Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations (1710.10121v3)

Abstract: In our work, we bridge deep neural network design with numerical differential equations. We show that many effective networks, such as ResNet, PolyNet, FractalNet and RevNet, can be interpreted as different numerical discretizations of differential equations. This finding brings us a brand new perspective on the design of effective deep architectures. We can take advantage of the rich knowledge in numerical analysis to guide us in designing new and potentially more effective deep networks. As an example, we propose a linear multi-step architecture (LM-architecture) which is inspired by the linear multi-step method solving ordinary differential equations. The LM-architecture is an effective structure that can be used on any ResNet-like networks. In particular, we demonstrate that LM-ResNet and LM-ResNeXt (i.e. the networks obtained by applying the LM-architecture on ResNet and ResNeXt respectively) can achieve noticeably higher accuracy than ResNet and ResNeXt on both CIFAR and ImageNet with comparable numbers of trainable parameters. In particular, on both CIFAR and ImageNet, LM-ResNet/LM-ResNeXt can significantly compress ($>50$\%) the original networks while maintaining a similar performance. This can be explained mathematically using the concept of modified equation from numerical analysis. Last but not least, we also establish a connection between stochastic control and noise injection in the training process which helps to improve generalization of the networks. Furthermore, by relating stochastic training strategy with stochastic dynamic system, we can easily apply stochastic training to the networks with the LM-architecture. As an example, we introduced stochastic depth to LM-ResNet and achieve significant improvement over the original LM-ResNet on CIFAR10.

• The paper establishes that popular neural architectures, such as ResNet and PolyNet, mimic numerical discretizations of ODEs, linking theory with practice.
• It introduces a Linear Multi-step architecture inspired by numerical methods, achieving higher accuracy and fewer parameters on benchmarks like CIFAR and ImageNet.
• The study unifies stochastic learning strategies with differential equation frameworks, providing new insights into noise injection techniques for enhanced generalization.

Bridging Deep Learning Architectures and Numerical Differential Equations

The paper, "Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations," offers a novel perspective by establishing a connection between the architectures of deep neural networks and the discretizations of ordinary differential equations (ODEs). This interdisciplinary approach introduces a new lens for understanding and designing deep network architectures, contributing to more theoretically grounded and efficient models.

Key Insights

The authors propose that many widely-used neural network architectures, such as ResNet, PolyNet, FractalNet, and RevNet, can be interpreted as various discretizations of ODEs. For instance, ResNet's architecture mirrors a forward Euler discretization scheme. This observation is pivotal as it suggests that insights from numerical analysis can guide the design of neural network architectures, potentially leading to improvements in performance and efficiency.

Numerical Differential Equation and Network Design

The paper emphasizes the utility of numerical schemes in constructing neural networks:

• ResNet and Variants: Interpreted as forward Euler discretizations, aligning their architecture with a basic ODE model.
• PolyNet: Approximating a backward Euler scheme, which inherently provides stability advantages.
• FractalNet and RevNet: Embodying more complex schemes like the Runge-Kutta method, showcasing sophisticated discretizations that potentially enhance network expressiveness.

Linear Multi-step Architecture

Drawing directly from numerical analysis, the authors introduce the Linear Multi-step Architecture (LM-architecture). This structure is an adaptation of the linear multi-step method from numerical ODEs, applied to ResNet-like models to form LM-ResNet and LM-ResNeXt. Empirical results demonstrate that these derivatives achieve higher accuracy on challenging datasets like CIFAR and ImageNet, with a marked reduction in the number of parameters—a crucial efficiency boon.

Theoretical Explanations and Modified Equations

From a theoretical standpoint, the paper discusses modified equations to explain the performance boosts observed with the LM-architecture. The concept of modified equations from numerical analysis explains why certain numerical discretizations can lead to faster convergence and better generalization due to the inherent properties of the underlying dynamical systems. This connection underscores the significance of choosing appropriate discretization schemes when designing network architectures.

Stochastic Learning Strategies

In addition to structural innovations, the authors explore stochastic learning tactics by equating stochastic depth and shake-shake regularization with stochastic differential equations (SDEs). By interpreting these strategies through the lens of stochastic control, they provide a unified view of noise injection techniques used to enhance the generalization capabilities of deep networks.

Practical Implications and Future Directions

The practical implications of this work are substantial, offering a new pathway to principled design of neural architectures. By leveraging the robust body of knowledge in numerical analysis, future deep learning models can be crafted with greater theoretical support, potentially leading to improvements in both accuracy and computational efficiency.

Future research may extend this framework to other types of differential equations, exploring the rich territory of partial differential equations (PDEs) and their potential applicability in more complex, dynamic systems. Additionally, further exploration of stochastic dynamic systems could yield new insights into the optimization and training processes of deep networks.

In summary, this paper effectively bridges the disciplines of deep learning and numerical analysis, providing a compelling new perspective that promises to inform and enhance the design and training of neural networks in future investigations.