A Unifying View on Implicit Bias in Training Linear Neural Networks
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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. For L-layer linear tensor networks that are orthogonally decomposable, we show that gradient flow on separable classification finds a stationary point of the ℓ_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 ℓ_1 and ℓ_2 norms in the transformed input space. Our theorems subsume existing results in the literature while removing most of the convergence assumptions. We also provide experiments that corroborate our analysis.
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