Mean-field Analysis on Two-layer Neural Networks from a Kernel Perspective (2403.14917v2)

Abstract: In this paper, we study the feature learning ability of two-layer neural networks in the mean-field regime through the lens of kernel methods. To focus on the dynamics of the kernel induced by the first layer, we utilize a two-timescale limit, where the second layer moves much faster than the first layer. In this limit, the learning problem is reduced to the minimization problem over the intrinsic kernel. Then, we show the global convergence of the mean-field Langevin dynamics and derive time and particle discretization error. We also demonstrate that two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than any kernel methods, and neural networks acquire data-dependent kernel which aligns with the target function. In addition, we develop a label noise procedure, which converges to the global optimum and show that the degrees of freedom appears as an implicit regularization.

• The paper demonstrates that two-layer neural networks outperform traditional kernel methods by dynamically learning data-dependent kernels via mean-field analysis.
• It employs mean-field Langevin dynamics to obtain rigorous convergence guarantees and quantify discretization errors, evidencing enhanced feature learning efficiency.
• The study introduces a novel label noise procedure that implicitly regularizes the network, thereby optimizing convergence and managing learning complexity.

Mean-field Analysis on Two-layer Neural Networks from a Kernel Perspective

The paper "Mean-field Analysis on Two-layer Neural Networks from a Kernel Perspective" by Shokichi Takakura and Taiji Suzuki explores the feature learning capabilities of two-layer neural networks operating under the mean-field regime via the interpretive lens of kernel methods. This work explores a nuanced subdivision of neural network dynamics, using a framework wherein the second layer adapts rapidly in comparison to the first layer, framed within a two-timescale limit. Extensive theoretical contributions elucidate how two-layer neural networks surpass traditional kernel-based approaches by effectively learning data-dependent kernels and optimizing functional spaces that encompass multiple Reproducing Kernel Hilbert Spaces (RKHS).

The primary focus lies in analyzing the optimization pathway within the mean-field framework, wherein the intrinsic kernel's role is foregrounded. Takakura and Suzuki employ the mean-field Langevin dynamics to show that two-layer neural networks hold a profound advantage in learning efficiency, particularly over unions of RKHS, which conventional kernel methods are not equipped to model as effectively due to limitations in sample complexity. This investigation demonstrates how neural architectures trained through mean-field convergence embody substantially enhanced data alignment properties in kernel and parameter spaces alike.

Key Contributions and Findings

• Convexity and Dynamics Analyses: The paper begins by framing the convexity of the objective function in terms of the kernel induced by the first layer, showing global convergence through the mean-field Langevin dynamics. It provides a rigorous assessment of time and particle discretization errors within this context.
• Feature Learning in Neural Networks: It is illustrated that neural networks achieve a compelling advantage in learning feature representations, as indicated by superior sample complexity for tasks pertaining to a variant of Barron spaces. This is attributed to the innate capacity of neural networks for feature adaptation—a characteristic less pronounced in static kernel methods.
• Quantitative Convergence Guarantees: Research into the mean-field regime underlines robust convergence guarantees, where quantitative analyses reveal uniform-in-time results spanning the particle discretization error. This positions the mean-field approach as a more formidable contender in managing learning complexity over broader input spaces.
• Implicit Regularization through Label Noise: The research also offers innovative insights into intrinsic noise dynamics, introducing a label noise procedure that notably adjusts the degrees of freedom within neural architectures, ensuring convergence to globally optimal solutions that are regularized intrinsically.

Implications and Future Directions

In terms of theoretical implications, the findings provide a critical underpinning for understanding how neural network models can dynamically align their operational kernels with target functions. This improves their performance beyond what is typically achievable with fixed-kernel methods. Practically, the paper opens pathways for developing neural architectures that are particularly adept at learning from complex, high-dimensional data structures by leveraging mean-field methods. The proposed label noise procedure emerges as a promising strategy for enhancing regularization and generalization performance, marking a shift towards automated noise handling within learning models.

Looking ahead, the exploration of broader applications in domains such as high-dimensional pattern recognition, where RKHS and kernel methods often find applications, seems plausible. Moreover, extending the analysis to deeper neural networks or integrative multi-layer architectures could yield further enhancements in model performance, potentially bridging existing gaps between theoretical optimality and empirical efficiency in learning tasks.

This paper provides a vital contribution to the understanding of neural network feature learning, particularly in high-dimensional contexts, using a theoretical lens that combines kernel perspectives with mean-field analyses. It heralds a significant advancement in the quest to elucidate the intricate dynamics of gradient-based learning in neural systems.