aiqbench analysis
This research identifies "boundary motion" in transformer training where weight updates effectively mimic bias shifts due to nonzero input means, revealing that standard optimizers like AdamW largely handle this displacement through weights rather than explicit biases. By decomposing parameters into shape and boundary components for independent optimization, the proposed SBO-AdamW method improves model accuracy on IMDb datasets while highlighting the need for stable centered-affine parameterizations in future general-purpose optimizers.
Hidden Boundary Motion in Transformer Optimization: Function-Space Orthogonalization of Affine Weight and Bias Updates
arXiv:2607.22927v1 Announce Type: new Abstract: 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 $\mu$, a weight update contains a sample-independent displacement $\Delta W\mu$ that is functionally indistinguishable from a bias update. We call this hidden contribution \emph{boundary motion} and decompose each update into a centered, sample-varying \emph{shape} component and a shared \emph{boundary} component. On a four-layer Transformer trained from scratch on IMDb, the bias-like term $g_b\mu^\top$ 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 $\norm{\Delta W\mu}/\norm{\Delta b}$ is 134.7, while $\norm{\Delta W\mu}/\norm{\Delta b+\Delta W\mu}$ 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 $g_W-g_b\mu^\top$ 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.
Abstract: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 $\mu$, a weight update contains a sample-independent displacement $\Delta W\mu$ that is functionally indistinguishable from a bias update. We call this hidden contribution \emph{boundary motion} and decompose each update into a centered, sample-varying \emph{shape} component and a shared \emph{boundary} component. On a four-layer Transformer trained from scratch on IMDb, the bias-like term $g_b\mu^\top$ 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 $\norm{\Delta W\mu}/\norm{\Delta b}$ is 134.7, while $\norm{\Delta W\mu}/\norm{\Delta b+\Delta W\mu}$ 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 $g_W-g_b\mu^\top$ 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.
Machine Learning (cs.LG)
arXiv:2607.22927 [cs.LG]
(or arXiv:2607.22927v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.22927 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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