Self-Distillation Amplifies Regularization in Hilbert Space (2002.05715v3)

Abstract: Knowledge distillation introduced in the deep learning context is a method to transfer knowledge from one architecture to another. In particular, when the architectures are identical, this is called self-distillation. The idea is to feed in predictions of the trained model as new target values for retraining (and iterate this loop possibly a few times). It has been empirically observed that the self-distilled model often achieves higher accuracy on held out data. Why this happens, however, has been a mystery: the self-distillation dynamics does not receive any new information about the task and solely evolves by looping over training. To the best of our knowledge, there is no rigorous understanding of this phenomenon. This work provides the first theoretical analysis of self-distillation. We focus on fitting a nonlinear function to training data, where the model space is Hilbert space and fitting is subject to $\ell_2$ regularization in this function space. We show that self-distillation iterations modify regularization by progressively limiting the number of basis functions that can be used to represent the solution. This implies (as we also verify empirically) that while a few rounds of self-distillation may reduce over-fitting, further rounds may lead to under-fitting and thus worse performance.

• The paper demonstrates that self-distillation enhances ℓ2 regularization in Hilbert spaces by progressively constraining the model’s capacity.
• It shows that initial self-distillation iterations reduce overfitting, while excessive iterations can lead to underfitting.
• Empirical results in deep neural networks confirm the theory, linking analytical insights with observable improvements in generalization.

Analysis of Self-Distillation in Hilbert Space Regularization

The paper "Self-Distillation Amplifies Regularization in Hilbert Space" offers a theoretical investigation into the phenomenon of self-distillation within the context of regularized function fitting in a Hilbert space framework. Self-distillation refers to the process wherein a model is retrained with its previous predictions used as target values, a technique noted for unexpectedly enhancing predictive performance without introducing new data insights. This work aims to illuminate the mechanics underpinning this improvement by focusing on $\ell_2$ regularized nonlinear function fitting.

Theoretical Contributions

The main results of the paper revolve around the regularization effect amplified through self-distillation iterations. The analysis uses a Hilbert space framework, where the learning model is regularized using an $\ell_2$ norm based method. The authors of the paper prove analytically that the self-distillation process can be viewed as a progressive constraint on the model's representational capacity, specifically by limiting the number of basis functions—which are essential to the solution representation—over iterations. This has several implications for both overfitting and underfitting:

• Reduction of Overfitting: A few rounds of self-distillation can effectively reduce the model's complexity and enhance its generalization by limiting over-complex models and thus, implicitly regularizing the solution.
• Induction of Underfitting: However, as more iterations are applied, the constriction becomes too severe, leading to underfitting as the model complexity is overly reduced, compromising its ability to capture the underlying data distribution accurately.

This theoretical framing is quite insightful as it establishes self-distillation as a non-standard regularization technique akin to a power iteration method with dynamic dependency, rather than solely data augmentation or traditional knowledge distillation from a more complex to a simpler model.

Practical Implications and Empirical Evidence

Experimentally, the authors establish that this theoretical construct holds in practice by extending their theoretical setup to experiments involving deep neural networks, specifically in scenarios that align with the Neural Tangent Kernel regime. The empirical results underscore the dual-phase behavior observed theoretically: initial improvement followed by degradation in model performance through prolonged self-distillation, demonstrating the real-world validity of the underfitting and overfitting balance.

Generalization Beyond Initial Setup

The authors speculate about potential extensions of their theoretical findings to more general settings in AI research, such as multi-class classification problems or yielding practical generalization bounds that can be informative for other machine learning tasks. The results further open the avenue for exploring different loss functions and settings beyond $\ell_2$, which is indicative of the foundational nature of their theoretical findings.

Conclusion and Future Directions

This comprehensive treatment of self-distillation sheds light on its potential role as a form of induced regularization, offering a clearer understanding of its empirical success despite self-distillation involving no new information introduction. The adaptation of this analysis to broader AI applications and complex neural network architectures forms an exciting avenue for future exploration, potentially leading to more robust and efficient learning paradigms in the field of deep learning.

Moving forward, rigorously extending these ideas to other regularization settings and exploring their applicability in non-Hilbert spaces or alternative neural architectures will enrich the understanding of knowledge distillation and its related phenomena in the broader AI landscape.

This paper presents a valuable intersection of theoretical rigor and practical insight that provides a robust framework for understanding and leveraging self-distillation in machine learning.