Weak-to-Strong Generalization (Burns et al.,2024) is the phenomenon whereby a strong student, say GPT-4, learns a task from a weak teacher, say GPT-2, and ends up significantly outperforming the teacher. We show that this phenomenon does not require a complex and pretrained learner like GPT-4, can arise even in simple non-pretrained models, simply due to the size advantage of the student. But, we also show that there are inherint limits to the extent of such weak to strong generalization. We consider students and teachers that are random feature models, described by two-layer networks with a random and fixed bottom layer and trained top layer. A ‘weak’ teacher, with a small number of units (i.e. random features), is trained on the population, and a ‘strong’ student, with a much larger number of units (i.e. random features), is trained only on labels generated by the weak teacher. We demonstrate, prove, and understand how the student can outperform the teacher, even though trained only on data labeled by the teacher. We also explain how such weak-to-strong generalization is enabled by early stopping. We then show the quantitative limits of weak-to-strong generalization in this model, and in fact in a much broader class of models, for arbitrary teacher and student feature spaces and a broad class of learning rules, including when the student features are pre-trained or otherwise more informative. In particular, we show that in such models the student’s error can only approach zero if the teacher’s error approaches zero, and a strong student cannot “boost” a slightly-better-then-chance teacher to obtain a small error.

We consider the overfitting behavior of minimum norm interpolating solutions of Gaussian kernel ridge regression (i.e. kernel ridgeless regression), when the bandwidth or input dimension varies with the sample size. For fixed dimensions, we show that even with varying or tuned bandwidth, the ridgeless solution is never consistent and, at least with large enough noise, always worse than the null predictor. For increasing dimension, we give a generic characterization of the overfitting behavior for any scaling of the dimension with sample size. We use this to provide the first example of benign overfitting using the Gaussian kernel with sub-polynomial scaling dimension. All our results are under the Gaussian universality ansatz and the (non-rigorous) risk predictions in terms of the kernel eigenstructure.