# An Active-Bottleneck Mechanism for Weak-to-Strong Generalization

arXiv: https://arxiv.org/abs/2609.33835 · HTML: https://mlsanity.com/paper/2609.33835/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 32 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Weak-to-Strong Generalization: Eliciting Strong Capabilities With Weak Supervision](https://mlsanity.com/paper/2312.09390/) (2023)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Distillation Scaling Laws](https://mlsanity.com/paper/2502.08606/) (2025)
- [Weak-to-Strong Generalization Even in Random Feature Networks, Provably](https://mlsanity.com/paper/2503.02877/) (2025)
- [Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension](https://mlsanity.com/paper/2502.05075/) (2025)
- [Representations Shape Weak-to-Strong Generalization: Theoretical Insights and Empirical Predictions](https://mlsanity.com/paper/2502.00620/) (2025)
- [Relating Misfit to Gain in Weak-to-Strong Generalization Beyond the Squared Loss](https://mlsanity.com/paper/2501.19105/) (2025)
- [Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models](https://mlsanity.com/paper/2502.05074/) (2025)
- [Weak-to-Strong Generalization via Bregman Bias-Variance Decomposition](https://mlsanity.com/paper/2505.24313/) (2025)
- [Improved Scaling Laws via Weak-to-Strong Generalization in Random Feature Ridge Regression](https://mlsanity.com/paper/2603.05691/) (2026)
- [Weak-to-Strong Generalization is Nearly Inevitable (in Linear Models)](https://mlsanity.com/paper/2605.05742/) (2026)
- [On the Blessing of Pre-training in Weak-to-Strong Generalization](https://mlsanity.com/paper/2605.05710/) (2026)

## Abstract (arXiv)

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 NS exceeds an explicit threshold. Both regimes are governed by a single "active-bottleneck" principle: whichever of m or NS 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.