## Gradient Descent Happens in a Tiny Subspace

27 Sep 2018 (modified: 21 Dec 2018)ICLR 2019 Conference Blind SubmissionReaders: Everyone
• Abstract: We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.
• Keywords: Gradient Descent, Hessian, Deep Learning
• TL;DR: For classification problems with k classes, we show that the gradient tends to live in a tiny, slowly-evolving subspace spanned by the eigenvectors corresponding to the k-largest eigenvalues of the Hessian.
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