Anomalous diffusion dynamics of learning in deep neural networks.

Learning in deep neural networks (DNNs) is implemented through minimizing a highly non-convex loss function, typically by a stochastic gradient descent (SGD) method. This learning process can effectively find generalizable solutions at flat minima. In this study, we present a novel account of how such effective deep learning emerges through the interactions of the SGD and the geometrical structure of the loss landscape. We find that the SGD exhibits rich, complex dynamics when navigating through the loss landscape; initially, the SGD exhibits superdiffusion, which attenuates gradually and changes to subdiffusion at long times when approaching a solution. Such learning dynamics happen ubiquitously in different DNN types such as ResNet, VGG-like networks and Vision Transformers; similar results emerge for various batch size and learning rate settings. The superdiffusion process during the initial learning phase indicates that the motion of SGD along the loss landscape possesses intermittent, big jumps; this non-equilibrium property enables the SGD to effectively explore the loss landscape. By adapting methods developed for studying energy landscapes in complex physical systems, we find that such superdiffusive learning processes are due to the interactions of the SGD and the fractal-like regions of the loss landscape. We further develop a phenomenological model to demonstrate the mechanistic role of the fractal-like loss landscape in enabling the SGD to effectively find flat minima. Our results reveal the effectiveness of SGD in deep learning from a novel perspective and have implications for designing efficient deep neural networks.