Task: Summarization
Input (English): 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.
Output (English): 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.