Rethinking State Tracking in Recurrent Models Through Error Control Dynamics

TL;DR

This paper challenges the focus on expressive capacity in recurrent models, emphasizing the importance of error control dynamics for robust state tracking, and demonstrates that affine recurrent networks have fundamental limitations in error correction.

Contribution

It introduces a theoretical framework highlighting error control as crucial for state tracking, revealing limitations of affine recurrent networks in error correction and robustness.

Findings

Affine recurrent networks cannot correct errors along state-separating subspaces.

Tracking failure occurs when within-class spread exceeds initial between-class separation.

Empirical results predict tracking collapse based on distinguishability ratio crossing a threshold.

Abstract

The theory of state tracking in recurrent architectures has predominantly focused on expressive capacity: whether a fixed architecture can theoretically realize a set of symbolic transition rules. We argue that equally important is error control, the dynamics governing hidden-state drift along the directions that distinguish symbolic states. We prove that affine recurrent networks, a class of models encompassing State-Space Models and Linear Attention, cannot correct errors along state-separating subspaces once they preserve state representations. Consequently, practical affine trackers do not learn robust state tracking; rather, they learn finite horizon solutions governed by accumulated state-relevant error. We characterize the mechanics of this failure, showing that tracking remains readable only while the accumulating within-class spread remains small relative to the initial between-class separation. We demonstrate empirically on group state-tracking tasks that this breakdown is predictable: tracking collapses when the distinguishability ratio crosses the readability threshold of the trained decoder. Across trained models, the point of this crossing predicts the horizon at which downstream accuracy fails. These results establish that robust state tracking is determined not only by an architecture's theoretical expressivity but crucially by its error control.