# Understanding the Parameter Space Geometry of Transformers Encoding Boolean Functions

arXiv: https://arxiv.org/abs/2606.08768 · HTML: https://mlsanity.com/paper/2606.08768/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 18 references that are in the map.
- First posted on arXiv 7 Jun 2026.

## Builds on (most cited first, 12 of 12)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Theoretical Limitations of Self-Attention in Neural Sequence Models](https://mlsanity.com/paper/1906.06755/) (2019)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [What Formal Languages Can Transformers Express? A Survey](https://mlsanity.com/paper/2311.00208/) (2023)
- [Overcoming a Theoretical Limitation of Self-Attention](https://mlsanity.com/paper/2202.12172/) (2022)
- [Why are Sensitive Functions Hard for Transformers?](https://mlsanity.com/paper/2402.09963/) (2024)
- [Generalization on the Unseen, Logic Reasoning and Degree Curriculum](https://mlsanity.com/paper/2301.13105/) (2023)
- [Simplicity Bias in Transformers and their Ability to Learn Sparse Boolean Functions](https://mlsanity.com/paper/2211.12316/) (2022)
- [Training Neural Networks as Recognizers of Formal Languages](https://mlsanity.com/paper/2411.07107/) (2024)
- [How Uniform Random Weights Induce Non-uniform Bias: Typical Interpolating Neural Networks Generalize with Narrow Teachers](https://mlsanity.com/paper/2402.06323/) (2024)
- [B2T Connection: Serving Stability and Performance in Deep Transformers](https://mlsanity.com/paper/2206.00330/) (2022)

## Built on by (most cited first, 1 of 1)

- [Exact-Solution Volume and Length Generalization in Transformers](https://mlsanity.com/paper/2610.07676/) (2026)

## Abstract (arXiv)

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.