← 研踪 PaperTrailarXiv:2611.01328 · v1
Same Function, Different Mechanism: Implementation-Local Geometry Beyond Accuracy
首次提交 2026-11-01 06:53
版本更新 2026-11-01 06:53
雷达观测 2026-11-04 23:43
摘要
Two models can make exactly the same predictions yet respond differently when their implementations are perturbed. A lookup table can change one case independently; a shared rule couples changes across cases. We compare these mechanisms through an implementation chart: a specified local family of logit perturbations around a fixed predictive distribution. Our main result counts the independent prediction changes allowed by exact finite-group sharing: the active dimension is the number of input-output orbits minus the number of input orbits. For softened modular addition, lookup and shared cyclic charts therefore have ranks $p^2(p-1)$ and $p-1$ despite identical predictions. The classical Fisher pullback connects these dimensions to local KL curvature; mechanism-preserving reparameterizations leave the rank unchanged. We also bound the spectral tail under approximate sharing and separate quadratic rank from higher-order learning coefficients. Trained modular-addition models verify that chart rank is measurable, while controlled compiled transformers distinguish compiler-supported intermediate charts from matched alternatives. Negative controls show where this discrimination fails. The framework evaluates specified mechanism hypotheses; it does not infer a unique mechanism from predictions or predict training dynamics.
作者备注：16 pages. Accepted at NeurIPS 2026 (poster). Includes technical appendices and ancillary reproduction code
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v1 · 2026-11-01 06:53 · 最新观测版本
Same Function, Different Mechanism: Implementation-Local Geometry Beyond Accuracy
Two models can make exactly the same predictions yet respond differently when their implementations are perturbed. A lookup table can change one case independently; a shared rule couples changes across cases. We compare these mechanisms through an implementation chart: a specified local family of logit perturbations around a fixed predictive distribution. Our main result counts the independent prediction changes allowed by exact finite-group sharing: the active dimension is the number of input-output orbits minus the number of input orbits. For softened modular addition, lookup and shared cyclic charts therefore have ranks $p^2(p-1)$ and $p-1$ despite identical predictions. The classical Fisher pullback connects these dimensions to local KL curvature; mechanism-preserving reparameterizations leave the rank unchanged. We also bound the spectral tail under approximate sharing and separate quadratic rank from higher-order learning coefficients. Trained modular-addition models verify that chart rank is measurable, while controlled compiled transformers distinguish compiler-supported intermediate charts from matched alternatives. Negative controls show where this discrimination fails. The framework evaluates specified mechanism hypotheses; it does not infer a unique mechanism from predictions or predict training dynamics.
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