Same Function, Different Trajectory: No Standard Optimizer Is Equivariant to Transformer Reparameterization

We analyze cumulative parameter trajectories of transformer training under AdamW and identify a dominant low-dimensional drift direction ("backbone") that captures 60--80% of long-horizon displacement from initialization. This direction is highly stable over rolling training windows yet reorients gradually across phases, particularly following objective reweighting. Per-batch gradients exhibit near-noise-floor alignment with the backbone, whereas optimizer-integrated updates align strongly with it, indicating that the structure emerges from accumulated optimizer dynamics rather than instantaneous gradient geometry. Replacing AdamW with SGD-family optimizers eliminates this structure, and reducing $β_2$ smoothly degrades backbone dominance and reheating recoverability. Reheating experiments show that transverse probe modes can be transiently re-excited without substantially altering accumulated backbone drift. These results provide a trajectory-level characterization of optimizer-induced geometric structure in transformer training and shift attention from instantaneous gradient properties to cumulative update dynamics.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce expert load imbalance in sparse MoE models, and in several cases control final vocabulary-logit growth, improve router stability, and overall training stability over the corresponding AdamW updates.

Task arithmetic, sequential fine-tuning, activation steering, and first-order random search all operate through relatively small perturbations around an already trained checkpoint, and they rely on different local approximations: individual perturbations should be first-order predictable, task updates should compose with controlled interference, useful tangent structure should be stable and possible to estimate, and weight edits should have counterparts in representation space. We measure 8 such properties with the same harness around a multitask LoRA operating point, on 9 transformers (82M-7B), with a prospectively registered property list, thresholds, and test split. We find a shared one-direction validity window up to the tested scale $10^{-2}$, but no universal radius for pairwise composition or update ordering. Along individual directions, changes of the probe loss remain first-order predictable throughout the grid: a perturbation's effect on the loss is essentially its projection onto the gradient, which is also what makes local random search work. Pairwise structure, however, proves to be far more fragile: on over a third of the measured (model, task pair) combinations, two-update order sensitivity sets in strictly inside that window; task-gradient subspaces rotate within tens of steps; additivity under our fixed activation probe fails at full task-vector scale on several models, including both held-out 7B models; and no model median passes the registered global mean-vector weight-to-steering correspondence bar. For two sequential task-gradient steps, the leading order-dependent term is the Lie bracket $H_B\textbf{g}_A-H_A\textbf{g}_B$; its normalized prediction $c(η)=ηκ+O(η^2)$ tracks the measured defect at median ratio 1.002, while the onset scale $η^\dagger\approx0.10/κ$ spans three orders of magnitude across models and task pairs.