# Integrating Symmetry into Differentiable Planning with Steerable Convolutions

arXiv: https://arxiv.org/abs/2206.03674 · HTML: https://mlsanity.com/paper/2206.03674/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 85 references that are in the map.
- First posted on arXiv 8 Jun 2022.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Value Iteration Networks](https://mlsanity.com/paper/1602.02867/) (2016)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Imagination-Augmented Agents for Deep Reinforcement Learning](https://mlsanity.com/paper/1707.06203/) (2017)
- [Value Prediction Network](https://mlsanity.com/paper/1707.03497/) (2017)
- [The Value Equivalence Principle for Model-Based Reinforcement Learning](https://mlsanity.com/paper/2011.03506/) (2020)
- [Exploring Model-based Planning with Policy Networks](https://mlsanity.com/paper/1906.08649/) (2019)
- [Temporal Difference Models: Model-Free Deep RL for Model-Based Control](https://mlsanity.com/paper/1802.09081/) (2018)
- [An investigation of model-free planning](https://mlsanity.com/paper/1901.03559/) (2019)
- [Model-Augmented Actor-Critic: Backpropagating through Paths](https://mlsanity.com/paper/2005.08068/) (2020)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [QMDP-Net: Deep Learning for Planning under Partial Observability](https://mlsanity.com/paper/1703.06692/) (2017)
- [Proper Value Equivalence](https://mlsanity.com/paper/2106.10316/) (2021)
- [Gated Path Planning Networks](https://mlsanity.com/paper/1806.06408/) (2018)
- [Universal Planning Networks](https://mlsanity.com/paper/1804.00645/) (2018)
- [The Differentiable Cross-Entropy Method](https://mlsanity.com/paper/1909.12830/) (2019)
- [Differentiable Spatial Planning using Transformers](https://mlsanity.com/paper/2112.01010/) (2021)

## Built on by (most cited first, 6 of 6)

- [A General Theory of Correct, Incorrect, and Extrinsic Equivariance](https://mlsanity.com/paper/2303.04745/) (2023)
- [Scaling up and Stabilizing Differentiable Planning with Implicit Differentiation](https://mlsanity.com/paper/2210.13542/) (2022)
- [E(2)-Equivariant Graph Planning for Navigation](https://mlsanity.com/paper/2309.13043/) (2023)
- [Equivariant Action Sampling for Reinforcement Learning and Planning](https://mlsanity.com/paper/2412.12237/) (2024)
- [Learning to Navigate in Mazes with Novel Layouts using Abstract Top-down Maps](https://mlsanity.com/paper/2412.12024/) (2024)
- [Can Euclidean Symmetry be Leveraged in Reinforcement Learning and Planning?](https://mlsanity.com/paper/2307.08226/) (2023)

## Abstract (arXiv)

We study how group symmetry helps improve data efficiency and generalization for end-to-end differentiable planning algorithms when symmetry appears in decision-making tasks. Motivated by equivariant convolution networks, we treat the path planning problem as textitsignals over grids. We show that value iteration in this case is a linear equivariant operator, which is a (steerable) convolution. This extends Value Iteration Networks (VINs) on using convolutional networks for path planning with additional rotation and reflection symmetry. Our implementation is based on VINs and uses steerable convolution networks to incorporate symmetry. The experiments are performed on four tasks: 2D navigation, visual navigation, and 2 degrees of freedom (2DOFs) configuration space and workspace manipulation. Our symmetric planning algorithms improve training efficiency and generalization by large margins compared to non-equivariant counterparts, VIN and GPPN.