The local geometry of high dimensional neural network loss landscapes can both challenge our cherished theoretical intuitions as well as dramatically impact the practical success of neural network training. Indeed recent works have observed 4 striking local properties of neural loss landscapes on classification tasks: (1) the landscape exhibits exactly $C$ directions of high positive curvature, where $C$ is the number of classes; (2) gradient directions are largely confined to this extremely low dimensional subspace of positive Hessian curvature, leaving the vast majority of directions in weight space unexplored; (3) gradient descent transiently explores intermediate regions of higher positive curvature before eventually finding flatter minima; (4) training can be successful even when confined to low dimensional {\it random} affine hyperplanes, as long as these hyperplanes intersect a Goldilocks zone of higher than average curvature. We develop a simple theoretical model of gradients and Hessians, justified by numerical experiments on architectures and datasets used in practice, that {\it simultaneously} accounts for all $4$ of these surprising and seemingly unrelated properties. Our unified model provides conceptual insights into the emergence of these properties and makes connections with diverse topics in neural networks, random matrix theory, and spin glasses, including the neural tangent kernel, BBP phase transitions, and Derrida's random energy model.

本文通过实验和理论研究了神经网络的波动，发现高维神经网络的损失函数曲面具有多方向高正曲率、梯度下降具有狭窄、随机位于此曲面中不同位置处的超平面理论能够解释背后的机理。

神经损失景观的局部几何的新兴特性