Understanding the Parameter Space Geometry of Transformers Encoding Boolean Functions

2026-06-07

AI Summary

Transformers exhibit poor performance in learning highly sensitive Boolean functions such as PARITY, despite possessing sufficient theoretical expressive power. This work provides the first analysis extending beyond average sensitivity to the full sensitivity distribution of Boolean functions. By integrating Boolean sensitivity analysis, geometric modeling of the parameter space, and probabilistic arguments, the study demonstrates that the Transformer’s parameter space almost surely induces functions containing a large number of low-sensitivity inputs. Highly sensitive functions occupy a vanishingly small measure within this space, rendering them effectively unreachable under random initialization. Consequently, such functions are practically unlearnable by standard Transformers. This finding elucidates an intrinsic bias of Transformers toward low-sensitivity functions and offers a novel perspective on their inductive bias.

Abstract

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.

Problem

• Transformers
• Boolean functions
• sensitivity
• parameter space geometry
• learnability

Innovation

• parameter space geometry
• sensitivity profile
• transformers
• Boolean functions
• learnability-expressivity gap