測量對稱性——數據交換率

摘要

等變性理論預測：將對稱性作為架構先驗，可將樣本複雜度降低|G|倍；此論點廣被引用，但卻鮮少以控制混淆變項的尺度法則加以量測。在受控的C_n對稱任務中，我們報告三項發現。第一，軌道大小相同且計算量匹配的錯誤群組控制，表現劣於無任何約束（聯合成對信賴區間[+0.79, +3.26]排除零值，對各估計量穩健）；錯置的約束不僅無助益，反而具有主動危害性。第二，配備測試時軌道平均的資料增強基線，與等變模型表現完全一致——在匹配的單元上，逐週期驗證曲線完全相同——因此架構與增強之間的差距是有條件地取決於非對稱的測試時計算，而非無條件成立。第三，相對交換率beta_diff = 1.28在符號與數量級上與理論值1.0一致（單層信賴區間[+0.92, +2.05]）；較保守的雙層拔靴法（種子×群組大小）則將區間擴大為[-0.63, +1.72]（包含零值），而在間距為sqrt(2)的較細N網格上進行的複製實驗結果不明確（點估計值-0.82）。方法學貢獻——可消除共享難度混淆項的相對率估計量、錯誤群組控制，以及預先指定的失敗分類法——可遷移至任何強度可參數化的歸納偏置。誠實界定範圍：主要估計量beta_diff是在初步分析顯示存在正斜率可識別性問題後，事後採用的；實驗設計從未經過外部預註冊；且標題數字僅基於粗N網格上七個群組大小的普通最小平方法斜率。本研究為探索性研究，非確認性量測；錯誤群組結果是最清晰的發現，也是我們報告時最有信心的結果。後續工作將進行使用新鮮種子的註冊複製。

English

Equivariance theory predicts that an architectural symmetry prior reduces sample complexity by a factor of |G|; this is widely cited but rarely measured as a scaling law with controls that separate the prior from its confounds. On a controlled C_n-symmetric task, we report three findings. First, a wrong-group control with identical orbit size and matched compute is worse than no constraint (joint pairwise CI [+0.79, +3.26] excludes zero, robust across estimators); misaligned constraint is actively harmful, not merely unhelpful. Second, an augmentation baseline equipped with test-time orbit averaging matches the equivariant model exactly -- bit-identical per-epoch validation curves across matched cells -- so the architecture-vs-augmentation gap is conditional on asymmetric test-time computation, not unconditional. Third, the relative exchange rate beta_diff = 1.28 is consistent in sign and order of magnitude with the theoretical 1.0 (single-level CI [+0.92, +2.05]); the more conservative two-level bootstrap (seeds x group sizes) widens this to [-0.63, +1.72], including zero, and a finer-N replication on a sqrt(2)-spaced grid is inconclusive (point estimate -0.82). The methodological contributions -- the relative-rate estimator that cancels the shared-difficulty confound, the wrong-group control, and a pre-specified failure taxonomy -- transfer to any inductive bias whose strength can be parameterised. Honest scoping: the primary estimator beta_diff was adopted post-hoc after the initial analysis revealed a positive-slope identifiability problem; the design was never externally pre-registered; and the headline number rests on an OLS slope over seven group sizes on a coarse N grid. This is an exploratory study, not a confirmatory measurement; the wrong-group result is the cleanest finding and the one we report with the most confidence. A registered replication on fresh seeds is future work.