Positional Attention: Expressivity and Learnability of Algorithmic Computation

TL;DR

This paper investigates how Transformers using only positional attention can execute algorithms, demonstrating their expressivity, learnability, and practical performance, especially for tasks relying solely on positional information.

Contribution

It proves that positional attention in Transformers retains the same expressivity as parallel models and analyzes the learnability and sample complexity trade-offs involved.

Findings

Positional Transformers match the expressivity of parallel computational models.

They exhibit a logarithmic depth cost for algorithmic execution.

Empirically, they perform well on tasks relying on positional information.

Abstract

There is a growing interest in the ability of neural networks to execute algorithmic tasks (e.g., arithmetic, summary statistics, and sorting). The goal of this work is to better understand the role of attention in Transformers for algorithmic execution. Its importance for algorithmic execution has been studied theoretically and empirically using parallel computational models. Notably, many parallel algorithms communicate between processors solely using positional information. Inspired by this observation, we investigate how Transformers can execute algorithms using positional attention, where attention weights depend exclusively on positional encodings. We prove that Transformers with positional attention (positional Transformers) maintain the same expressivity of parallel computational models, incurring a logarithmic depth cost relative to the input length. We analyze their in-distribution learnability and explore how parameter norms in positional attention affect sample complexity. Our results show that positional Transformers introduce a learning trade-off: while they exhibit better theoretical dependence on parameter norms, certain tasks may require more layers, which can, in turn, increase sample complexity. Finally, we empirically explore the out-of-distribution performance of positional Transformers and find that they perform well in tasks where their underlying algorithmic solution relies on positional information.