Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks

TL;DR

This paper develops a theoretical framework using Lagrangian mechanics to understand how symmetry breaking influences the learning dynamics, stability, and efficiency of neural networks, especially with normalization layers.

Contribution

It introduces the concept of kinetic symmetry breaking and generalizes Noether's theorem to neural network learning, revealing mechanisms of implicit adaptive optimization.

Findings

Kinetic symmetry breaking (KSB) explains stability and efficiency in neural networks.

Normalization layers induce implicit adaptive optimization similar to RMSProp.

The framework provides geometric principles for neural network learning dynamics.

Abstract

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the "geometry of learning dynamics" in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify "kinetic symmetry breaking" (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: "Noether's Learning Dynamics" (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of "implicit adaptive optimization", establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.