Transformers Can Do Arithmetic with the Right Embeddings
Authors: name: Sean McLeish, name: Arpit Bansal, name: Alex Stein, name: Neel Jain, name: John Kirchenbauer, name: Brian R. Bartoldson, name: Bhavya Kailkhura, name: Abhinav Bhatele, name: Jonas Geiping, name: Avi Schwarzschild, name: Tom Goldstein
Organization: Tom Goldstein's Lab at University of Maryland, College Park

The poor performance of transformers on arithmetic tasks seems to stem in
large part from their inability to keep track of the exact position of each
digit inside of a large span of digits. We mend this problem by adding an
embedding to each digit that encodes its position relative to the start of the
number. In addition to the boost these embeddings provide on their own, we show
that this fix enables architectural modifications such as input injection and
recurrent layers to improve performance even further.
  With positions resolved, we can study the logical extrapolation ability of
transformers. Can they solve arithmetic problems that are larger and more
complex than those in their training data? We find that training on only 20
digit numbers with a single GPU for one day, we can reach state-of-the-art
performance, achieving up to 99% accuracy on 100 digit addition problems.
Finally, we show that these gains in numeracy also unlock improvements on other
multi-step reasoning tasks including sorting and multiplication.