RoPE attention is an exact forward-pass gradient step with softmax intact

Julie Huang; Maggie Chlon; Leon Chlon

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arXiv · 2609.06685

RoPE attention is an exact forward-pass gradient step with softmax intact

Julie Huang·Maggie Chlon·Leon Chlon

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Abstract

We derive an exact gradient-step representation of the RoPE-softmax forward pass. For every deterministic RoPE-softmax attention head with arbitrary affine projection weights, we construct a query-dependent effective matrix $\Delta M_i$ satisfying $y_i = \mu_i + u_i^\top \Delta M_i$, where $\mu_i$ is the uniform mean of the attended values and $u_i$ is the augmented query input. The construction applies the classical exponential divided difference $\rho = \phi_1$ to retain the softmax exactly. Its positive coefficients give a unit gradient-step representation on a query-conditioned quadratic objective. The same function connects the RoPE generator to exact positional finite differences. We derive a tokenwise formula for the error of reusing one query's matrix and prove that a nonconstant finite-cache head cannot admit a globally exact affine query readout. Reconstruction checks and frozen-reuse calibration on one pretrained Qwen2.5-0.5B layer verify the representation and quantify the correction required when one query's matrix is reused.

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Julie Huang, Maggie Chlon, Leon Chlon. 2026-09-06. RoPE attention is an exact forward-pass gradient step with softmax intact. https://arxiv.org/abs/2609.06685

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