# Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data

arXiv: https://arxiv.org/abs/2308.04428 · HTML: https://mlsanity.com/paper/2308.04428/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 60 references that are in the map.
- First posted on arXiv 8 Aug 2023.

## Builds on (most cited first, 30 of 42)

- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [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)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Multi-task Deep Reinforcement Learning with PopArt](https://mlsanity.com/paper/1809.04474/) (2018)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Online Control with Adversarial Disturbances](https://mlsanity.com/paper/1902.08721/) (2019)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [The Gap Between Model-Based and Model-Free Methods on the Linear Quadratic Regulator: An Asymptotic Viewpoint](https://mlsanity.com/paper/1812.03565/) (2018)
- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [Logarithmic Regret for Online Control](https://mlsanity.com/paper/1909.05062/) (2019)
- [Least-Squares Temporal Difference Learning for the Linear Quadratic Regulator](https://mlsanity.com/paper/1712.08642/) (2017)
- [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)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Sample Efficient Linear Meta-Learning by Alternating Minimization](https://mlsanity.com/paper/2105.08306/) (2021)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Finite-time Analysis of Approximate Policy Iteration for the Linear Quadratic Regulator](https://mlsanity.com/paper/1905.12842/) (2019)
- [Meta-Adaptive Nonlinear Control: Theory and Algorithms](https://mlsanity.com/paper/2106.06098/) (2021)
- [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022)
- [Scalable Multi-Task Imitation Learning with Autonomous Improvement](https://mlsanity.com/paper/2003.02636/) (2020)
- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
- [FedSysID: A Federated Approach to Sample-Efficient System Identification](https://mlsanity.com/paper/2211.14393/) (2022)
- [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021)
- [Provable Multi-Task Representation Learning by Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/2307.06887/) (2023)

## Built on by (most cited first, 13 of 13)

- [Meta-Learning Linear Quadratic Regulators: A Policy Gradient MAML Approach for Model-free LQR](https://mlsanity.com/paper/2401.14534/) (2024)
- [Federated Representation Learning in the Under-Parameterized Regime](https://mlsanity.com/paper/2406.04596/) (2024)
- [Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification](https://mlsanity.com/paper/2401.00073/) (2023)
- [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)
- [Learning Stabilizing Policies via an Unstable Subspace Representation](https://mlsanity.com/paper/2505.01348/) (2025)
- [Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples](https://mlsanity.com/paper/2410.11227/) (2024)
- [Latent Linear Quadratic Regulator for Robotic Control Tasks](https://mlsanity.com/paper/2407.11107/) (2024)
- [Learning with Shared Representations: Statistical Rates and Efficient Algorithms](https://mlsanity.com/paper/2409.04919/) (2024)
- [Adversarially Robust Multitask Adaptive Control](https://mlsanity.com/paper/2511.05444/) (2025)
- [How Does Preconditioning Guide Feature Learning in Deep Neural Networks?](https://mlsanity.com/paper/2509.25637/) (2025)
- [On the Power of Source Screening for Learning Shared Feature Extractors](https://mlsanity.com/paper/2602.16125/) (2026)
- [Feed m Birds with One Scone: Accelerating Multi-task Gradient Balancing via Bi-level Optimization](https://mlsanity.com/paper/2603.07389/) (2026)

## Abstract (arXiv)

A powerful concept behind much of the recent progress in machine learning is the extraction of common features across data from heterogeneous sources or tasks. Intuitively, using all of one's data to learn a common representation function benefits both computational effort and statistical generalization by leaving a smaller number of parameters to fine-tune on a given task. Toward theoretically grounding these merits, we propose a general setting of recovering linear operators M from noisy vector measurements y = Mx + w, where the covariates x may be both non-i.i.d. and non-isotropic. We demonstrate that existing isotropy-agnostic representation learning approaches incur biases on the representation update, which causes the scaling of the noise terms to lose favorable dependence on the number of source tasks. This in turn can cause the sample complexity of representation learning to be bottlenecked by the single-task data size. We introduce an adaptation, textttDe-bias & Feature-Whiten (textttDFW), of the popular alternating minimization-descent scheme proposed independently in Collins et al., (2021) and Nayer and Vaswani (2022), and establish linear convergence to the optimal representation with noise level scaling down with the textittotal source data size. This leads to generalization bounds on the same order as an oracle empirical risk minimizer. We verify the vital importance of textttDFW on various numerical simulations. In particular, we show that vanilla alternating-minimization descent fails catastrophically even for iid, but mildly non-isotropic data. Our analysis unifies and generalizes prior work, and provides a flexible framework for a wider range of applications, such as in controls and dynamical systems.