The Loss Surfaces of Neural Networks with General Activation Functions

Abstract

The loss surfaces of deep neural networks have been the subject of several studies, theoretical and experimental, over the last few years. One strand of work considers the complexity, in the sense of local optima, of high dimensional random functions with the aim of informing how local optimisation methods may perform in such complicated settings. Prior work of Choromanska et al (2015) established a direct link between the training loss surfaces of deep multi-layer perceptron networks and spherical multi-spin glass models under some very strong assumptions on the network and its data. In this work, we test the validity of this approach by removing the undesirable restriction to ReLU activation functions. In doing so, we chart a new path through the spin glass complexity calculations using supersymmetric methods in Random Matrix Theory which may prove useful in other contexts. Our results shed new light on both the strengths and the weaknesses of spin glass models in this context.

50 pages, 11 figures; references added for Kac-Rice reduction to RMT method; updates following JSTAT review and publication

References in corpus (5)

• On Calibration of Modern Neural Networks
• The Loss Surfaces of Multilayer Networks
• An Investigation into Neural Net Optimization via Hessian Eigenvalue Density
• Explorations on high dimensional landscapes
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

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• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
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• A spin-glass model for the loss surfaces of generative adversarial networks
• Local convexity of the TAP free energy and AMP convergence for Z2-synchronization
• Universal characteristics of deep neural network loss surfaces from random matrix theory
• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
• Appearance of Random Matrix Theory in Deep Learning
• Optimization landscape in the simplest constrained random least-square problem