INTEGRATING SYMMETRY INTO DIFFERENTIABLE PLANNING WITH STEERABLE CONVOLUTIONS

Abstract

In this paper, we study a principled approach on incorporating group symmetry into end-to-end differentiable planning algorithms and explore the benefits of symmetry in planning. To achieve this, we draw inspiration from equivariant convolution networks and model the path planning problem as a set of signals over grids. We demonstrate that value iteration can be treated as a linear equivariant operator, which is effectively a steerable convolution. Building upon Value Iteration Networks (VIN), we propose a new Symmetric Planning (SymPlan) framework that incorporates rotation and reflection symmetry using steerable convolution networks. We evaluate our approach on four tasks: 2D navigation, visual navigation, 2 degrees of freedom (2-DOF) configuration space manipulation, and 2-DOF workspace manipulation. Our experimental results show that our symmetric planning algorithms significantly improve training efficiency and generalization performance compared to non-equivariant baselines, including VINs and GPPN.

1. INTRODUCTION

Model-based planning algorithms can struggle to find solutions for complex problems, and one solution is to apply planning in a more structured and reduced space (Sutton and Barto, 2018; Li et al., 2006; Ravindran and Barto, 2004; Fox and Long, 2002) . When a task exhibits symmetry, this structure can be used to effectively reduce the search space for planning. However, existing planning algorithms often assume perfect knowledge of dynamics and require building equivalence classes, which can be inefficient and limit their applicability to specific tasks (Fox and Long, 1999; 2002; Pochter et al., 2011; Zinkevich and Balch, 2001; Narayanamurthy and Ravindran, 2008) . In this paper, we study the path-planning problem and its symmetry structure, as shown in Figure 1 . Given a map M (top row), the objective is to find optimal actions A = SymPlan(M ) (bottom row) to a given position (red dots). If we rotated the map g.M (top right), its solution g.A (shortest path) can also be connected by a rotation with the original solution A. Specifically, we say the task has symmetry since the solutions SymPlan(g.M ) = g.SymPlan(M ) are related by a ⟲ 90 • rotation. As a more concrete example, the action in the NW corner of A is the same as the action in the SW corner of g.A, after also rotating the arrow ⟲ 90 • . This is an example of symmetry appearing in a specific task, which can be observed before solving the task or assuming other domain knowledge. Recently, symmetry in model-free deep reinforcement learning (RL) has also been studied (Mondal et al., 2020; van der Pol et al., 2020a; Wang et al., 2021) . A core benefit of model-free RL



Figure 1: Symmetry in path planning.Symmetric Planning approach guarantees the solutions are same up to rotations.

If we can use the rotation (and reflection) symmetry in this task, we effectively reduce the search space by |C 4 | = 4 (or |D 4 | = 8) times. Instead, classic planning algorithms like A* would require searching symmetric states (NP-hard) with known dynamics (Pochter et al., 2011).

