No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training
Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of the maximal chordal distance sqrt(2r), at Pythia-160M with r=128): the apparent rotation at each refresh is dominated by estimator noise. This holds across four model families in three architecture classes from 70M to 6.9B parameters, strengthening wi...
| Authors | Noel Thomas |
|---|---|
| Primary category | cs.LG |
| Published | 7 July 2026 |
| Last updated at source | 19 July 2026 |
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