Generalization Bounds for Neural Networks: Kernels, Symmetry, and Sample Compression
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Though Deep Neural Networks (DNNs) are widely celebrated for their practical performance, they demonstrate many intriguing phenomena related to depth that are difficult to explain both theoretically and intuitively. Understanding how weights in deep networks coordinate together across layers to form useful learners has proven somewhat intractable, in part because of the repeated composition of nonlinearities induced by depth. We present a reparameterization of DNNs as a linear function of a particular feature map that is locally independent of the weights. This feature map transforms depth-dependencies into simple tensor products and maps each input to a discrete subset of the feature space. Then, in analogy with logistic regression, we propose a max-margin assumption that enables us to present a so-called sample compression representation of the neural network in terms of the discrete activation state of neurons induced by s "support vectors". We show how the number of support vectors relate to learning guarantees for neural networks through sample compression bounds, yielding a sample complexity O(ns/ϵ) for networks with n neurons. Additionally, this number of support vectors has monotonic dependence on width, depth, and label noise for simple networks trained on the MNIST dataset.
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