An empirical investigation of generalization dynamics in deep ReLU networks via nonlinear mode decomposition
Keywords: learning, generalization, statistical mechanics, teacher-student, svd
TL;DR: We measure nonlinear mode dynamics that fully capture learning and generalization of deep ReLU networks in a teacher-student framework
Abstract: The ability of deep networks to generalize, effectively learning some underlying nonlinear transform from noisy data, has long been investigated. The generalization dynamics of deep linear networks have previously been solved analytically and shown to qualitatively capture some aspects of how a linear transform is learned by a nonlinear network. Here we explore zero-bias deep ReLU networks playing both roles in a teacher-student framework, where measurement of the Jacobian of the network transform with respect to every input point allows for a complete description of the transform at those points, given the piecewise-linear nature of the ReLU network transform. The singular value/mode decomposition (SVD) of the Jacobian is computed at every input point for both teacher and student networks. The evolution over training of the singular values and vectors, averaged over all inputs, provides measurements of the globally nonlinear behavior of the network transform. For a deep ReLU student network trained on data from a deep ReLU teacher network with a user-specified singular value spectrum, we show over the course of training increasing student singular value magnitudes and increasing alignments of student singular vectors with teacher singular vectors, as observed in deep linear networks. We decompose the loss over training by singular mode, and directly observe nonlinear coupling of noise to student signal singular modes as well as coupling due to competition between signal modes. Nonlinear modes are shown to occur in teacher-student scenarios for deep convolutional ReLU networks with MNIST data, where their singular vectors reveal interpretable features that the network learns at local and global levels.
Primary Area: learning theory
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Submission Number: 8473
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