408.14336/) (2024)
- [Relational Reasoning via Set Transformers: Provable Efficiency and Applications to MARL](https://mlsanity.com/paper/2209.09845/) (2022)
- [On Robot Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2306.06489/) (2023)
- [Efficient Equivariant Transfer Learning from Pretrained Models](https://mlsanity.com/paper/2305.09900/) (2023)
- [Continual Vision-based Reinforcement Learning with Group Symmetries](https://mlsanity.com/paper/2210.12301/) (2022)
- [Learning from 10 Demos: Generalisable and Sample-Efficient Policy Learning with Oriented Affordance Frames](https://mlsanity.com/paper/2410.12124/) (2024)
- [SE(3)-Equivariant Robot Learning and Control: A Tutorial Survey](https://mlsanity.com/paper/2503.09829/) (2025)
- [SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space](https://mlsanity.com/paper/2507.01723/) (2025)
- [Learning Fabric Manipulation in the Real World with Human Videos](https://mlsanity.com/paper/2211.02832/) (2022)
- [SO(2)-Equivariant Downwash Models for Close Proximity Flight](https://mlsanity.com/paper/2305.18983/) (2023)
- [Understanding when Dynamics-Invariant Data Augmentations Benefit Model-Free Reinforcement Learning Updates](https://mlsanity.com/paper/2310.17786/) (2023)
- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)

## Abstract (arXiv)

Recently, a variety of new equivariant neural network model architectures have been proposed that generalize better over rotational and reflectional symmetries than standard models. These models are relevant to robotics because many robotics problems can be expressed in a rotationally symmetric way. This paper focuses on equivariance over a visual state space and a spatial action space -- the setting where the robot action space includes a subset of rmSE(2). In this situation, we know a priori that rotations and translations in the state image should result in the same rotations and translations in the spatial action dimensions of the optimal policy. Therefore, we can use equivariant model architectures to make Q learning more sample efficient. This paper identifies when the optimal Q function is equivariant and proposes Q network architectures for this setting. We show experimentally that this approach outperforms standard methods in a set of challenging manipulation problems.