We present a new approach to understanding the relationship between loss curvature and generalisation in deep learning. Specifically, we use existing empirical analyses of the spectrum of deep network loss Hessians to ground an ansatz tying together the loss Hessian and the input-output Jacobian of a deep neural network. We then prove a series of theoretical results which quantify the degree to which the input-output Jacobian of a model approximates its Lipschitz norm over a data distribution, and deduce a novel generalisation bound in terms of the empirical Jacobian. We use our ansatz, together with our theoretical results, to give a new account of the recently observed progressive sharpening phenomenon, as well as the generalisation properties of flat minima. Experimental evidence is provided to validate our claims.

翻译：我们提出了一种理解深度学习损失曲率与泛化之间关系的新方法。具体而言，我们利用对深度网络损失海森矩阵谱的现有实证分析，为将深度神经网络的损失海森矩阵与输入-输出雅可比矩阵联系起来的假设提供了基础。随后，我们证明了一系列理论结果，量化了模型输入-输出雅可比矩阵在其数据分布上逼近利普希茨范数的程度，并推导出一个基于经验雅可比矩阵的新型泛化界。结合我们的假设与理论结果，我们为最近观察到的渐进锐化现象以及平坦极小值的泛化特性提供了新的解释。实验证据验证了我们的论断。