Subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.
Abstract
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 with scale, and more weakly in a vision transformer. Only ~39 of 128 directions are reproducible across minibatches, and averaging cannot recover the rest: under N-fold averaging the gradient's spectral tail shrinks as N^(-1/4) rather than the N^(-1/2) of pure noise, so no averaging budget makes the subspace well defined. What helps instead follows from treating each refresh as a change of coordinates for Adam's state. Carrying the second moment blindly is provably about (r-k*)/2 worse than the best rotation-blind estimator, while the first moment transports exactly through the rotation, the optimal linear map under isotropic gradients and the rule LDAdam uses. At 1B over 40k steps (3 seeds), full LDAdam reaches 18.7 perplexity at beta2=0.999, beating untransported GaLore after its best beta2 fix (19.3); shortening the second-moment memory to beta2=0.99 helps the refreshing optimizers, though for canonical GaLore the effect is small and a full-rank control reverses it. One measurable fact, subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.
Experiments show that SubZero+, an improved SubZero framework that improves stability in three complementary ways, consistently outperforms prior ZO baselines, enlarges the stable learning-rate range, and narrows the gap to first-order methods with minimal extra memory overhead.
Ziming Yu, Shu-Yao Xiao, Xingyu Zhao et al.· 0 citations
This work intervenes on one factor at a time inside a fixed pipeline, holding candidate scoring and voting constant while they vary the search dimension, the subspace that carries the perturbation, and its norm.
Taeyeon Kim, Ahhyun Kim, Taehyeon Kim et al.· 0 citations
It is shown that a principled design space for LoRA initialization and curvature preconditioning should be treated as a tunable dimension rather than a fixed design decision, and a deployable, search-free variant, ULoRA-Auto, selects per-layer exponents from measured spectral statistics, approaches this upper bound at no additional search cost, and ranks at or near the top among deployable LoRA methods.
Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum,"shared-scalar"Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A"spectral schedule"reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.
Low-rank adaptation (LoRA) has become the standard for parameter-efficient fine-tuning of large language models. Most LoRA variants follow a uniform-LR convention, applying a single global learning rate across every rank-one component of every adapter. We show that this convention overlooks substantial within-module heterogeneity, where the rank-one components of a LoRA adapter update at highly uneven rates and low-velocity modules converge to concentrated singular spectra that underutilize the nominal rank budget. To address this, we propose an adaptive anisotropic learning-rate model that assigns each rank-one component its own effective learning rate, computed online from training-time signals and mean-normalized per module to preserve the global LR budget. AnLR-LoRA instantiates this model with two signals available during AdamW optimization, namely function-space velocity and Adam SNR, as a lightweight scheme with no extra trainable parameters. Across commonsense reasoning, natural language generation and visual instruction-tuning benchmarks, AnLR-LoRA consistently improves over LoRA while encouraging broader use of rank capacity, with gains that remain robust across a wide range of global learning rates and transfer cleanly to other LoRA variants.
Huiyi Wang, Daijiao Liu, Lina Yao et al.· 0 citations
This work proposes extracting mechanism mounts directly from linear sites by column-tiled SVD: each mount is a triple read as trigger, write, and strength, and evaluates mounts with a pre-registered suite judged on full-write energy lift rather than tile-local lift.
Ash Manvi, Samreena Tajreen· 0 citations
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