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Early Directional Convergence in Deep Homogeneous Neural Networks for Small Initializations

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arxiv 2403.08121 v3 pith:UWYAUCWT submitted 2024-03-12 cs.LG math.OCstat.ML

classification cs.LGmath.OCstat.ML
keywords neuralnetworkspointssmallcorrelationdeepearlyfunction
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This paper studies the gradient flow dynamics that arise when training deep homogeneous neural networks assumed to have locally Lipschitz gradients and an order of homogeneity strictly greater than two. It is shown here that for sufficiently small initializations, during the early stages of training, the weights of the neural network remain small in (Euclidean) norm and approximately converge in direction to the Karush-Kuhn-Tucker (KKT) points of the recently introduced neural correlation function. Additionally, this paper also studies the KKT points of the neural correlation function for feed-forward networks with (Leaky) ReLU and polynomial (Leaky) ReLU activations, deriving necessary and sufficient conditions for rank-one KKT points.

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Cited by 2 Pith papers

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  1. The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks

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  2. An overview of condensation phenomenon in deep learning

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    Neural networks exhibit condensation of neurons into clusters with similar outputs whose number increases monotonically during training, facilitated by small initializations or dropout, providing insights into general...

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