Curvature in the Looking-Glass: Optimal Methods to Exploit Curvature of Expectation in the Loss Landscape

Abstract

Harnessing the local topography of the loss landscape is a central challenge in advanced optimization tasks. By accounting for the effect of potential parameter changes, we can alter the model more efficiently. Contrary to standard assumptions, we find that the Hessian does not always approximate loss curvature well, particularly near gradient discontinuities, which commonly arise in deep learning architectures. We present a new conceptual framework to understand how curvature of expected changes in loss emerges in architectures with many rectified linear units. Each ReLU creates a parameter boundary that, when crossed, induces a pseudorandom gradient perturbation. Our derivations show how these discontinuities combine to form a glass-like structure, similar to amorphous solids that contain microscopic domains of strong, but random, atomic alignment. By estimating the density of the resulting gradient variations, we can bound how the loss may change with parameter movement. Our analysis includes the optimal kernel and sample distribution for approximating glass density from ordinary gradient evaluations. We also derive the optimal modification to quasi-Newton steps that incorporate both glass and Hessian terms, as well as certain exactness properties that are possible with Nesterov-accelerated gradient updates. Our algorithm, Alice, tests these techniques to determine which curvature terms are most impactful for training a given architecture and dataset. Additional safeguards enforce stable exploitation through step bounds that expand on the functionality of Adam. These theoretical and experimental tools lay groundwork to improve future efforts (e.g., pruning and quantization) by providing new insight into the loss landscape.