AlgebraNets

Jordan Hoffmann, Simon Schmitt, Simon Osindero, Karen Simonyan, Erich Elsen

Keywords: Sparsity, Pruning, Efficiency, Mathematics

Abstract: Neural networks have historically been built layerwise from the set of functions in ${f: \mathbb{R}^n \to \mathbb{R}^m }$, i.e. with activations and weights/parameters represented by real numbers, $\mathbb{R}$. Our work considers a richer set of objects for activations and weights, and undertakes a comprehensive study of alternative algebras as number representations by studying their performance on two challenging problems: large-scale image classification using the ImageNet dataset and language modeling using the enwiki8 and WikiText-103 datasets. We denote this broader class of models as AlgebraNets. Our findings indicate that the conclusions of prior work, which explored neural networks constructed from $\mathbb{C}$ (complex numbers) and $\mathbb{H}$ (quaternions) on smaller datasets, do not always transfer to these challenging settings. However, our results demonstrate that there are alternative algebras which deliver better parameter and computational efficiency compared with $\mathbb{R}$. We consider $\mathbb{C}$, $\mathbb{H}$, $M_{2}(\mathbb{R})$ (the set of $2\times2$ real-valued matrices), $M_{2}(\mathbb{C})$, $M_{3}(\mathbb{R})$, $M_{4}(\mathbb{R})$, dual numbers and the $\mathbb{R}^3$ cross product. Additionally, we note that multiplication in these algebras has higher compute density than real multiplication, a useful property in situations with inherently limited parameter reuse such as auto-regressive inference and sparse neural networks. We therefore investigate how to induce sparsity within AlgebraNets. We hope that our strong results on large-scale, practical benchmarks will spur further exploration of these unconventional architectures which challenge the default choice of using real numbers for neural network weights and activations.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

One-sentence Summary: We investigate a wide range of alternatives to real-valued weights in neural networks and find promising alternatives.

Community Implementations: [ 3 code implementations](https://www.catalyzex.com/paper/arxiv:2006.07360/code)

Reviewed Version (pdf): https://openreview.net/references/pdf?id=iaVGYcsIw