Same Function, Different Trajectory: No Standard Optimizer Is Equivariant to Transformer Reparameterization
We show that two Transformers representing the same function can follow different training trajectories. For example, doubling a query matrix and halving its key matrix preserves the model's function but changes the gradient scales seen by the optimizer. Our theoretical analysis establishes that standard block-local first-order optimizers fail to scale updates with parameters under a shared learning rate. For AdamW, transforming the first- and second-moment estimates alone does not correct the mismatch between parameter and update scales. To make runs restarted from equivalent checkpoints follow the same trajectory, we balance matrix pairs and transform their moment estimates together, br...
| Authors | Bum Jun Kim |
|---|---|
| Primary category | cs.LG |
| Published | 13 October 2026 |
| Last updated at source | 13 October 2026 |
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