测量对称性——数据交换速率

摘要

等变理论预测，架构对称性先验可将样本复杂度降低|G|倍；这一观点被广泛引用，但很少作为缩放定律进行测量，且未通过控制变量将先验与其混杂因素分离。针对受控的C_n对称任务，我们报告三项发现。第一，具有相同轨道大小和匹配计算量的错误群组控制比无约束更差（联合配对CI [+0.79， +3.26]排除零，估计量稳健）；错位约束不仅无益，反而有害。第二，配备测试时轨道平均的数据增强基线完全匹配等变模型——跨匹配单元的逐周期验证曲线比特级一致——因此架构与增强之间的差距取决于非对称测试时计算，而非无条件存在。第三，相对交换率β_diff = 1.28在符号和量级上与理论值1.0一致（单层CI [+0.92， +2.05]）；更保守的两层引导（种子×群组大小）将其扩大至[-0.63， +1.72]（包含零），而在sqrt(2)间隔网格上的更细N复制无明确结论（点估计-0.82）。方法学贡献——抵消共享难度混杂因素的相对率估计量、错误群组控制以及预设的失败分类法——可迁移至任何强度可参数化的归纳偏置。诚实说明：主要估计量β_diff是在初步分析发现正斜率可识别性问题后事后采用的；该设计从未进行外部预注册；且首要数据基于粗N网格上七个群组大小的普通最小二乘法斜率。这是一项探索性研究，而非确证性测量；错误群组结果是最清晰的发现，也是我们报告时最有信心的结果。基于新种子的注册复制留待未来工作。

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.