# Multi-Task Imitation Learning for Linear Dynamical Systems

arXiv: https://arxiv.org/abs/2212.00186 · HTML: https://mlsanity.com/paper/2212.00186/

## Key facts

- Cited by 23 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 33 of its 52 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## 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 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [RL2: Fast Reinforcement Learning via Slow Reinforcement Learning](https://mlsanity.com/paper/1611.02779/) (2016)
- [End-to-end Driving via Conditional Imitation Learning](https://mlsanity.com/paper/1710.02410/) (2017)
- [Efficient Off-Policy Meta-Reinforcement Learning via Probabilistic Context Variables](https://mlsanity.com/paper/1903.08254/) (2019)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Distral: Robust Multitask Reinforcement Learning](https://mlsanity.com/paper/1707.04175/) (2017)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Multi-task Deep Reinforcement Learning with PopArt](https://mlsanity.com/paper/1809.04474/) (2018)
- [Representation Matters: Offline Pretraining for Sequential Decision Making](https://mlsanity.com/paper/2102.05815/) (2021)
- [Conservative Data Sharing for Multi-Task Offline Reinforcement Learning](https://mlsanity.com/paper/2109.08128/) (2021)
- [Provable Representation Learning for Imitation Learning via Bi-level Optimization](https://mlsanity.com/paper/2002.10544/) (2020)
- [How Fine-Tuning Allows for Effective Meta-Learning](https://mlsanity.com/paper/2105.02221/) (2021)
- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [Learning with Square Loss: Localization through Offset Rademacher Complexity](https://mlsanity.com/paper/1502.06134/) (2015)
- [Meta-Learning Dynamics Forecasting Using Task Inference](https://mlsanity.com/paper/2102.10271/) (2021)
- [On the Power of Multitask Representation Learning in Linear MDP](https://mlsanity.com/paper/2106.08053/) (2021)
- [Provable Benefit of Multitask Representation Learning in Reinforcement Learning](https://mlsanity.com/paper/2206.05900/) (2022)
- [Knowledge Transfer in Multi-Task Deep Reinforcement Learning for Continuous Control](https://mlsanity.com/paper/2010.07494/) (2020)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Learning Internal Representations (COLT 1995)](https://mlsanity.com/paper/1911.05781/) (2019)
- [Meta-Adaptive Nonlinear Control: Theory and Algorithms](https://mlsanity.com/paper/2106.06098/) (2021)
- [Scalable Multi-Task Imitation Learning with Autonomous Improvement](https://mlsanity.com/paper/2003.02636/) (2020)
- [TaSIL: Taylor Series Imitation Learning](https://mlsanity.com/paper/2205.14812/) (2022)
- [Control-oriented meta-learning](https://mlsanity.com/paper/2204.06716/) (2022)
- [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021)

## Built on by (most cited first, 23 of 23)

- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Inverse Dynamics Pretraining Learns Good Representations for Multitask Imitation](https://mlsanity.com/paper/2305.16985/) (2023)
- [Meta-Learning Linear Quadratic Regulators: A Policy Gradient MAML Approach for Model-free LQR](https://mlsanity.com/paper/2401.14534/) (2024)
- [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023)
- [Learning Personalized Models with Clustered System Identification](https://mlsanity.com/paper/2304.01395/) (2023)
- [Robot Fleet Learning via Policy Merging](https://mlsanity.com/paper/2310.01362/) (2023)
- [Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification](https://mlsanity.com/paper/2401.00073/) (2023)
- [Computational Teaching for Driving via Multi-Task Imitation Learning](https://mlsanity.com/paper/2410.01608/) (2024)
- [Coreset-Based Task Selection for Sample-Efficient Meta-Reinforcement Learning](https://mlsanity.com/paper/2502.02332/) (2025)
- [Asynchronous Heterogeneous Linear Quadratic Regulator Design](https://mlsanity.com/paper/2404.09061/) (2024)
- [Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control](https://mlsanity.com/paper/2407.05781/) (2024)
- [Combining Federated Learning and Control: A Survey](https://mlsanity.com/paper/2407.11069/) (2024)
- [DIDA: Denoised Imitation Learning based on Domain Adaptation](https://mlsanity.com/paper/2404.03382/) (2024)
- [Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples](https://mlsanity.com/paper/2410.11227/) (2024)
- [Redefining Clustered Federated Learning for System Identification: The Path of ClusterCraft](https://mlsanity.com/paper/2505.16857/) (2025)
- [A Statistical Guarantee for Representation Transfer in Multitask Imitation Learning](https://mlsanity.com/paper/2311.01589/) (2023)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Finite Sample Analysis of Tensor Decomposition for Learning Mixtures of Linear Systems](https://mlsanity.com/paper/2412.10615/) (2024)
- [Harnessing Data from Clustered LQR Systems: Personalized and Collaborative Policy Optimization](https://mlsanity.com/paper/2511.17489/) (2025)
- [Shared-unique Features and Task-aware Prioritized Sampling on Multi-task Reinforcement Learning](https://mlsanity.com/paper/2406.00761/) (2024)
- [Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem](https://mlsanity.com/paper/2411.15972/) (2024)
- [A State Alignment-Centric Approach to Federated System Identification: The FedAlign Framework](https://mlsanity.com/paper/2503.12137/) (2025)
- [How to Mitigate the Distribution Shift Problem in Robotics Control: A Robust and Adaptive Approach Based on Offline to Online Imitation Learning](https://mlsanity.com/paper/2605.25414/) (2026)

## Abstract (arXiv)

We study representation learning for efficient imitation learning over linear systems. In particular, we consider a setting where learning is split into two phases: (a) a pre-training step where a shared k-dimensional representation is learned from H source policies, and (b) a target policy fine-tuning step where the learned representation is used to parameterize the policy class. We find that the imitation gap over trajectories generated by the learned target policy is bounded by tildeOleft( frack nxHNshared + frack nuNtargetright), where nx > k is the state dimension, nu is the input dimension, Nshared denotes the total amount of data collected for each policy during representation learning, and Ntarget is the amount of target task data. This result formalizes the intuition that aggregating data across related tasks to learn a representation can significantly improve the sample efficiency of learning a target task. The trends suggested by this bound are corroborated in simulation.