We examine two in context learning (ICL) tasks with mathematical functions in several train and test settings for transformer models. Our study generalizes work on linear functions by showing that small transformers, even models with attention layers only, can approximate arbitrary polynomial functions and hence continuous functions under certain conditions. Our models also can approximate previously unseen classes of polynomial functions, as well as the zeros of complex functions. Our models perform far better on this task than LLMs like GPT4 and involve complex reasoning when provided with suitable training data and methods. Our models also have important limitations; they fail to generalize outside of training distributions and so don't learn class forms of functions. We explain why this is so.

In-context learning (ICL) has emerged as a powerful paradigm for easily adapting Large Language Models (LLMs) to various tasks. However, our understanding of how ICL works remains limited. We explore a simple model of ICL in a controlled setup with synthetic training data to investigate ICL of univariate linear functions. We experiment with a range of GPT-2-like transformer models trained from scratch. Our findings challenge the prevailing narrative that transformers adopt algorithmic approaches like linear regression to learn a linear function in-context. These models fail to generalize beyond their training distribution, highlighting fundamental limitations in their capacity to infer abstract task structures. Our experiments lead us to propose a mathematically precise hypothesis of what the model might be learning.

This paper explores the much discussed, possible explanatory link between attention weights (AW) in transformer models and predicted output. Contrary to intuition and early research on attention, more recent prior research has provided formal arguments and empirical evidence that AW are not explanatorily relevant. We show that the formal arguments are incorrect. We introduce and effectively compute efficient attention, which isolates the effective components of attention matrices in tasks and models in which AW play an explanatory role. We show that efficient attention has a causal role (provides minimally necessary and sufficient conditions) for predicting model output in NLP tasks requiring contextual information, and we show, contrary to [7], that efficient attention matrices are probability distributions and are effectively calculable. Thus, they should play an important part in the explanation of attention based model behavior. We offer empirical experiments in support of our method illustrating various properties of efficient attention with various metrics on four datasets.