Implicit regularization of deep residual networks towards neural ODEs (2309.01213v3)

Abstract: Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous models still lacks a solid mathematical foundation. In this article, we take a step in this direction by establishing an implicit regularization of deep residual networks towards neural ODEs, for nonlinear networks trained with gradient flow. We prove that if the network is initialized as a discretization of a neural ODE, then such a discretization holds throughout training. Our results are valid for a finite training time, and also as the training time tends to infinity provided that the network satisfies a Polyak-Lojasiewicz condition. Importantly, this condition holds for a family of residual networks where the residuals are two-layer perceptrons with an overparameterization in width that is only linear, and implies the convergence of gradient flow to a global minimum. Numerical experiments illustrate our results.

• The paper shows that residual networks, when properly scaled and initialized, converge to neural ODE trajectories even within finite training times.
• The analysis proves that under a Polyak-Łojasiewicz condition for wide networks, gradient flow achieves global minimization, ensuring stable convergence.
• Numerical experiments on synthetic Gaussian data and CIFAR-10 validate the theoretical claims, highlighting practical benefits for network design and training stability.

Implicit Regularization of Deep Residual Networks towards Neural ODEs

The paper under discussion explores the relationship between residual neural networks and neural ordinary differential equations (ODEs), focusing on implicit regularization. The authors investigate the convergence of deep residual networks toward neural ODEs under specific training regimes and initialization conditions, providing both theoretical insights and experimental validations.

Residual networks (ResNets) are widely recognized for their ability to train very deep networks successfully, primarily due to their skip connections. The concept of neural ODEs extends ResNets to the continuous domain, offering a sophisticated framework for understanding deep learning models with infinitely many layers. However, a formal mathematical connection between these discrete and continuous frameworks has remained largely unexplored until now.

This paper provides a rigorous mathematical foundation showing that residual networks, when properly scaled and initialized, converge to neural ODEs during training. To achieve this, the authors use the notion of implicit regularization in gradient-based learning and establish conditions under which the weights of a residual network adhere to a Lipschitz continuous trajectory, aligning with neural ODE dynamics.

Key contributions of the paper include:

1. Finite Training Time Limit: The authors demonstrate that residual networks trained with a gradient flow, for a fixed time horizon, maintain a structure akin to neural ODEs. Even with finite training duration, they establish convergence in the large-depth limit, where the networks' weights discretize a neural ODE trajectory.
2. Polyak-Łojasiewicz Condition for Long-Time Limit: For sufficiently wide networks, the paper extends the analysis to the infinite training time scenario, proving that gradient flow finds a global minimum under a Polyak-Łojasiewicz (PL) condition. This is a significant result, providing reassurance on convergence properties without falling into local minima traps.
3. Numerical Validation: The theoretical results are supported by numerical experiments on both synthetic Gaussian data and real-world datasets such as CIFAR-10. These experiments confirm the theoretical findings and demonstrate that neural networks with smooth activations and specific initialization schemes maintain their discretized neural ODE structure post-training.

The paper reveals implications for both theory and practice. Theoretically, it provides a framework to understand deep residual networks in terms of dynamical systems, enriching the discussion of network expressivity, generalization, and optimization dynamics. Practically, the results suggest initialization and scaling strategies for designing residual networks that can potentially harness the theoretical benefits of neural ODEs, such as improved interpretability, memory efficiency, and training stability.

Although the experimental results raise questions about the performance trade-offs when adhering strictly to a neural ODE structure, they offer a compelling case for further investigation. For future work, extending these effects to i.i.d. initialization or non-smooth activations such as ReLU could widen the applicability of these theoretical insights. Moreover, exploring other architectures like convolutional or transformer networks under this framework could reveal additional benefits and considerations.

In summary, this paper advances our understanding of the implicit regularization landscape in deep learning by positioning deep residual networks within the neural ODE paradigm. It opens up avenues for potential enhancements in the design and training of deep networks, grounded in a solid mathematical foundation.