aiqbench analysis
This research investigates the internal algebraic mechanisms of neural networks, specifically how Transformers and recurrent models track group elements through sequential updates. The study reveals that while standard Transformers converge to solutions based on normal subgroups, parameter-matched recurrent networks can encode more complex non-normal cosets during training. These findings provide a rigorous mathematical framework for understanding what models actually learn beyond simple accuracy metrics in state tracking tasks.
Tracking States or Tracking Cosets? An Algebraic Account of Learned State Tracking
arXiv:2609.29951v1 Announce Type: new Abstract: State tracking requires composing a sequence of updates, but accuracy alone does not reveal what a model has learned. We study neural networks trained to predict the running product of group elements. We identify quotient solutions in Transformers, where models recover the quotient class while predicting nearly uniformly among its members. The reciprocal of class size predicts partial accuracy without a fitted parameter, extending parity-based accounts to non-parity quotients. Our baseline Transformers' predictions change little under prefix reordering beyond the exact-tracking frontier. We prove that, for finite groups under uniform i.i.d. full-group inputs, optimal order-blind exact accuracy converges to the reciprocal of abelianization class size as prefix length grows, consistent with the observed abelianization plateaus. Sequential updates permit more: any partition into right cosets of a subgroup, normal or not, survives sequential updates. In our census of standard Transformers, every recovered coset partition comes from a normal subgroup, whereas parameter-matched recurrent networks pass through both normal and non-normal right-coset stages during training. On $A_5$, we identify low-dimensional subspaces of the recurrent state that encode non-normal cosets. In the three-dimensional cases, coset mean vectors form approximate dodecahedra, and swapping the state components in these subspaces transfers the donor's coset state through a shared input suffix. Our results connect partial accuracy, learning stages, and internal computation through the subgroup cosets that models learn to track.
Title:Tracking States or Tracking Cosets? An Algebraic Account of Learned State Tracking
Abstract:State tracking requires composing a sequence of updates, but accuracy alone does not reveal what a model has learned. We study neural networks trained to predict the running product of group elements. We identify quotient solutions in Transformers, where models recover the quotient class while predicting nearly uniformly among its members. The reciprocal of class size predicts partial accuracy without a fitted parameter, extending parity-based accounts to non-parity quotients. Our baseline Transformers' predictions change little under prefix reordering beyond the exact-tracking frontier. We prove that, for finite groups under uniform i.i.d. full-group inputs, optimal order-blind exact accuracy converges to the reciprocal of abelianization class size as prefix length grows, consistent with the observed abelianization plateaus. Sequential updates permit more: any partition into right cosets of a subgroup, normal or not, survives sequential updates. In our census of standard Transformers, every recovered coset partition comes from a normal subgroup, whereas parameter-matched recurrent networks pass through both normal and non-normal right-coset stages during training. On $A_5$, we identify low-dimensional subspaces of the recurrent state that encode non-normal cosets. In the three-dimensional cases, coset mean vectors form approximate dodecahedra, and swapping the state components in these subspaces transfers the donor's coset state through a shared input suffix. Our results connect partial accuracy, learning stages, and internal computation through the subgroup cosets that models learn to track.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2609.29951 [cs.LG]
(or arXiv:2609.29951v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.29951 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 332 words
