# Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery

arXiv: https://arxiv.org/abs/2412.07544 · HTML: https://mlsanity.com/paper/2412.07544/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Imitation learning methods, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 32 of its 51 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Placed in

- Offline RL and imitation learning from data / [Imitation learning methods](https://mlsanity.com/offlinerl/imitation-learning-methods/)

## Builds on (most cited first, 30 of 32)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [Behavioral Cloning from Observation](https://mlsanity.com/paper/1805.01954/) (2018)
- [Learning Robust Rewards with Adversarial Inverse Reinforcement Learning](https://mlsanity.com/paper/1710.11248/) (2017)
- [Orbit: A Unified Simulation Framework for Interactive Robot Learning Environments](https://mlsanity.com/paper/2301.04195/) (2023)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [DART: Noise Injection for Robust Imitation Learning](https://mlsanity.com/paper/1703.09327/) (2017)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Learning to Generalize Across Long-Horizon Tasks from Human Demonstrations](https://mlsanity.com/paper/2003.06085/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Variational Discriminator Bottleneck: Improving Imitation Learning, Inverse RL, and GANs by Constraining Information Flow](https://mlsanity.com/paper/1810.00821/) (2018)
- [EnsembleDAgger: A Bayesian Approach to Safe Imitation Learning](https://mlsanity.com/paper/1807.08364/) (2018)
- [Neural Dynamic Policies for End-to-End Sensorimotor Learning](https://mlsanity.com/paper/2012.02788/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Learning Contracting Vector Fields For Stable Imitation Learning](https://mlsanity.com/paper/1804.04878/) (2018)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations](https://mlsanity.com/paper/2304.02976/) (2023)
- [Learning Deep Dynamical Systems using Stable Neural ODEs](https://mlsanity.com/paper/2404.10622/) (2024)
- [Learning to optimize with convergence guarantees using nonlinear system theory](https://mlsanity.com/paper/2403.09389/) (2024)
- [A PAC-Bayesian Framework for Optimal Control with Stability Guarantees](https://mlsanity.com/paper/2403.17790/) (2024)
- [Learning Lyapunov-Stable Polynomial Dynamical Systems through Imitation](https://mlsanity.com/paper/2310.20605/) (2023)

## Built on by (most cited first, 5 of 5)

- [R2DN: Scalable Parameterization of Contracting and Lipschitz Recurrent Deep Networks](https://mlsanity.com/paper/2504.01250/) (2025)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Contractive Diffusion Policies: Robust Action Diffusion via Contractive Score-Based Sampling with Differential Equations](https://mlsanity.com/paper/2601.01003/) (2026)
- [Denoising-based Contractive Imitation Learning](https://mlsanity.com/paper/2503.15918/) (2025)
- [IL-ACT: Imitation Learning with Adaptive Cartesian Tracking Control for a 30-ton Excavator](https://mlsanity.com/paper/2609.16696/) (2026)

## Abstract (arXiv)

Imitation learning is a data-driven approach to learning policies from expert behavior, but it is prone to unreliable outcomes in out-of-sample (OOS) regions. While previous research relying on stable dynamical systems guarantees convergence to a desired state, it often overlooks transient behavior. We propose a framework for learning policies modeled by contractive dynamical systems, ensuring that all policy rollouts converge regardless of perturbations, and in turn, enable efficient OOS recovery. By leveraging recurrent equilibrium networks and coupling layers, the policy structure guarantees contractivity for any parameter choice, which facilitates unconstrained optimization. We also provide theoretical upper bounds for worst-case and expected loss to rigorously establish the reliability of our method in deployment. Empirically, we demonstrate substantial OOS performance improvements for simulated robotic manipulation and navigation tasks.