A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks

TL;DR

This paper provides a mathematical explanation for why autoregressive language models excel at downstream tasks like text classification, linking next word prediction to classification performance through formal analysis and empirical validation.

Contribution

It formalizes the connection between language modeling and classification, showing that near-optimal language models learn features useful for classification with quantifiable error bounds.

Findings

Language modeling benefits downstream classification tasks.

Models with low cross-entropy learn linearly solvable features.

A new objective function improves classification performance.

Abstract

Autoregressive language models, pretrained using large text corpora to do well on next word prediction, have been successful at solving many downstream tasks, even with zero-shot usage. However, there is little theoretical understanding of this success. This paper initiates a mathematical study of this phenomenon for the downstream task of text classification by considering the following questions: (1) What is the intuitive connection between the pretraining task of next word prediction and text classification? (2) How can we mathematically formalize this connection and quantify the benefit of language modeling? For (1), we hypothesize, and verify empirically, that classification tasks of interest can be reformulated as sentence completion tasks, thus making language modeling a meaningful pretraining task. With a mathematical formalization of this hypothesis, we make progress towards (2) and show that language models that are $\epsilon$-optimal in cross-entropy (log-perplexity) learn features that can linearly solve such classification tasks with $\mathcal{O}(\sqrt{\epsilon})$ error, thus demonstrating that doing well on language modeling can be beneficial for downstream tasks. We experimentally verify various assumptions and theoretical findings, and also use insights from the analysis to design a new objective function that performs well on some classification tasks.