Reinforcement learning (RL) is a powerful framework for robotic control, yet its practical application is often hindered by high sample complexity. This is particularly restrictive in physical domains where interaction data is costly. While the world often exhibits geometric and physical symmetries, standard RL algorithms typically fail to exploit this structure. In this paper, we demonstrate that exploiting group symmetries significantly reduces the sample complexity of RL. Focusing on finite-horizon Markov decision processes, we find that leveraging homomorphisms induced by group symmetries significantly reduces the theoretical upper and lower bounds on the number of environment interactions required to reach an optimal return. We further extend these bounds to continuous state and action spaces, providing corresponding sample-complexity guarantees under appropriate regularity assumptions. Beyond theory, we validate our findings through controlled experiments and demonstrate the advantages of symmetry-aware policy learning on high-dimensional continuous robotic simulations. Our results show that integrating symmetry into the learning pipeline yields substantial gains in sample efficiency and performance, offering a principled path toward more data-efficient robotics.
Figures & tables
Figure 1: The homomorphism h maps equivalent states in M to representative states in the quotient MDP Mˉ , reducing the effective sample complexity from O~(kM) to O~(kMˉ) with kMˉ≪kM .
Figure 2: Example where homomorphism h maps the original MDP M to the quotient MDP Mˉ .
Figure 3: Theoretical comparison between unstructured and symmetry-aware sample complexity for a graph-based MDP with ∣S∣=105 states before quotienting by symmetry. For the ring graph, the symmetry group has full state-size symmetry, ∣G∣=105 , yielding a potential five-order-of-magnitude reduction in the number of episodes required to reach 95% optimality with high probability.
Figure 4: Performance across the 1D cyclic ring, 3D cyclic ring, and Rubik’s cube experiments. Symmetry-aware learning reduces the number of episodes needed for better performance.
Table 5
Figure 5: Continuous-control manipulation benchmarks comparing symmetry-aware and non-equivariant learning across multiple random seeds.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Illustrations of the 1D cyclic ring, 3D cyclic ring, and Rubik’s cube experiments.
Figure 7: Rubik’s cube canonicalization. A scrambled state s (left) is replaced by the lexicographically smallest representative of its orbit O(s)={g⋅s:g∈Oh} under the 24 -element rotation group Oh≅S4
Figure 8: Illustrations of the block stacking and nut assembly robotic manipulation benchmarks.