Feature learning as alignment: a structural property of gradient descent in non-linear neural networks (2402.05271v4)

Abstract: Understanding the mechanisms through which neural networks extract statistics from input-label pairs through feature learning is one of the most important unsolved problems in supervised learning. Prior works demonstrated that the gram matrices of the weights (the neural feature matrices, NFM) and the average gradient outer products (AGOP) become correlated during training, in a statement known as the neural feature ansatz (NFA). Through the NFA, the authors introduce mapping with the AGOP as a general mechanism for neural feature learning. However, these works do not provide a theoretical explanation for this correlation or its origins. In this work, we further clarify the nature of this correlation, and explain its emergence. We show that this correlation is equivalent to alignment between the left singular structure of the weight matrices and the newly defined pre-activation tangent features at each layer. We further establish that the alignment is driven by the interaction of weight changes induced by SGD with the pre-activation features, and analyze the resulting dynamics analytically at early times in terms of simple statistics of the inputs and labels. We prove the derivative alignment occurs almost surely in specific high dimensional settings. Finally, we introduce a simple optimization rule motivated by our analysis of the centered correlation which dramatically increases the NFA correlations at any given layer and improves the quality of features learned.

• The paper shows that gradient descent induces alignment between the singular structure of network weights and the empirical NTK, bridging the NFA with linear algebraic interpretations.
• It demonstrates that early training dynamics, predicted by input-label statistics, reliably forecast the speed of feature alignment development.
• The study introduces strategic interventions, such as scaled initialization and layer-wise gradient normalization, that enhance the quality of feature learning.

An Examination of Gradient Descent's Role in Weight and NTK Alignment in Deep Networks

The paper, "Gradient Descent Induces Alignment Between Weights and the Empirical NTK for Deep Non-linear Networks," provides a comprehensive analysis of how gradient descent influences the alignment between neural network weights and empirical Neural Tangent Kernels (NTKs). This paper sheds light on the intricate dynamics of neural network training, enhancing our understanding of feature learning in deep networks.

Overview

Neural networks have revolutionized computational tasks across various domains. However, the precise mechanisms that allow these models to generalize well remain obscure. One notable approach in understanding neural networks revolves around the Neural Tangent Kernel (NTK) framework. The NTK has been pivotal in linking neural networks to kernel methods. Despite this, NTK approximations miss critical dynamics of deep learning, especially in contexts where feature learning is substantial.

In prior literature, the Neural Feature Ansatz (NFA) has been a cornerstone for understanding weight dynamics. The NFA suggests that, post-training, the Gram matrices of weights align proportionally with the average gradient outer product (AGOP). Nonetheless, the underlying mechanisms leading to this alignment during training were not well-defined. The current research seeks to elucidate this phenomenon by linking it to the singular structures of weight matrices and NTKs.

Key Contributions

1. Equivalence of NFA and Singular Structure Alignment: The paper posits that the NFA can be interpreted through the lens of alignment between the singular value structure of network weights and the empirical NTK. This realization connects the empirical observations of NFA with canonical matrix alignments in mathematical theory.
2. Early-time Predictability: Analyzing early training dynamics, this paper shows that the development speed of the NFA can be anticipated using simple input-label statistics. This predictability opens new avenues for optimizing neural network initialization strategies.
3. Interventions for Enhanced Feature Learning: The researchers introduce novel interventions, such as modifying initialization scales and employing a layer-wise gradient normalization scheme, called \OptName{}, which enhance the correlation between NFM and AGOP. These interventions significantly improve feature quality across the layers.

Numerical Analysis and Results

The research presents robust numerical simulations to validate theoretical predictions. Notably, experiments demonstrate that the strength of NFA alignment, evidenced by high cosine similarity between NFMs and AGOPs, can be manipulated effectively via strategic interventions. These results support the notion that inducing stronger alignment can correlate with improved model performance on various tasks.

Implications and Future Directions

This work bears both theoretical and practical implications. Theoretically, it clarifies the role of gradient descent in orchestrating complex weight dynamics, contributing to the broader understanding of deep learning algorithms. Practically, the insights gleaned from this research suggest potential improvements in neural network architectures and training protocols. Given these results, future research could explore generalizations to more complex architectures like convolutional and transformer networks.

Moreover, understanding the extent of layer-wise contributions in deeper networks remains an open challenge. Future explorations could include detailed investigations into how different learning schedules or optimizer modifications can enhance alignment and consequentially, model performance.

In conclusion, this paper significantly advances our grasp of the nuanced mechanisms in neural network training, particularly regarding the alignment properties facilitated by gradient descent. The findings present substantial opportunities for enhancing current methodologies and computational understanding in deep learning.