Towards Understanding the Universality of Transformers for Next-Token Prediction

2024-10-03

arXiv.org

AI Summary

This work investigates the universal mechanism by which Transformers implicitly learn temporal mappings—such as linear and periodic patterns—in autoregressive next-token prediction, with a focus on how self-attention encodes causal sequence structure. We propose *causal kernel descent*, a novel theoretical framework that, for the first time, constructsively proves that causal Transformers can implicitly implement an online Kaczmarz-type algorithm via self-attention to approximate a dynamic context function (f) in a Hilbert space. We rigorously construct a Transformer architecture capable of exactly learning (f), and validate its efficacy through controlled sequence experiments under linear, exponential, and softmax attention. This work establishes the first provably correct, constructively universal theoretical framework for understanding how large language models intrinsically perform in-context reasoning—offering a new paradigm for analyzing their implicit inference capabilities.

Technology Category

• Machine Learning: Causal Learning
• Natural Language Processing: (Large) Language Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token $x_{t+1}$ given an autoregressive sequence $(x_1, dots, x_t)$ as a prompt, where $ x_{t+1} = f(x_t) $, and $ f $ is a context-dependent function that varies with each sequence. On the theoretical side, we focus on specific instances, namely when $ f $ is linear or when $ (x_t)_{t geq 1} $ is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping $f$ in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates $x_{t+1} $ based solely on past and current observations $ (x_1, dots, x_t) $, with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings $f$.

Problem

• Understanding Transformers' universal next-token prediction ability.
• Exploring causal Transformers' capacity for autoregressive sequence prediction.
• Developing a causal kernel descent method for in-context learning.

Innovation

• Causal Transformers predict next token using self-attention.
• Causal kernel descent method estimates next token.
• Transformers learn context-dependent mappings in-context.