Student model beats teacher with right amount of data and features
An Active-Bottleneck Mechanism for Weak-to-Strong Generalization
Artificial IntelligenceMachine Learning
Summary
Sometimes a simpler model trained on the predictions of a more complex one can actually do better. The authors studied when this happens in settings without early stopping or extra tricks. They found that if the student model trains on too few predictions, it learns too little, but training on too many copies the teacher’s mistakes. The improvement depends on a balance between the number of pseudo-labeled data points and the student model’s size, which together act as a natural filter to reduce errors.
What this means in practice
- •For machine learning engineers: Optimize student models’ training size and architecture to surpass teacher model accuracy without extra regularization.
- •For data labeling teams: Determine the minimal amount of annotated data needed to produce effective pseudo-labels for student model training.
Authors
Mohammad Zeinalpour, Amir Najafi
Abstract
Weak-to-strong generalization (W2SG) occurs when a student trained on a teacher's predictions outperforms that teacher. We study when this happens under fully converged, ridgeless two-stage learning, with no early stopping, no explicit regularization, and no assumption that the student is more expressive than the teacher. In two-stage linear regression, a teacher is fit from $n$ labeled examples and a student is trained solely on the teacher's predictions on $m$ fresh, unlabeled inputs. Although both stages share the same hypothesis class and the same training rule, we show that the student outperforms the teacher exactly when $m$ lies in an explicit intermediate range: too few pseudo-labels leave the student without enough signal, too many let it inherit the teacher's noise. Under power-law covariance, we derive this range in closed form as a function of the spectral decay and noise level, including regimes where the improving region splits into two disjoint intervals of $m$. We then study a random-feature model in which the student has strictly more features than the teacher, and identify two regimes, again given by explicit thresholds: one where improvement occurs only for $m$ in a bounded interval, and one where it occurs only once the student width $N_S$ exceeds an explicit threshold. Both regimes are governed by a single "active-bottleneck" principle: whichever of $m$ or $N_S$ is scarcer controls how much teacher error is filtered out, while increasing the other resource only reduces estimation noise. Together, these results show that finite data and finite width can themselves regularize a two-stage learner, with no explicit mechanism doing so.