揭示Transformer学习敏感函数失败的几何根源
Understanding the Parameter Space Geometry of Transformers Encoding Boolean Functions
- 从平均敏感度扩展到全敏感度分布，分析参数空间几何
- 随机初始化几乎必然错过高敏感函数所在区域
- 解释为何某些可表达函数仍无法被学习
Transformer在学习某些简单但理论上可表达的函数时持续失败，这种可表达性与可学习性之间的差距在敏感函数（如PARITY）中尤为明显。已有研究指出Transformer对低平均敏感度函数存在偏好，但其机制尚不明确。本文通过分析Transformer参数空间的几何结构发现：即使敏感函数在理论上可表示，其对应的参数区域也极小，随机初始化极大概率无法覆盖。研究将关注点从平均敏感度扩展至完整的敏感度分布——即所有输入下的敏感度分布情况，并证明随机初始化的Transformer几乎必然计算出包含低敏感字符串的函数。因此，若某函数不存在此类低敏感字符串，则该函数在随机初始化下必定不可学习。
原文摘要 · Abstract (English)
Transformers consistently fail to learn certain simple functions that are provably expressible with specific parameter settings. This gap between learnability and expressivity is particularly prominent for sensitive functions -- functions whose output is likely to change if a single bit of the input is flipped -- for example, PARITY. While prior work has established that transformers exhibit a bias toward functions with low average sensitivity, the precise mechanism underlying this bias remains poorly understood. To shed light on this phenomenon, we study the geometry of transformers' parameter space. We show that sensitive functions -- even when representable -- occupy a vanishingly small region that random initialization is very likely to miss. Specifically, we shift the focus from average sensitivity to the full sensitivity profile -- the distribution of sensitivity values across all inputs -- and prove that randomly initialized transformers almost surely compute functions which have low-sensitivity strings. Consequently, any function that lacks such strings is provably unlearnable.
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