# The Initialization Determines Whether In-Context Learning Is Gradient Descent

arXiv: https://arxiv.org/abs/2512.04268 · HTML: https://mlsanity.com/paper/2512.04268/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [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)
- [Making Monolingual Sentence Embeddings Multilingual using Knowledge Distillation](https://mlsanity.com/paper/2004.09813/) (2020)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (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 Learning through the Bayesian Prism](https://mlsanity.com/paper/2306.04891/) (2023)
- [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)
- [Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective](https://mlsanity.com/paper/2406.00793/) (2024)
- [Transformers Learn to Achieve Second-Order Convergence Rates for In-Context Linear Regression](https://mlsanity.com/paper/2310.17086/) (2023)
- [Induction Heads as an Essential Mechanism for Pattern Matching in In-context Learning](https://mlsanity.com/paper/2407.07011/) (2024)
- [In-Context Learning of a Linear Transformer Block: Benefits of the MLP Component and One-Step GD Initialization](https://mlsanity.com/paper/2402.14951/) (2024)
- [How Well Can Transformers Emulate In-context Newton's Method?](https://mlsanity.com/paper/2403.03183/) (2024)
- [Superiority of Multi-Head Attention in In-Context Linear Regression](https://mlsanity.com/paper/2401.17426/) (2024)
- [How Transformers Utilize Multi-Head Attention in In-Context Learning? A Case Study on Sparse Linear Regression](https://mlsanity.com/paper/2408.04532/) (2024)
- [On Understanding Attention-Based In-Context Learning for Categorical Data](https://mlsanity.com/paper/2405.17248/) (2024)

## Built on by (most cited first, 3 of 3)

- [Rethinking Zero-Shot Time Series Classification: From Task-specific Classifiers to In-Context Inference](https://mlsanity.com/paper/2602.00620/) (2026)
- [A Bayesian Perspective on the Role of Epistemic Uncertainty for Delayed Generalization in In-Context Learning](https://mlsanity.com/paper/2604.12434/) (2026)
- [ALICE: In-context, Zero-shot, Mutual Information Estimation](https://mlsanity.com/paper/2609.34962/) (2026)

## Abstract (arXiv)

In-context learning (ICL) in large language models (LLMs) is a striking phenomenon, yet its underlying mechanisms remain only partially understood. Previous work connects linear self-attention (LSA) to gradient descent (GD), this connection has primarily been established under simplified conditions with zero-mean Gaussian priors and zero initialization for GD. However, subsequent studies have challenged this simplified view by highlighting its overly restrictive assumptions, demonstrating instead that under conditions such as multi-layer or nonlinear attention, self-attention performs optimization-like inference, akin to but distinct from GD. We investigate how multi-head LSA approximates GD under more realistic conditions specifically when incorporating non-zero Gaussian prior means in linear regression formulations of ICL. We first extend multi-head LSA embedding matrix by introducing an initial estimation of the query, referred to as the initial guess. We prove an upper bound on the number of heads needed for ICL linear regression setup. Our experiments confirm this result and further observe that a performance gap between one-step GD and multi-head LSA persists. To address this gap, we introduce yq-LSA, a simple generalization of single-head LSA with a trainable initial guess yq. We theoretically establish the capabilities of yq-LSA and provide experimental validation on linear regression tasks, thereby extending the theory that bridges ICL and GD. Finally, inspired by our findings in the case of linear regression, we consider widespread LLMs augmented with initial guess capabilities, and show that their performance is improved on a semantic similarity task.