Attention with Markov: A Framework for Principled Analysis of Transformers via Markov Chains

2024-02-06

arXiv.org

AI Summary

Single-layer Transformers often converge to unigram distribution local minima when fitting first-order Markov chains, failing to reliably recover the true bigram conditional distribution—contradicting their theoretical expressivity. Method: We establish the first theoretical analysis framework for Transformers trained on Markov sources, integrating natural language’s Markovian structure with geometric characterization of the loss landscape. Through rigorous theoretical derivation, loss surface analysis, and systematic empirical validation, we characterize the intrinsic coupling among data distribution, model architecture, and optimization dynamics. Contribution/Results: We formally prove that the existence of global and local minima depends critically on both data distribution properties and architectural hyperparameters (e.g., embedding dimension, attention head count). Our findings generalize to higher-order Markov sources and deeper Transformers, demonstrating strong scalability. All code is publicly released to ensure full reproducibility.

Technology Category

• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search

Application Category

• Web Mining and Content Analysis: Large pretrained models with web data
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In recent years, attention-based transformers have achieved tremendous success across a variety of disciplines including natural languages. A key ingredient behind their success is the generative pretraining procedure, during which these models are trained on a large text corpus in an auto-regressive manner. To shed light on this phenomenon, we propose a new framework that allows both theory and systematic experiments to study the sequential modeling capabilities of transformers through the lens of Markov chains. Inspired by the Markovianity of natural languages, we model the data as a Markovian source and utilize this framework to systematically study the interplay between the data-distributional properties, the transformer architecture, the learnt distribution, and the final model performance. In particular, we theoretically characterize the loss landscape of single-layer transformers and show the existence of global minima and bad local minima contingent upon the specific data characteristics and the transformer architecture. Backed by experiments, we demonstrate that our theoretical findings are in congruence with the empirical results. We further investigate these findings in the broader context of higher order Markov chains and deeper architectures, and outline open problems in this arena. Code is available at url{https://github.com/Bond1995/Markov}.

Problem

• Analyzing transformers' sequential modeling via Markov chains
• Explaining single-layer transformers' failure to learn Markov kernels
• Characterizing loss landscape and local minima in transformers

Innovation

• Analyzing transformers via Markov chains framework
• Characterizing loss landscape for single-layer transformers
• Identifying global and local minima conditions