Transformers Efficiently Perform In-Context Logistic Regression via Normalized Gradient Descent

TL;DR

This paper demonstrates how transformers can perform in-context logistic regression by executing normalized gradient descent steps, providing theoretical insights into their in-context learning capabilities.

Contribution

It constructs a class of transformers that perform in-context logistic regression through gradient descent, with training and generalization guarantees.

Findings

Transformers can perform exact in-context logistic regression.

A single self-attention layer trained via gradient descent suffices.

The looped model generalizes out-of-distribution.

Abstract

Transformers have demonstrated remarkable in-context learning (ICL) capabilities. The strong ICL performance of transformers is commonly believed to arise from their ability to implicitly execute certain algorithms on the context, thereby enhancing prediction and generation. In this work, we investigate how transformers with softmax attention perform in-context learning on linear classification data. We first construct a class of multi-layer transformers that can perform in-context logistic regression, with each layer exactly performing one step of normalized gradient descent on an in-context loss. Then, we show that our constructed transformer can be obtained through (i) training a single self-attention layer supervised by one-step gradient descent, and (ii) recurrently applying the trained layer to obtain a looped model. Training convergence guarantees of the self-attention layer and out-of-distribution generalization guarantees of the looped model are provided. Our results advance the theoretical understanding of ICL mechanism by showcasing how softmax transformers can effectively act as in-context learners.