Optimizer-Induced Low-Dimensional Drift and Transverse Dynamics in Transformer Training

Weights and biases are normally optimized as separate parameter tensors, yet they do not represent separate functions when the input to an affine layer has nonzero mean. For an affine map $z=Wx+b$ with input mean $μ$, a weight update contains a sample-independent displacement $ΔWμ$ that is functionally indistinguishable from a bias update. We call this hidden contribution boundary motion and decompose each update into a centered, sample-varying shape component and a shared boundary component. On a four-layer Transformer trained from scratch on IMDb, the bias-like term $g_bμ^\top$ has a median norm equal to 0.664 of the raw weight-gradient norm across affine layers and training checkpoints. More strikingly, the median ratio $\norm{ΔWμ}/\norm{Δb}$ is 134.7, while $\norm{ΔWμ}/\norm{Δb+ΔWμ}$ is 0.994. Thus, under AdamW, the observed boundary motion is almost entirely realized through the weight matrix rather than the explicit bias. We implement a diagnostic optimizer, Shape--Boundary Orthogonal AdamW (SBO-AdamW), that optimizes $g_W-g_bμ^\top$ and $g_b$ with independent Adam states and compensates the weight-induced boundary displacement. In a single-seed experiment, SBO-AdamW raises validation accuracy from 81.68% to 85.81% and validation-selected test accuracy from 78.73% to 82.73%, with the best validation checkpoint occurring at step 800 instead of step 3000. However, the moving-batch-center compensation produces severe bias-coordinate drift and strongly reduces boundary energy. The present evidence therefore supports hidden boundary motion as an important optimization mechanism, but it does not yet establish a final general-purpose optimizer. A stable centered-affine parameterization is identified as the required next step.

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, bringing distinct but equivalent training states to the same starting point. To correct scale imbalance between paired Transformer matrices without changing predictions, we propose Symmetry-Aware Gradient Geometry Stabilization (SAGGS), which periodically equalizes the Frobenius norms of exact symmetry pairs, and its damped variant, Soft SAGGS. SAGGS reduces perplexity (PPL) by $0.374\pm0.026$ at 101M parameters on WikiText-103, and Soft SAGGS reduces PPL by $0.780\pm0.040$ in a 31M modern-architecture replication, both under scale imbalance. In a 123.8M checkpoint-resume experiment, reciprocally rescaling matrix pairs barely changes validation loss at restart but increases final PPL by $3.78\pm0.42$; hard and soft rebalancing recover to within $0.02$ PPL of the control.

We study the large-depth limit of transformers trained with AdamW, by modelling the hidden-state dynamics as an interacting particle system (IPS) coupled through the attention mechanism. Under appropriate scaling of the attention heads, we prove that the joint dynamics of the hidden states and backpropagated variables converge in $L^2$, uniformly over the initial condition, to the solution of a forward--backward system of ODEs at rate $\mathcal O(L^{-1}+L^{-1/3}H^{-1/2})$. Here, $L$ and $H$ denote the depth and number of heads of the transformer, respectively. The limiting system of ODEs can be identified with a McKean--Vlasov ODE (MVODE) when the attention heads do not incorporate causal masking. By using the flow maps associated with this MVODE and applying concentration of measure techniques, we obtain bounds on the difference between the discrete and continuous models that are uniform over compact sets of initial conditions. As this is achieved without resorting to a covering argument, the constants in our bounds are independent of the number of tokens. Furthermore, under a suitable adaptation to AdamW, the bounds become independent of the token embedding dimension.