# High-dimensional Analysis of Knowledge Distillation: Weak-to-Strong Generalization and Scaling Laws

arXiv: https://arxiv.org/abs/2410.18837 · HTML: https://mlsanity.com/paper/2410.18837/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Placed in General knowledge distillation methods, one of 12 directions in Knowledge distillation.
- Builds on 42 of its 55 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Placed in

- Knowledge distillation / [General knowledge distillation methods](https://mlsanity.com/distill/general-knowledge-distillation-methods/)

## Builds on (most cited first, 30 of 42)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Self-Instruct: Aligning Language Models with Self-Generated Instructions](https://mlsanity.com/paper/2212.10560/) (2022)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Weak-to-Strong Generalization: Eliciting Strong Capabilities With Weak Supervision](https://mlsanity.com/paper/2312.09390/) (2023)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Theoretical Analysis of Self-Training with Deep Networks on Unlabeled Data](https://mlsanity.com/paper/2010.03622/) (2020)
- [Self-Distillation Amplifies Regularization in Hilbert Space](https://mlsanity.com/paper/2002.05715/) (2020)
- [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020)
- [Rho-1: Not All Tokens Are What You Need](https://mlsanity.com/paper/2404.07965/) (2024)
- [Learning curves of generic features maps for realistic datasets with a teacher-student model](https://mlsanity.com/paper/2102.08127/) (2021)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [On the Optimal Weighted ℓ2 Regularization in Overparameterized Linear Regression](https://mlsanity.com/paper/2006.05800/) (2020)
- [Generalization Error Rates in Kernel Regression: The Crossover from the Noiseless to Noisy Regime](https://mlsanity.com/paper/2105.15004/) (2021)
- [More Than a Toy: Random Matrix Models Predict How Real-World Neural Representations Generalize](https://mlsanity.com/paper/2203.06176/) (2022)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [4+3 Phases of Compute-Optimal Neural Scaling Laws](https://mlsanity.com/paper/2405.15074/) (2024)
- [Asymptotics of Ridge (less) Regression under General Source Condition](https://mlsanity.com/paper/2006.06386/) (2020)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [Dimension free ridge regression](https://mlsanity.com/paper/2210.08571/) (2022)
- [Theoretical Analysis of Weak-to-Strong Generalization](https://mlsanity.com/paper/2405.16043/) (2024)
- [Quantifying the Gain in Weak-to-Strong Generalization](https://mlsanity.com/paper/2405.15116/) (2024)
- [Provable Benefits of Overparameterization in Model Compression: From Double Descent to Pruning Neural Networks](https://mlsanity.com/paper/2012.08749/) (2020)

## Built on by (most cited first, 22 of 22)

- [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)
- [The Road to Artificial SuperIntelligence: A Comprehensive Survey of Superalignment](https://mlsanity.com/paper/2412.16468/) (2024)
- [Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension](https://mlsanity.com/paper/2502.05075/) (2025)
- [Preventing Model Collapse Under Overparametrization: Optimal Mixing Ratios for Interpolation Learning and Ridge Regression](https://mlsanity.com/paper/2509.22341/) (2025)
- [On the Mechanisms of Weak-to-Strong Generalization: A Theoretical Perspective](https://mlsanity.com/paper/2505.18346/) (2025)
- [High-dimensional Analysis of Synthetic Data Selection](https://mlsanity.com/paper/2510.08123/) (2025)
- [Prediction-Powered Semi-Supervised Learning with Online Power Tuning](https://mlsanity.com/paper/2510.22586/) (2025)
- [Weak-to-Strong Generalization via Bregman Bias-Variance Decomposition](https://mlsanity.com/paper/2505.24313/) (2025)
- [Weak-to-Strong Generalization under Distribution Shifts](https://mlsanity.com/paper/2510.21332/) (2025)
- [Provably Improving Generalization of Few-Shot Models with Synthetic Data](https://mlsanity.com/paper/2505.24190/) (2025)
- [Does Weak-to-strong Generalization Happen under Spurious Correlations?](https://mlsanity.com/paper/2509.24005/) (2025)
- [Optimal Regularization for Performative Learning](https://mlsanity.com/paper/2510.12249/) (2025)
- [Why Self-Training Helps and Hurts: Denoising vs. Signal Forgetting](https://mlsanity.com/paper/2602.14029/) (2026)
- [SNCE: Geometry-Aware Supervision for Scalable Discrete Image Generation](https://mlsanity.com/paper/2603.15150/) (2026)
- [On the Blessing of Pre-training in Weak-to-Strong Generalization](https://mlsanity.com/paper/2605.05710/) (2026)
- [The Mechanism of Weak-to-Strong Generalization: Feature Elicitation from Latent Knowledge](https://mlsanity.com/paper/2605.12908/) (2026)
- [Optimal Representation Size: High-Dimensional Analysis of Pretraining and Linear Probing](https://mlsanity.com/paper/2605.20105/) (2026)
- [Information-Theoretic Criteria for Knowledge Distillation in Multimodal Learning](https://mlsanity.com/paper/2510.13182/) (2025)
- [What Makes a Strong Model? A Unified Spectral Analysis of Knowledge Transfer over High-dimensional Linear Regression](https://mlsanity.com/paper/2606.01292/) (2026)
- [Prediction-Only Distillation in Linear and Logistic Regression](https://mlsanity.com/paper/2607.15450/) (2026)
- [Scaling Properties of Same-Family On-Policy Distillation](https://mlsanity.com/paper/2609.32722/) (2026)

## Abstract (arXiv)

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: (i) model shift, where the surrogate model is arbitrary, and (ii) distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that (i) W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but (ii) it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures.