# Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection

arXiv: https://arxiv.org/abs/2306.04637 · HTML: https://mlsanity.com/paper/2306.04637/

## Key facts

- Cited by 298 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 97 references that are in the map.
- First posted on arXiv 7 Jun 2023.

## Builds on (most cited first, 30 of 41)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [What Makes Good In-Context Examples for GPT-3?](https://mlsanity.com/paper/2101.06804/) (2021)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [A Generalist Agent](https://mlsanity.com/paper/2205.06175/) (2022)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [A Survey on In-context Learning](https://mlsanity.com/paper/2301.00234/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Learning To Retrieve Prompts for In-Context Learning](https://mlsanity.com/paper/2112.08633/) (2021)
- [MetaICL: Learning to Learn In Context](https://mlsanity.com/paper/2110.15943/) (2021)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Larger language models do in-context learning differently](https://mlsanity.com/paper/2303.03846/) (2023)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [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)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [Why Can GPT Learn In-Context? Language Models Implicitly Perform Gradient Descent as Meta-Optimizers](https://mlsanity.com/paper/2212.10559/) (2022)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [Noisy Channel Language Model Prompting for Few-Shot Text Classification](https://mlsanity.com/paper/2108.04106/) (2021)
- [Impact of Pretraining Term Frequencies on Few-Shot Reasoning](https://mlsanity.com/paper/2202.07206/) (2022)
- [On the Turing Completeness of Modern Neural Network Architectures](https://mlsanity.com/paper/1901.03429/) (2019)
- [ProGen: Language Modeling for Protein Generation](https://mlsanity.com/paper/2004.03497/) (2020)
- [Unveiling Transformers with LEGO: a synthetic reasoning task](https://mlsanity.com/paper/2206.04301/) (2022)

## Built on by (most cited first, 30 of 298)

- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Looped Transformers are Better at Learning Learning Algorithms](https://mlsanity.com/paper/2311.12424/) (2023)
- [The mechanistic basis of data dependence and abrupt learning in an in-context classification task](https://mlsanity.com/paper/2312.03002/) (2023)
- [How Transformers Learn Causal Structure with Gradient Descent](https://mlsanity.com/paper/2402.14735/) (2024)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [Can Mamba Learn How to Learn? A Comparative Study on In-Context Learning Tasks](https://mlsanity.com/paper/2402.04248/) (2024)
- [The Expressive Power of Low-Rank Adaptation](https://mlsanity.com/paper/2310.17513/) (2023)
- [Uncovering mesa-optimization algorithms in Transformers](https://mlsanity.com/paper/2309.05858/) (2023)
- [How Many Pretraining Tasks Are Needed for In-Context Learning of Linear Regression?](https://mlsanity.com/paper/2310.08391/) (2023)
- [A Survey of Graph Retrieval-Augmented Generation for Customized Large Language Models](https://mlsanity.com/paper/2501.13958/) (2025)
- [Transformers as Decision Makers: Provable In-Context Reinforcement Learning via Supervised Pretraining](https://mlsanity.com/paper/2310.08566/) (2023)
- [Can large language models explore in-context?](https://mlsanity.com/paper/2403.15371/) (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 as Support Vector Machines](https://mlsanity.com/paper/2308.16898/) (2023)
- [Can Looped Transformers Learn to Implement Multi-step Gradient Descent for In-context Learning?](https://mlsanity.com/paper/2410.08292/) (2024)
- [Understanding In-Context Learning in Transformers and LLMs by Learning to Learn Discrete Functions](https://mlsanity.com/paper/2310.03016/) (2023)
- [Transformers Implement Functional Gradient Descent to Learn Non-Linear Functions In Context](https://mlsanity.com/paper/2312.06528/) (2023)
- [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)
- [Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective](https://mlsanity.com/paper/2406.00793/) (2024)
- [Asymptotic theory of in-context learning by linear attention](https://mlsanity.com/paper/2405.11751/) (2024)
- [JoMA: Demystifying Multilayer Transformers via JOint Dynamics of MLP and Attention](https://mlsanity.com/paper/2310.00535/) (2023)
- [Transformers Learn to Achieve Second-Order Convergence Rates for In-Context Linear Regression](https://mlsanity.com/paper/2310.17086/) (2023)
- [CausalPFN: Amortized Causal Effect Estimation via In-Context Learning](https://mlsanity.com/paper/2506.07918/) (2025)
- [Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscape](https://mlsanity.com/paper/2402.01258/) (2024)
- [Training Dynamics of In-Context Learning in Linear Attention](https://mlsanity.com/paper/2501.16265/) (2025)
- [Is Mamba Capable of In-Context Learning?](https://mlsanity.com/paper/2402.03170/) (2024)
- [Why Larger Language Models Do In-context Learning Differently?](https://mlsanity.com/paper/2405.19592/) (2024)
- [Replacing softmax with ReLU in Vision Transformers](https://mlsanity.com/paper/2309.08586/) (2023)
- [A Theoretical Understanding of Self-Correction through In-context Alignment](https://mlsanity.com/paper/2405.18634/) (2024)
- [Dual Operating Modes of In-Context Learning](https://mlsanity.com/paper/2402.18819/) (2024)

## Abstract (arXiv)

Neural sequence models based on the transformer architecture have demonstrated remarkable in-context learning (ICL) abilities, where they can perform new tasks when prompted with training and test examples, without any parameter update to the model. This work first provides a comprehensive statistical theory for transformers to perform ICL. Concretely, we show that transformers can implement a broad class of standard machine learning algorithms in context, such as least squares, ridge regression, Lasso, learning generalized linear models, and gradient descent on two-layer neural networks, with near-optimal predictive power on various in-context data distributions. Using an efficient implementation of in-context gradient descent as the underlying mechanism, our transformer constructions admit mild size bounds, and can be learned with polynomially many pretraining sequences. Building on these ``base'' ICL algorithms, intriguingly, we show that transformers can implement more complex ICL procedures involving in-context algorithm selection, akin to what a statistician can do in real life -- A single transformer can adaptively select different base ICL algorithms -- or even perform qualitatively different tasks -- on different input sequences, without any explicit prompting of the right algorithm or task. We both establish this in theory by explicit constructions, and also observe this phenomenon experimentally. In theory, we construct two general mechanisms for algorithm selection with concrete examples: pre-ICL testing, and post-ICL validation. As an example, we use the post-ICL validation mechanism to construct a transformer that can perform nearly Bayes-optimal ICL on a challenging task -- noisy linear models with mixed noise levels. Experimentally, we demonstrate the strong in-context algorithm selection capabilities of standard transformer architectures.