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Learning Dynamics of Two-Layer Neural Networks: Incremental Approximation of Target Functions


Core Concepts
The learning dynamics of two-layer neural networks exhibits a separation of timescales and incremental approximation of the target function, where the network learns simpler polynomial approximations before converging to the full target function.
Abstract
The key insights from the content are: Gradient-based learning in multi-layer neural networks displays striking features, including non-monotone decrease in empirical risk, long plateaus followed by rapid decreases, and separation of timescales in the learning process. The authors study the gradient flow dynamics of a wide two-layer neural network in high-dimensional settings, where the target function depends on a one-dimensional projection of the input features (single-index model). Through a mixture of rigorous results, non-rigorous derivations, and numerical simulations, the authors propose a scenario for the learning dynamics in this setting. The proposed evolution exhibits separation of timescales and intermittency, which arise naturally because the population gradient flow can be recast as a singularly perturbed dynamical system. The authors show that the learning dynamics constructs a sequence of polynomial approximations of the target function, with each phase of learning corresponding to a more accurate polynomial approximation. This incremental learning behavior is formalized as the "canonical learning order". The authors provide a detailed analysis of the gradient flow dynamics using tools from dynamical systems theory, including singular perturbation theory and matched asymptotic expansions. They are able to prove the proposed scenario in several special cases and provide a heuristic argument for its generality. The authors also discuss the implications of their findings for understanding generalization in deep learning, as the notion of complexity corresponds to the order in which the solution space is explored.
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Quotes
"Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely any progress alternate with intervals of rapid decrease. These successive phases of learning often take place on very different time scales." "We propose a scenario for the learning dynamics in this setting. In particular, the proposed evolution exhibits separation of timescales and intermittency. These behaviors arise naturally because the population gradient flow can be recast as a singularly perturbed dynamical system."

Key Insights Distilled From

by Raph... at arxiv.org 04-19-2024

https://arxiv.org/pdf/2303.00055.pdf
Learning time-scales in two-layers neural networks

Deeper Inquiries

What are the implications of the incremental learning behavior for the generalization performance of two-layer neural networks

The incremental learning behavior observed in two-layer neural networks has significant implications for the generalization performance of these networks. As the network learns incrementally, starting with simpler models and gradually increasing in complexity, it has been observed that the models learned in the early phases of training are typically easier to learn. This incremental learning process plays a crucial role in controlling the complexity of the model learned at different stages of training. By stopping the learning process at a certain time, one can effectively control the complexity of the model and prevent overfitting. The implications of this incremental learning behavior for generalization performance are profound. Models that are learned incrementally tend to generalize better to unseen data. By learning simpler models first and gradually increasing in complexity, the network is able to capture the underlying patterns in the data in a more structured and organized manner. This leads to improved generalization performance, as the network is better equipped to make accurate predictions on new, unseen data points. Furthermore, the incremental learning behavior helps in avoiding overfitting, as the complexity of the model is controlled throughout the learning process. By incrementally learning more complex patterns in the data, the network is less likely to memorize noise or irrelevant details in the training data, leading to better generalization to new data points.

How would the learning dynamics change if the target function depended on a higher-dimensional projection of the input features (multi-index model)

In the context of a multi-index model where the target function depends on a higher-dimensional projection of the input features, the learning dynamics of two-layer neural networks would undergo significant changes. In a multi-index model, the target function is defined as a function of multiple projections of the input features, leading to a more complex relationship between the input and output variables. The higher-dimensional projection of the input features introduces additional complexity to the learning process. The network would need to learn and capture the dependencies between multiple projections of the input features and the target function. This increased complexity may require a larger network capacity or more sophisticated learning algorithms to effectively model the relationships in the data. Additionally, the higher-dimensional nature of the multi-index model may lead to a more intricate optimization landscape, potentially resulting in slower convergence and more challenging learning dynamics. The network would need to navigate through a higher-dimensional parameter space, which could impact the speed and efficiency of the learning process. Overall, the learning dynamics of two-layer neural networks in a multi-index model setting would be influenced by the increased complexity and dimensionality of the data, requiring adaptations in the network architecture and learning algorithms to effectively capture the relationships in the data.

Can the insights from the analysis of two-layer networks be extended to understand the learning dynamics of deeper neural network architectures

The insights gained from the analysis of two-layer neural networks can be extended to understand the learning dynamics of deeper neural network architectures. While the analysis in the context of two-layer networks provides valuable insights into the incremental learning behavior, time scales separation, and generalization performance, similar principles can be applied to deeper architectures. Deeper neural networks exhibit more complex learning dynamics due to the presence of multiple hidden layers and non-linear activations. The incremental learning behavior observed in two-layer networks, where simpler models are learned first before progressing to more complex ones, can also be observed in deeper architectures. By understanding the incremental learning process in deeper networks, researchers can gain insights into how information is processed and represented at different layers of the network. Additionally, the analysis of time scales separation in two-layer networks can be extended to deeper architectures to study how learning progresses at different layers and time scales. By examining the dynamics of gradient flow in deeper networks, researchers can uncover how information flows through the network and how the representations evolve over time. Overall, the insights from the analysis of two-layer neural networks can serve as a foundation for understanding the learning dynamics of deeper architectures, providing valuable insights into how neural networks learn and generalize in more complex settings.
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