A unifying view on implicit bias in training linear neural networksDownload PDF

Published: 12 Jan 2021, Last Modified: 05 May 2023ICLR 2021 PosterReaders: Everyone
Keywords: implicit bias, implicit regularization, convergence, gradient flow, gradient descent
Abstract: We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) on linear neural network training. We propose a tensor formulation of neural networks that includes fully-connected, diagonal, and convolutional networks as special cases, and investigate the linear version of the formulation called linear tensor networks. With this formulation, we can characterize the convergence direction of the network parameters as singular vectors of a tensor defined by the network. For $L$-layer linear tensor networks that are orthogonally decomposable, we show that gradient flow on separable classification finds a stationary point of the $\ell_{2/L}$ max-margin problem in a "transformed" input space defined by the network. For underdetermined regression, we prove that gradient flow finds a global minimum which minimizes a norm-like function that interpolates between weighted $\ell_1$ and $\ell_2$ norms in the transformed input space. Our theorems subsume existing results in the literature while removing standard convergence assumptions. We also provide experiments that corroborate our analysis.
One-sentence Summary: We propose a unifying framework for analyzing implicit bias of linear networks and show theorems that extend existing results with less convergence assumptions.
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