On the Blessing of Pre-training in Weak-to-Strong Generalization

TL;DR

This paper investigates the critical role of pre-training in enabling Weak-to-Strong Generalization (W2SG), providing theoretical proofs and empirical evidence that pre-training acts as a geometric warm start necessary for W2SG to occur.

Contribution

It formalizes the W2SG problem within a high-dimensional model, proving that pre-training enables W2SG through a phase transition observed in large language models.

Findings

Pre-training provides a geometric warm start crucial for W2SG.

W2SG emerges via a phase transition during pre-training.

A generalization bound captures the optimization dynamics of W2SG.

Abstract

The paradigm of Weak-to-Strong Generalization (W2SG) suggests that a pre-trained strong model can surpass its weak supervisor, yet the decisive role of pre-training remains theoretically and empirically under-explored. In this work, we identify pre-training as the essential prerequisite for the emergence of W2SG. Theoretically, we formalize the W2SG problem within a high-dimensional single-index model framework using spiked Gaussian data, modeling pre-training as a spectral initialization step. Building upon prior impossibility results regarding the failure of learning under random initialization, we prove that W2SG is achievable when pre-training provides a geometric warm start that places the model within an "effective region" characterized by a perturbed strong-convexity geometry. Within this region, we derive a rigorous generalization bound that naturally captures the optimization dynamics: an initial performance improvement followed by a saturation bottleneck dictated by the weak supervisor's bias. Empirically, we first validate all our assumptions and theoretical insights through controlled synthetic simulations. Finally, through a massive-scale evaluation of hundreds of intermediate pre-training checkpoints from large language models, we demonstrate that W2SG is not an innate capability but emerges via a phase transition tightly coupled with the progression of pre-training.