# Linear Transformers are Versatile In-Context Learners

arXiv: https://arxiv.org/abs/2402.14180 · HTML: https://mlsanity.com/paper/2402.14180/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 42 references that are in the map.
- First posted on arXiv 21 Feb 2024.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [PaLM 2 Technical Report](https://mlsanity.com/paper/2305.10403/) (2023)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Larger language models do in-context learning differently](https://mlsanity.com/paper/2303.03846/) (2023)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Data Distributional Properties Drive Emergent In-Context Learning in Transformers](https://mlsanity.com/paper/2205.05055/) (2022)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Risks from Learned Optimization in Advanced Machine Learning Systems](https://mlsanity.com/paper/1906.01820/) (2019)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Large Language Models as General Pattern Machines](https://mlsanity.com/paper/2307.04721/) (2023)
- [One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention](https://mlsanity.com/paper/2307.03576/) (2023)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [Scan and Snap: Understanding Training Dynamics and Token Composition in 1-layer Transformer](https://mlsanity.com/paper/2305.16380/) (2023)
- [Uncovering mesa-optimization algorithms in Transformers](https://mlsanity.com/paper/2309.05858/) (2023)
- [In-Context Language Learning: Architectures and Algorithms](https://mlsanity.com/paper/2401.12973/) (2024)
- [How Do Transformers Learn In-Context Beyond Simple Functions? A Case Study on Learning with Representations](https://mlsanity.com/paper/2310.10616/) (2023)
- [Transformers Implement Functional Gradient Descent to Learn Non-Linear Functions In Context](https://mlsanity.com/paper/2312.06528/) (2023)

## Built on by (most cited first, 28 of 28)

- [From Words to Numbers: Your Large Language Model Is Secretly A Capable Regressor When Given In-Context Examples](https://mlsanity.com/paper/2404.07544/) (2024)
- [Training Dynamics of In-Context Learning in Linear Attention](https://mlsanity.com/paper/2501.16265/) (2025)
- [Theory of Scaling Laws for In-Context Regression: Depth, Width, Context and Time](https://mlsanity.com/paper/2510.01098/) (2025)
- [XLand-100B: A Large-Scale Multi-Task Dataset for In-Context Reinforcement Learning](https://mlsanity.com/paper/2406.08973/) (2024)
- [Ask, and it shall be given: On the Turing completeness of prompting](https://mlsanity.com/paper/2411.01992/) (2024)
- [Contextualize-then-Aggregate: Circuits for In-Context Learning in Gemma-2 2B](https://mlsanity.com/paper/2504.00132/) (2025)
- [Understanding In-Context Learning of Linear Models in Transformers Through an Adversarial Lens](https://mlsanity.com/paper/2411.05189/) (2024)
- [Transformer learns the cross-task prior and regularization for in-context learning](https://mlsanity.com/paper/2505.12138/) (2025)
- [Transformers Meet In-Context Learning: A Universal Approximation Theory](https://mlsanity.com/paper/2506.05200/) (2025)
- [On Mesa-Optimization in Autoregressively Trained Transformers: Emergence and Capability](https://mlsanity.com/paper/2405.16845/) (2024)
- [In-Context Learning of Linear Dynamical Systems with Transformers: Approximation Bounds and Depth-Separation](https://mlsanity.com/paper/2502.08136/) (2025)
- [Learning Causal Graphs at Scale: A Foundation Model Approach](https://mlsanity.com/paper/2506.18285/) (2025)
- [Softmax ≥ Linear: Transformers may learn to classify in-context by kernel gradient descent](https://mlsanity.com/paper/2510.10425/) (2025)
- [From Unstructured Data to In-Context Learning: Exploring What Tasks Can Be Learned and When](https://mlsanity.com/paper/2406.00131/) (2024)
- [Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery](https://mlsanity.com/paper/2505.23106/) (2025)
- [In-Context Optimization for Retrieval-Augmented Generation: A Gradient-Descent Perspective](https://mlsanity.com/paper/2605.26356/) (2026)
- [Multi-Layer Attention is the Amplifier of Demonstration Effectiveness](https://mlsanity.com/paper/2508.00385/) (2025)
- [Linear Transformers Implicitly Discover Unified Numerical Algorithms](https://mlsanity.com/paper/2509.19702/) (2025)
- [Bayesian Optimality of In-Context Learning with Selective State Spaces](https://mlsanity.com/paper/2602.17744/) (2026)
- [Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression](https://mlsanity.com/paper/2605.08475/) (2026)
- [Transformers Can Learn Posterior Predictive Distributions In-Context](https://mlsanity.com/paper/2605.26713/) (2026)
- [Toward generalizable learning of all (linear) first-order methods via memory augmented Transformers](https://mlsanity.com/paper/2410.07263/) (2024)
- [How Private is Your Attention? Bridging Privacy with In-Context Learning](https://mlsanity.com/paper/2504.16000/) (2025)
- [From Text to Trajectories: GPT-2 as an ODE Solver via In-Context](https://mlsanity.com/paper/2508.03031/) (2025)
- [Data Distribution as a Lever for Guiding Optimizers Toward Superior Generalization in LLMs](https://mlsanity.com/paper/2602.00576/) (2026)
- [Ordinary Least Squares is a Special Case of Transformer](https://mlsanity.com/paper/2604.13656/) (2026)
- [Neural Events: Discrete Asynchronous Autoencoders for Event-Based Vision](https://mlsanity.com/paper/2606.19835/) (2026)
- [Dendritic In-Context Learning in a Single-Layer Spiking Neural Network](https://mlsanity.com/paper/2607.02283/) (2026)

## Abstract (arXiv)

Recent research has demonstrated that transformers, particularly linear attention models, implicitly execute gradient-descent-like algorithms on data provided in-context during their forward inference step. However, their capability in handling more complex problems remains unexplored. In this paper, we prove that each layer of a linear transformer maintains a weight vector for an implicit linear regression problem and can be interpreted as performing a variant of preconditioned gradient descent. We also investigate the use of linear transformers in a challenging scenario where the training data is corrupted with different levels of noise. Remarkably, we demonstrate that for this problem linear transformers discover an intricate and highly effective optimization algorithm, surpassing or matching in performance many reasonable baselines. We analyze this algorithm and show that it is a novel approach incorporating momentum and adaptive rescaling based on noise levels. Our findings show that even linear transformers possess the surprising ability to discover sophisticated optimization strategies.