In this work we explore the limiting dynamics of deep neural networks trained with stochastic gradient descent (SGD). We find empirically that long after performance has converged, networks continue to move through parameter space by a process of anomalous diffusion in which distance travelled grows as a power law in the number of gradient updates with a nontrivial exponent. We reveal an intricate interaction between the hyperparameters of optimization, the structure in the gradient noise, and the Hessian matrix at the end of training that explains this anomalous diffusion. To build this understanding, we first derive a continuous-time model for SGD with finite learning rates and batch sizes as an underdamped Langevin equation. We study this equation in the setting of linear regression, where we can derive exact, analytic expressions for the phase space dynamics of the parameters and their instantaneous velocities from initialization to stationarity. Using the Fokker-Planck equation, we show that the key ingredient driving these dynamics is not the original training loss, but rather the combination of a modified loss, which implicitly regularizes the velocity, and probability currents, which cause oscillations in phase space. We identify qualitative and quantitative predictions of this theory in the dynamics of a ResNet-18 model trained on ImageNet. Through the lens of statistical physics, we uncover a mechanistic origin for the anomalous limiting dynamics of deep neural networks trained with SGD.

研究SGD训练的深度神经网络在性能收敛后的步长限制动态，揭示了优化超参数、梯度噪声结构及训练结束时Hessian矩阵之间错综复杂的相互作用，通过统计物理学的视角解释这种异常扩散现象并在ImageNet数据集的ResNet-18模型上得到了实证验证。

SGD的极限动力学:修改的损失，相空间振荡和反常扩散