Sparse Linear Networks with a Fixed Butterfly Structure: Theory and Practice
This work addresses the computational inefficiency of dense layers in neural networks for practitioners, though it is incremental as it builds on known butterfly network properties.
The authors tackled the problem of reducing the quadratic weight count in dense linear layers of neural networks by replacing them with a fixed butterfly structure, achieving nearly linear weight scaling with little loss in expressibility and matching or outperforming existing architectures in NLP and vision tasks while offering faster training and prediction.
A butterfly network consists of logarithmically many layers, each with a linear number of non-zero weights (pre-specified). The fast Johnson-Lindenstrauss transform (FJLT) can be represented as a butterfly network followed by a projection onto a random subset of the coordinates. Moreover, a random matrix based on FJLT with high probability approximates the action of any matrix on a vector. Motivated by these facts, we propose to replace a dense linear layer in any neural network by an architecture based on the butterfly network. The proposed architecture significantly improves upon the quadratic number of weights required in a standard dense layer to nearly linear with little compromise in expressibility of the resulting operator. In a collection of wide variety of experiments, including supervised prediction on both the NLP and vision data, we show that this not only produces results that match and at times outperform existing well-known architectures, but it also offers faster training and prediction in deployment. To understand the optimization problems posed by neural networks with a butterfly network, we also study the optimization landscape of the encoder-decoder network, where the encoder is replaced by a butterfly network followed by a dense linear layer in smaller dimension. Theoretical result presented in the paper explains why the training speed and outcome are not compromised by our proposed approach.