Knowledge Cascade: Reverse Knowledge Distillation on Nonparametric Multivariate Functional Estimation

2026-06-24

AI Summary

This work addresses the computational burden of constructing teacher models in nonparametric multivariate function estimation, which arises from the need to select multiple smoothing parameters. To overcome this challenge, the authors propose a Knowledge Cascading (KCas) framework that reverses the conventional direction of knowledge distillation: instead of compressing a large teacher into a small student, a lightweight student model guides the training of a more complex teacher. By integrating smoothing splines in reproducing kernel Hilbert spaces with asymptotic scaling laws, KCas enables efficient cross-scale parameter transfer. This approach transcends the traditional role of distillation as merely a model compression tool, significantly reducing computational costs on high-dimensional, large-scale data while maintaining or even surpassing the statistical performance of full-sample estimation methods.

Abstract

As machine learning models and datasets continue to grow, developing complex models has become increasingly computationally demanding. Knowledge distillation reduces deployment cost by compressing a large, well-trained teacher model into a compact student model, but it does not address settings where constructing the teacher itself is the bottleneck. Motivated by this challenge, we introduce Knowledge Cascade (KCas), a reverse knowledge distillation framework that uses information from a small, inexpensive student model to guide the development of a more complex teacher model. Although this direction is counterintuitive because the teacher typically has greater representational capacity, we show that student-to-teacher transfer can be principled when supported by statistical scaling relationships. We first develop KCas for nonparametric multivariate functional estimation in reproducing kernel Hilbert spaces via smoothing splines, where selecting multiple smoothing parameters is a major computational bottleneck. KCas transfers student-selected smoothing parameters to the full-sample regime through asymptotic scaling laws, substantially reducing computational cost for high-dimensional and large-scale datasets while retaining theoretical guarantees. Beyond smoothing splines, we illustrate the same principle through kernel density estimation and deep learning hyperparameter transfer. Simulations and real-data experiments show that KCas achieves substantial computational savings while maintaining strong statistical performance, and can sometimes outperform the corresponding full-sample procedure.

Problem

• nonparametric estimation
• smoothing parameters
• computational bottleneck
• multivariate functional estimation
• knowledge distillation

Innovation

• reverse knowledge distillation
• nonparametric estimation
• smoothing splines
• statistical scaling laws
• hyperparameter transfer