Asymptotic Analysis of Two-Layer Neural Networks after One Gradient Step under Gaussian Mixtures Data with Structure (2503.00856v3)

Abstract: In this work, we study the training and generalization performance of two-layer neural networks (NNs) after one gradient descent step under structured data modeled by Gaussian mixtures. While previous research has extensively analyzed this model under isotropic data assumption, such simplifications overlook the complexities inherent in real-world datasets. Our work addresses this limitation by analyzing two-layer NNs under Gaussian mixture data assumption in the asymptotically proportional limit, where the input dimension, number of hidden neurons, and sample size grow with finite ratios. We characterize the training and generalization errors by leveraging recent advancements in Gaussian universality. Specifically, we prove that a high-order polynomial model performs equivalent to the nonlinear neural networks under certain conditions. The degree of the equivalent model is intricately linked to both the "data spread" and the learning rate employed during one gradient step. Through extensive simulations, we demonstrate the equivalence between the original model and its polynomial counterpart across various regression and classification tasks. Additionally, we explore how different properties of Gaussian mixtures affect learning outcomes. Finally, we illustrate experimental results on Fashion-MNIST classification, indicating that our findings can translate to realistic data.

Overview of "Asymptotic Analysis of Two-Layer Neural Networks after One Gradient Step under Gaussian Mixtures Data with Structure"

This paper presents a rigorous analysis of two-layer neural networks (NNs) under conditions that more accurately reflect real-world data complexities compared to previous models. Specifically, this paper evaluates the training and generalization performance of two-layer NNs after a single gradient descent step when the input data is modeled by structured Gaussian mixtures. Prior studies often simplified analyses by assuming isotropic data; however, such assumptions overlook critical structural intricacies inherent in practical datasets. The authors address this gap by exploring neural network performance under the assumption that data follows a Gaussian mixture model in the asymptotic limit, where input dimensions, sample sizes, and hidden neurons increase proportionally.

Key Contributions

• Theoretical Framework: The paper establishes a comprehensive theoretical framework characterizing training and generalization errors for two-layer NNs under Gaussian mixtures with covariances that contain low-dimensional structures. This work leverages recent advances in Gaussian universality to offer insights into generalization capacities.
• Equivalent Polynomial Model: The authors demonstrate that under specific conditions, a high-degree polynomial model—referred to as the "Hermite model"—achieves performance equivalent to nonlinear neural networks. This equivalence hinges on the data spread and learning rate.
• Extensive Simulation Validation: Extensive simulations, including Fashion-MNIST classification, validate findings, showcasing the capabilities of their model in translating theoretical predictions to realistic datasets.

Methodological Insights

• Conditional Gaussian Equivalence: A notable methodological advancement is proving a conditional Gaussian equivalence, where feature maps from activation functions can be replaced with Gaussian counterparts, maintaining equivalent performance metrics. This significantly simplifies complex nonlinear settings into more tractable forms.
• Scaling Dynamics: By introducing parameters like the strength parameter $\beta$ and the weighting parameter $\alpha$, the paper effectively analyzes interdependencies between data spread and learning rates. Variations in these parameters elucidate how the intricacies of structured data influence learning outcomes.
• Hermite Expansion: This approach utilizes a Hermite polynomial expansion to approximate nonlinear activation functions, creating an equivalent performance model with reduced complexity, providing an insightful bridge between traditional neural network paradigms and polynomial approaches.

Simulation and Results

• Varying Complexity: The simulations reveal that increased data structure complexity, characterized by higher data spread, generally improves model performance, emphasizing the value of incorporating realistic data structures into model assumptions.
• Impact of Learning Rate: Interestingly, the results indicate that larger generalization benefits are observed with a higher data spread compared to solely increasing the learning rate. This finding indicates that structured data, as it informs feature learning, provides substantial improvements in neural network generalization.
• Realistic Application: The translation of theoretical results to Fashion-MNIST—a real data-driven task—demonstrates the robustness and applicability of their theoretical findings in a practical setting, further underscoring the model’s relevance beyond simulated environments.

Future Implications

The findings prompt further exploration into broader ranges of the strength parameter $\beta$, potentially extending analyses to efficaciously interpret neural network behavior in high-dimensional, structured data contexts beyond the current constraint of $\beta \leq 1$. Such expansions could provide more nuanced insights into model dynamics under differing scales of data complexity and learning paradigms.

Overall, this work contributes significantly to understanding feature learning dynamics in neural networks, particularly under realistic data representations. It bridges theoretical analysis with practical applicability, offering a guidepost for future studies in structured data environments.