# Geometric Entropy: When Trajectory Diversity Helps and Hurts in Imitation Learning

arXiv: https://arxiv.org/abs/2606.20871 · HTML: https://mlsanity.com/paper/2606.20871/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Robot manipulation imitation learning, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 35 of its 40 references that are in the map.
- First posted on arXiv 18 Jun 2026.

## Placed in

- Offline RL and imitation learning from data / [Robot manipulation imitation learning](https://mlsanity.com/offlinerl/robot-manipulation-imitation-learning/)

## Builds on (most cited first, 30 of 35)

- [RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control](https://mlsanity.com/paper/2307.15818/) (2023)
- [OpenVLA: An Open-Source Vision-Language-Action Model](https://mlsanity.com/paper/2406.09246/) (2024)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [PaLM-E: An Embodied Multimodal Language Model](https://mlsanity.com/paper/2303.03378/) (2023)
- [RT-1: Robotics Transformer for Real-World Control at Scale](https://mlsanity.com/paper/2212.06817/) (2022)
- [π0.5: a Vision-Language-Action Model with Open-World Generalization](https://mlsanity.com/paper/2504.16054/) (2025)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [Octo: An Open-Source Generalist Robot Policy](https://mlsanity.com/paper/2405.12213/) (2024)
- [Open X-Embodiment: Robotic Learning Datasets and RT-X Models](https://mlsanity.com/paper/2310.08864/) (2023)
- [DROID: A Large-Scale In-The-Wild Robot Manipulation Dataset](https://mlsanity.com/paper/2403.12945/) (2024)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [BC-Z: Zero-Shot Task Generalization with Robotic Imitation Learning](https://mlsanity.com/paper/2202.02005/) (2022)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [VIMA: General Robot Manipulation with Multimodal Prompts](https://mlsanity.com/paper/2210.03094/) (2022)
- [MimicGen: A Data Generation System for Scalable Robot Learning using Human Demonstrations](https://mlsanity.com/paper/2310.17596/) (2023)
- [Behavior Transformers: Cloning k modes with one stone](https://mlsanity.com/paper/2206.11251/) (2022)
- [ManiSkill3: GPU Parallelized Robotics Simulation and Rendering for Generalizable Embodied AI](https://mlsanity.com/paper/2410.00425/) (2024)
- [DART: Noise Injection for Robust Imitation Learning](https://mlsanity.com/paper/1703.09327/) (2017)
- [Behavior Generation with Latent Actions](https://mlsanity.com/paper/2403.03181/) (2024)
- [RoboCat: A Self-Improving Generalist Agent for Robotic Manipulation](https://mlsanity.com/paper/2306.11706/) (2023)
- [Discriminator-Weighted Offline Imitation Learning from Suboptimal Demonstrations](https://mlsanity.com/paper/2207.10050/) (2022)
- [BAKU: An Efficient Transformer for Multi-Task Policy Learning](https://mlsanity.com/paper/2406.07539/) (2024)
- [Data Quality in Imitation Learning](https://mlsanity.com/paper/2306.02437/) (2023)
- [DemoGen: Synthetic Demonstration Generation for Data-Efficient Visuomotor Policy Learning](https://mlsanity.com/paper/2502.16932/) (2025)
- [Multi-Modal Imitation Learning from Unstructured Demonstrations using Generative Adversarial Nets](https://mlsanity.com/paper/1705.10479/) (2017)
- [Towards Diverse Behaviors: A Benchmark for Imitation Learning with Human Demonstrations](https://mlsanity.com/paper/2402.14606/) (2024)
- [AW-Opt: Learning Robotic Skills with Imitation and Reinforcement at Scale](https://mlsanity.com/paper/2111.05424/) (2021)
- [CUPID: Curating Data your Robot Loves with Influence Functions](https://mlsanity.com/paper/2506.19121/) (2025)
- [Is Diversity All You Need for Scalable Robotic Manipulation?](https://mlsanity.com/paper/2507.06219/) (2025)

## Abstract (arXiv)

We study how trajectory-shape diversity in demonstrations affects imitation learning (IL) performance across models, tasks, and data scales. We introduce Geometric Entropy (H_G), a task-agnostic metric that quantifies the intrinsic diversity of transit trajectories after normalizing away extrinsic variation, such as goal pose and workspace scale, via target-frame alignment. Across multiple IL architectures and both simulated and real-robot contact-rich manipulation tasks, we observe a consistent inverted-U relationship between success and H_G: increasing geometric diversity improves robustness in low-diversity regimes but degrades performance once diversity induces strategy ambiguity. Moreover, the optimal entropy shifts toward lower values as task mastery increases through more data, easier tasks, or stronger priors, and for a pretrained vision-language-action model the trend becomes effectively monotonic decreasing. Practically, H_G enables fast pre-training auditing of demonstration datasets and offers a simple guideline for calibrating demonstrations toward the learnable regime.