No fine-tuning, no cry: Robust svd for compressing deep networks

A common technique for compressing a neural network is to compute the k-rank ℓ₂ approximation A_k of the matrix A ∈ R^n×d via SVD that corresponds to a fully connected layer (or embedding layer). Here, d is the number of input neurons in the layer, n is the number in the next one, and A_k is stored in O((n + d)k) memory instead of O(nd). Then, a fine-tuning step is used to improve this initial compression. However, end users may not have the required computation resources, time, or budget to run this fine-tuning stage. Furthermore, the original training set may not be available. In this paper, we provide an algorithm for compressing neural networks using a similar initial compression time (to common techniques) but without the fine-tuning step. The main idea is replacing the k-rank ℓ₂ approximation with ℓ_p, for p ∈ [1, 2], which is known to be less sensitive to outliers but much harder to compute. Our main technical result is a practical and provable approximation algorithm to compute it for any p ≥ 1, based on modern techniques in computational geometry. Extensive experimental results on the GLUE benchmark for compressing the networks BERT, DistilBERT, XLNet, and RoBERTa confirm this theoretical advantage.