# Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control

arXiv: https://arxiv.org/abs/2407.05781 · HTML: https://mlsanity.com/paper/2407.05781/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 40 references that are in the map.
- First posted on arXiv 8 Jul 2024.

## Builds on (most cited first, 28 of 28)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control](https://mlsanity.com/paper/2307.15818/) (2023)
- [RT-1: Robotics Transformer for Real-World Control at Scale](https://mlsanity.com/paper/2212.06817/) (2022)
- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [FedProto: Federated Prototype Learning across Heterogeneous Clients](https://mlsanity.com/paper/2105.00243/) (2021)
- [RVT: Robotic View Transformer for 3D Object Manipulation](https://mlsanity.com/paper/2306.14896/) (2023)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [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)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Improper Learning for Non-Stochastic Control](https://mlsanity.com/paper/2001.09254/) (2020)
- [Pre-Training for Robots: Offline RL Enables Learning New Tasks from a Handful of Trials](https://mlsanity.com/paper/2210.05178/) (2022)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [The Nonstochastic Control Problem](https://mlsanity.com/paper/1911.12178/) (2019)
- [Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently](https://mlsanity.com/paper/2002.08095/) (2020)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Regret Bounds for Adaptive Nonlinear Control](https://mlsanity.com/paper/2011.13101/) (2020)
- [Sample Efficient Linear Meta-Learning by Alternating Minimization](https://mlsanity.com/paper/2105.08306/) (2021)
- [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022)
- [Model-free Learning with Heterogeneous Dynamical Systems: A Federated LQR Approach](https://mlsanity.com/paper/2308.11743/) (2023)
- [Meta-Learning Linear Quadratic Regulators: A Policy Gradient MAML Approach for Model-free LQR](https://mlsanity.com/paper/2401.14534/) (2024)
- [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021)
- [Minimal Expected Regret in Linear Quadratic Control](https://mlsanity.com/paper/2109.14429/) (2021)
- [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023)
- [Imitation and Transfer Learning for LQG Control](https://mlsanity.com/paper/2303.09002/) (2023)
- [Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification](https://mlsanity.com/paper/2401.00073/) (2023)
- [Robust Online Control with Model Misspecification](https://mlsanity.com/paper/2107.07732/) (2021)

## Built on by (most cited first, 4 of 4)

- [Coreset-Based Task Selection for Sample-Efficient Meta-Reinforcement Learning](https://mlsanity.com/paper/2502.02332/) (2025)
- [Learning Stabilizing Policies via an Unstable Subspace Representation](https://mlsanity.com/paper/2505.01348/) (2025)
- [Adversarially Robust Multitask Adaptive Control](https://mlsanity.com/paper/2511.05444/) (2025)
- [Harnessing Data from Clustered LQR Systems: Personalized and Collaborative Policy Optimization](https://mlsanity.com/paper/2511.17489/) (2025)

## Abstract (arXiv)

Representation learning is a powerful tool that enables learning over large multitudes of agents or domains by enforcing that all agents operate on a shared set of learned features. However, many robotics or controls applications that would benefit from collaboration operate in settings with changing environments and goals, whereas most guarantees for representation learning are stated for static settings. Toward rigorously establishing the benefit of representation learning in dynamic settings, we analyze the regret of multi-task representation learning for linear-quadratic control. This setting introduces unique challenges. Firstly, we must account for and balance the textitmisspecification introduced by an approximate representation. Secondly, we cannot rely on the parameter update schemes of single-task online LQR, for which least-squares often suffices, and must devise a novel scheme to ensure sufficient improvement. We demonstrate that for settings where exploration is "benign", the regret of any agent after T timesteps scales as tilde O(√T/H), where H is the number of agents. In settings with "difficult" exploration, the regret scales as tilde O(√du dθ √T + T3/4/H1/5), where dx is the state-space dimension, du is the input dimension, and dθ is the task-specific parameter count. In both cases, by comparing to the minimax single-task regret O(√dx du2√T), we see a benefit of a large number of agents. Notably, in the difficult exploration case, by sharing a representation across tasks, the effective task-specific parameter count can often be small dθ < dx du. Lastly, we provide numerical validation of the trends we predict.