A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
It provides a foundational mathematical explanation for a key problem in machine learning, though it is incremental in formalizing existing empirical observations.
The paper tackles the lack of theoretical understanding of why autoregressive language models help solve downstream tasks, showing that language models optimal in cross-entropy can learn features for text classification with error scaling as the square root of the cross-entropy error.
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 $ε$-optimal in cross-entropy (log-perplexity) learn features that can linearly solve such classification tasks with $\mathcal{O}(\sqrtε)$ 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.