Rethinking State Tracking in Recurrent Models Through Error Control Dynamics
For researchers designing recurrent architectures for long-range state tracking, this work identifies a fundamental limitation of affine models that is not captured by expressivity analysis alone.
The paper proves that affine recurrent networks (including State-Space Models and Linear Attention) cannot correct errors along state-separating subspaces, leading to finite-horizon tracking that collapses when the distinguishability ratio crosses a readability threshold. Empirical results on group state-tracking tasks show that this breakdown predicts downstream accuracy failure.
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.