A Riemannian Optimization Perspective of the Gauss-Newton Method for Feedforward Neural Networks (2412.14031v4)

Abstract: We analyze the convergence of Gauss-Newton dynamics for training neural networks with smooth activation functions. In the underparameterized regime, the Gauss-Newton gradient flow induces a Riemannian gradient flow on a low-dimensional, smooth, embedded submanifold of the Euclidean output space. Using tools from Riemannian optimization, we prove \emph{last-iterate} convergence of the Riemannian gradient flow to the optimal in-class predictor at an \emph{exponential rate} that is independent of the conditioning of the Gram matrix, \emph{without} requiring explicit regularization. We further characterize the critical impacts of the neural network scaling factor and the initialization on the convergence behavior. In the overparameterized regime, we show that the Levenberg-Marquardt dynamics with an appropriately chosen damping schedule yields fast convergence rate despite potentially ill-conditioned neural tangent kernel matrices, analogous to the underparameterized regime. These findings demonstrate the potential of Gauss-Newton methods for efficiently optimizing neural networks in the near-initialization regime, particularly in ill-conditioned problems where kernel and Gram matrices have small singular values.

• The paper demonstrates that in the underparameterized regime, the Gauss-Newton gradient flow acts as a Riemannian gradient flow, achieving exponential convergence without explicit regularization.
• For overparameterized networks, the research proposes using Levenberg-Marquardt dynamics, showing convergence rate independence from neural tangent kernel singular values.
• These findings suggest that leveraging Gauss-Newton and Riemannian optimization improves neural network training efficiency, especially in ill-conditioned scenarios, and opens avenues for future geometric method research.

Gauss-Newton Dynamics for Neural Networks: A Riemannian Optimization Perspective

The research paper titled "Gauss-Newton Dynamics for Neural Networks: A Riemannian Optimization Perspective" offers a detailed exploration of the implementation and efficiency of the Gauss-Newton method within the domain of neural network training. Specifically, the paper examines the convergence properties of this approach in the context of smooth activation functions, focusing on both underparameterized and overparameterized regimes.

Key Contributions

The paper makes several significant contributions to the current understanding of optimization in neural networks:

1. Underparameterized Regime and Riemannian Optimization:
   • The work demonstrates that in an underparameterized regime, the Gauss-Newton gradient flow exhibits Riemannian gradient flow behavior by influencing a low-dimensional, smooth submanifold of the Euclidean output space. Importantly, exponential convergence to the optimal in-class predictor is established without explicit regularization.
   • The research elucidates the role of neural network scaling factors and initialization, showing that convergence rates are unaffected by the conditioning of the Gram matrix.
2. Overparameterized Regime and Levenberg-Marquardt Dynamics:
   • For overparameterized networks, where the Jacobian Gram matrix is rank-deficient, the paper proposes that the Levenberg-Marquardt dynamics with appropriately chosen damping factors provide robustness to ill-conditioned kernels.
   • A significant result is the independence of the convergence rate from the singular values of the neural tangent kernel matrix, a property not shared by first-order methods.

These findings suggest that Gauss-Newton methods can substantially improve the efficiency of neural network optimization, particularly in scenarios involving ill-conditioned problems.

Theoretical Implications

The paper capitalizes on the concepts from Riemannian optimization to establish new convergence results which are not only theoretical nuances but also potential improvements over traditional approaches like stochastic gradient descent (SGD). The use of Riemannian tools to analyze underparameterized networks is a distinctive approach, demonstrating applications beyond conventional Euclidean spaces and providing insights into the intrinsic geometry of neural networks.

Numerical Results and Insights

The paper supports its theoretical claims with numerical experiments, showing the practicality of the Gauss-Newton dynamics under various scaling factors. The experiments correlate with the theoretical assertions, illustrating the efficiency of the proposed method by comparing it to standard gradient descent techniques without preconditioning.

Implications for Future Research

This research opens several avenues for future work. The exploration of Gauss-Newton methods in other training regimes, such as the rich regime, and their implications on optimization performance, presents an interesting direction. Moreover, extending these techniques to multi-layer networks or networks with discrete activation functions could expand their applicability.

The paper convincingly argues for the benefit of considering manifold geometry in neural network optimization, encouraging further investigation of geometric methods in artificial intelligence. As the field advances, such approaches may prove crucial in harnessing the full potential of deep learning models, especially in complex, high-dimensional spaces where traditional methods fall short.

In summary, this paper presents a robust analysis of Gauss-Newton methods within neural network training, highlighting significant theoretical and practical implications. This approach's ability to offer a controlled convergence without explicit regularization sets a foundation for leveraging manifold structures in machine learning, potentially transforming how optimization in neural networks is approached.