An Uncertainty Principle for Linear Recurrent Neural Networks

TL;DR

This paper establishes an uncertainty principle for linear recurrent neural networks, showing a fundamental trade-off between the range of past information they can effectively utilize and the accuracy of the approximation.

Contribution

It provides a theoretical characterization of the limitations and capabilities of linear RNNs on a core copy task, including lower bounds and explicit optimal filters.

Findings

Derived lower bounds for approximation error in linear RNNs.

Constructed explicit filters that achieve the theoretical bounds.

Discovered an uncertainty principle relating filter range and accuracy.

Abstract

We consider linear recurrent neural networks, which have become a key building block of sequence modeling due to their ability for stable and effective long-range modeling. In this paper, we aim at characterizing this ability on a simple but core copy task, whose goal is to build a linear filter of order $S$ that approximates the filter that looks $K$ time steps in the past (which we refer to as the shift-$K$ filter), where $K$ is larger than $S$. Using classical signal models and quadratic cost, we fully characterize the problem by providing lower bounds of approximation, as well as explicit filters that achieve this lower bound up to constants. The optimal performance highlights an uncertainty principle: the optimal filter has to average values around the $K$-th time step in the past with a range~(width) that is proportional to $K/S$.