The Geometry of Updates: Fisher Alignment at Vocabulary Scale

arXiv:2606.27242 · cs.LG, cs.CL, stat.ML · Submitted 2026-06-25 · Read on arXiv

cs.LG, cs.CL, stat.ML

Submitted: 2026-06-25

Updated: 2026-06-25

Comments: Accepted at the 43rd International Conference on Machine Learning (ICML 2026), PMLR 306. 64 pages total (main paper plus appendix), 4 figures, 29 tables

Code: https://github.com/google/svcca

License: http://creativecommons.org/licenses/by/4.0/

The gist: Training-free source selection for LLM families with shared vocabularies arises in scientific string domains such as SMILES, protein, and genomic sequences, where candidate corpora share a tokenizer

Terminology

Abstract

Training-free source selection for LLM families with shared vocabularies arises in scientific string domains such as SMILES, protein, and genomic sequences, where candidate corpora share a tokenizer but differ in prediction targets. This creates an activation-dark regime: representation-similarity metrics can be uninformative without assumptions about label-conditioned error geometry, while classical update-geometry metrics are computationally prohibitive at vocabulary scale. We show that, in a shared-output head setting, representation metrics (e.g., CKA) are non-identifiable for transfer; models can share identical representations yet have orthogonal head updates. The key identity is that head Fisher alignment is exactly a cosine between kernel mean embeddings in the joint activation-error space, exposing activation, error, and coupling factors rather than requiring a materialized Fisher matrix. FisherSketch estimates this cosine directly in a single streaming pass, making K=128,256 head Fisher alignment practical with a 16 KB task signature (m=4096) and a 192 KB per-task streaming state, small enough to store next to a model hash, but encoding transfer-relevant update structure. Beyond source selection, the same signatures and marginals provide a diagnostic instrument for studying whether LLM task similarity is driven by activations, errors, or their coupling; shared-parameter and internal-layer validations, together with Llama-3.1-8B verbalizer-shift experiments, show that FisherSketch remains informative when activation similarity cannot distinguish tasks.

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