Stochastic Gradient Descent for Two-layer Neural Networks (2407.07670v1)

Abstract: This paper presents a comprehensive study on the convergence rates of the stochastic gradient descent (SGD) algorithm when applied to overparameterized two-layer neural networks. Our approach combines the Neural Tangent Kernel (NTK) approximation with convergence analysis in the Reproducing Kernel Hilbert Space (RKHS) generated by NTK, aiming to provide a deep understanding of the convergence behavior of SGD in overparameterized two-layer neural networks. Our research framework enables us to explore the intricate interplay between kernel methods and optimization processes, shedding light on the optimization dynamics and convergence properties of neural networks. In this study, we establish sharp convergence rates for the last iterate of the SGD algorithm in overparameterized two-layer neural networks. Additionally, we have made significant advancements in relaxing the constraints on the number of neurons, which have been reduced from exponential dependence to polynomial dependence on the sample size or number of iterations. This improvement allows for more flexibility in the design and scaling of neural networks, and will deepen our theoretical understanding of neural network models trained with SGD.

• The paper establishes sharp convergence rates for the last SGD iterate using NTK approximations in an RKHS framework.
• It reduces the neuron dependency from exponential to polynomial relative to sample size or iterations, enhancing scalability.
• Numerical and theoretical insights bridge neural network models with kernel methods to guide efficient training in overparameterized settings.

Stochastic Gradient Descent for Two-layer Neural Networks

This paper provides a comprehensive analysis of the convergence rates for the Stochastic Gradient Descent (SGD) algorithm applied to overparameterized two-layer neural networks. The paper combines the Neural Tangent Kernel (NTK) approximation with convergence analysis in the Reproducing Kernel Hilbert Space (RKHS) generated by the NTK. The analysis aims to elucidate the convergence behavior of SGD in such networks, achieving sharp convergence rates for the last iterate of the algorithm and relaxing constraints on the number of neurons by reducing their exponential dependence to polynomial dependence on the sample size or number of iterations.

Theoretical Framework and Key Contributions

The theoretical framework adopted in the paper involves formulating the estimation of the regression function in an overparameterized two-layer neural network under the NTK regime. This perspective aids in bridging the neural network models with kernel methods, providing insights into their dynamics and generalization properties. The NTK, introduced by Jacot et al., is instrumental in understanding the behavior of deep neural networks. It approximates the network's objective function linearly within a small neighborhood, facilitating the convergence analysis in RKHS.

The primary contributions of this paper can be summarized as follows:

1. Convergence Analysis: The paper presents a robust convergence analysis of SGD in the NTK-generated RKHS. It incorporates a priori conditions from the convergence analysis of SGD in RKHS, like the capacity of the space and the regularity conditions on the objective function's smoothness.
2. Sharp Convergence Rates: The paper proves that the convergence rates of the last iterate of SGD are ${O}(T^{-\frac{2r}{2r + 1}})$ or ${O}(T^{-\frac{2r}{2r + 1} + \epsilon})$ in two distinct scenarios, where $0 < \epsilon < \frac{2r}{2r + 1}$ and $r$ characterizes the regularity of the target function.
3. Reduction in Neuron Dependency: The number of neurons required in the network is reduced from exponential to polynomial dependence on the sample size or number of iterations.

Numerical and Theoretical Implications

The paper’s numerical results and theoretical implications signify a breakthrough in the optimization of overparameterized neural networks trained with SGD. The main results demonstrate significant improvements over existing literature concerning the dependency on the number of neurons. For instance, previous works indicated an exponential dependency of neurons for achieving similar convergence rates. The current results achieve near-optimal rates with a polynomial dependency, making it feasible to implement these strategies in practical, large-scale neural network applications without the need for an infeasibly large number of neurons.

Future Directions

While the current results are promising, several avenues for future research can be pursued:

1. Extension to Deep Networks: Extending the developed theoretical framework from two-layer neural networks to deeper architectures, such as ResNet or Transformer models, could provide insights into optimization strategies for more complex network structures.
2. Empirical Validation: Conducting empirical studies to validate the theoretical convergence rates and neuron dependency constraints in practical, real-world datasets and applications could further substantiate these findings.
3. Algorithmic Variants: Exploring other variants of the SGD algorithm, such as the Averaged SGD (ASGD) or mini-batch SGD, might reveal improved convergence properties and practical performance enhancements.

Conclusion

The paper establishes a foundational theoretical framework and sets the stage for future advancements in the optimization of overparameterized neural networks. By achieving sharp convergence rates and significantly relaxing the neuron dependency constraints, this paper deepens our understanding of the interplay between kernel methods and optimization processes within neural networks trained by SGD.

This research is poised to influence both theoretical studies and practical applications in the field of machine learning, particularly in optimizing large-scale neural network models efficiently.