Length Generalization in Arithmetic Transformers

TL;DR

This paper investigates how transformer models can generalize to longer arithmetic sequences, proposing relative position embeddings for addition and a priming method for multiplication, with implications for broader sequence tasks.

Contribution

The paper introduces a priming technique that enables transformers to generalize to longer sequences in arithmetic tasks, extending beyond previous limitations.

Findings

Relative position embeddings enable length generalization in addition.

Priming with a few long sequences improves multiplication generalization.

Priming sample size scales logarithmically with training set size.

Abstract

We examine how transformers cope with two challenges: learning basic integer arithmetic, and generalizing to longer sequences than seen during training. We find that relative position embeddings enable length generalization for simple tasks, such as addition: models trained on $5$-digit numbers can perform $15$-digit sums. However, this method fails for multiplication, and we propose train set priming: adding a few ($10$ to $50$) long sequences to the training set. We show that priming allows models trained on $5$-digit $\times$ $3$-digit multiplications to generalize to $35\times 3$ examples. We also show that models can be primed for different generalization lengths, and that the priming sample size scales as the logarithm of the training set size. Finally, we discuss potential applications of priming beyond arithmetic.