Two in context learning tasks with complex functions

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.