On the Emergence of Weak-to-Strong Generalization: A Bias-Variance Perspective

TL;DR

This paper provides a theoretical analysis of weak-to-strong generalization, showing how a student model trained on a weak teacher's labels can outperform the teacher, especially when approximating the posterior mean and avoiding overfitting.

Contribution

It offers a bias-variance decomposition analysis of W2SG without restrictive assumptions and demonstrates the benefits of reverse cross-entropy loss empirically.

Findings

W2SG is linked to the misfit between student and teacher models.

Approximating the posterior mean enhances W2SG emergence.

Reverse cross-entropy loss improves student performance.

Abstract

Weak-to-strong generalization (W2SG) refers to the phenomenon where a strong student model, trained on a dataset labeled by a weak teacher, ultimately outperforms the teacher on the target task. Recent studies attribute this performance gain to the prediction misfit between the student and teacher models. In this work, we theoretically investigate the emergence of W2SG through a generalized bias-variance decomposition of Bregman divergence. Specifically, we show that the expected population risk gap between the student and teacher is quantified by the expected misfit between the two models. While this aligns with previous results, our analysis removes several restrictive assumptions, most notably, the convexity of the student's hypothesis class, required in earlier works. Moreover, we show that W2SG is more likely to emerge when the student model approximates its posterior mean teacher, rather than mimicking an individual teacher. Using a concrete example, we demonstrate that if the student model size is sufficiently large, it can indeed converge to the posterior mean teacher in expectation. Our analysis also suggests that avoiding overfitting to the teacher's supervision and reducing the entropy of student's prediction further facilitate W2SG. In addition, we show that the reverse cross-entropy loss, unlike the standard forward cross-entropy, is less sensitive to the predictive uncertainty of the teacher. Finally, we empirically verify our theoretical insights and demonstrate that incorporating the reverse cross-entropy loss consistently improves student performance.