On progressive sharpening, flat minima and generalisation
This work addresses the problem of understanding generalization in deep learning for researchers, offering incremental theoretical insights based on existing empirical analyses.
The authors tackled the relationship between loss curvature and model behavior in deep neural networks, proving theoretical results that link the input-output Jacobian to generalization and providing a new explanation for progressive sharpening and flat minima, with experimental validation.
We present a new approach to understanding the relationship between loss curvature and input-output model behaviour 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 over training samples during the training of deep neural networks. 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.